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Fw(\()p Fo(p)p Fw(\))97 b Fs(j)g Fo(p)28 b Fs(2)g Fo(B)5 b Fs(g)17 b Fo(:)0 1888 y Fw(Then,)39 b(with)c(a)i(sligh)m(t)e (abuse)i(of)f(notation)f(\(whic)m(h)i(has)f(b)s(ecome)h(standard)f(in)g (the)h(literature\),)0 2004 y(w)m(e)d(will)c(sometimes)i(write)g Fo(u)1106 2019 y Fp(p)1173 2004 y Fw(:=)c Fo(u)p Fw(\()p Fo(p)p Fw(\),)33 b(for)f Fo(p)c Fs(2)g Fw(\003)k(and)h(tak)m(e)h(deriv) -5 b(ativ)m(es)32 b(of)h Fo(H)3045 2019 y Fp(B)3138 2004 y Fw(with)f(resp)s(ect)0 2120 y(to)41 b Fo(u)184 2135 y Fp(p)264 2120 y Fw(\(indeed,)g Fo(H)727 2135 y Fp(B)828 2120 y Fw(will)e(b)s(e)i(assumed)g(to)g(ha)m(v)m(e)h(at)f(least)f(t)m (w)m(o)i(deriv)-5 b(ativ)m(es,)43 b(see)f Fz(H8)f Fw(b)s(elo)m(w\).)0 2237 y(W)-8 b(e)36 b(exp)s(ect)g(that)f(no)g(confusion)g(arises)g(b)s (et)m(w)m(een)i(the)f Fo(p)p Fw(-index)f(in)f Fo(u)2606 2252 y Fp(p)2680 2237 y Fw(and)i(indices)e(of)h(di\013eren)m(t)0 2353 y(t)m(yp)s(es)26 b(that)f(w)m(e)g(use)h(in)e(the)h(pap)s(er)g 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b(assumptions)f(are)f(v)m(ery)j(common)c (in)h(Statistical)f(Mec)m(hanics)0 3167 y(\(see)k([Rue99]\).)51 b(Later)35 b(in)f(Assumption)h Fz(H8)g Fw(w)m(e)h(will)d(mak)m(e)i (some)g(other)g(assumptions)g(on)g(the)0 3283 y(con)m(v)m(ergence)e(of) d(deriv)-5 b(ativ)m(es)31 b(of)f(the)g(in)m(teraction)g(with)g(resp)s (ect)i(to)e(the)g(p)s(oin)m(ts)h(as)f(w)m(ell)g(as)h(some)0 3399 y(requiremen)m(ts)i(that)g(the)g(con)m(v)m(ergence)i(is)d(uniform) f(in)h(some)g(sets)i(of)e(in)m(teractions.)0 3601 y(W)-8 b(e)34 b(no)m(w)g(list)f(some)g(further)h(assumptions)f(required)i(on)e (the)h(in)m(teraction)f(p)s(oten)m(tial.)45 b(F)-8 b(or)33 b(this,)0 3717 y(w)m(e)h(in)m(tro)s(duce)e(some)h(auxiliary)d(notation) h(ab)s(out)h(a)h(giv)m(en)f(con\014guration)g Fo(u)p Fw(.)0 3833 y(First,)24 b(observ)m(e)h(that)e(the)h(action)f(of)f Fo(G)i Fw(on)f(\003)g(extends)i(naturally)d(to)h(the)g(space)i(of)e (con\014gurations.)0 3949 y(Namely)-8 b(,)32 b(if)f Fo(\015)i Fs(2)28 b Fo(G)p Fw(,)k(w)m(e)i(de\014ne)1417 4068 y Fs(T)1471 4083 y Fp(\015)1516 4068 y Fo(u)p Fw(\()p Fo(p)p Fw(\))44 b(:=)g Fo(u)p Fw(\()p Fo(\015)5 b Fw(\()p Fo(p)p Fw(\)\))17 b Fo(;)0 4206 y Fw(for)32 b(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003.)43 b(Giv)m(en)33 b Fo(`)27 b Fs(2)h Fq(R)5 b Fw(,)39 b(w)m(e)34 b(also)d(set)1518 4365 y Fs(R)1602 4380 y Fp(`)1636 4365 y Fo(u)44 b Fw(:=)g Fo(u)21 b Fw(+)h Fo(`)17 b(:)0 4524 y Fw(With)32 b(this)g(notation,)f(w)m(e)j (assume:)98 4662 y Fz(H3:)42 b Fw(The)k(p)s(oten)m(tial)f Fo(H)57 b Fw(=)51 b Fs(f)p Fo(H)1350 4677 y Fp(B)1410 4662 y Fs(g)46 b Fw(is)f Fo(G)p Fr(-invariant)p Fw(,)j(i.e.)83 b(if)45 b Fo(B)56 b Fs(2)50 b(S)7 b Fw(,)50 b(then)d Fo(\015)5 b(B)55 b Fs(2)c(S)j Fw(for)315 4778 y(an)m(y)33 b Fo(\015)g Fs(2)28 b Fo(G)k Fw(and)1406 4897 y Fo(H)1487 4912 y Fp(\015)t(B)1588 4897 y Fw(\()p Fo(u)p Fw(\))44 b(=)g Fo(H)1965 4912 y Fp(B)2025 4897 y Fw(\()p Fs(T)2117 4912 y Fp(\015)2162 4897 y Fo(u)p Fw(\))315 5035 y(for)32 b(all)e Fo(\015)j Fs(2)28 b Fo(G)p Fw(,)k(all)f Fo(B)38 b Fw(and)32 b(all)f(con\014gurations)h Fo(u)g Fw(on)g(\003.)98 5151 y Fz(H4:)42 b Fo(H)e Fw(has)33 b Fr(p)-5 b(erio)g(dic)33 b(phase)p Fw(,)f(i.e.,)1432 5310 y Fo(H)1513 5325 y Fp(B)1573 5310 y Fw(\()p Fs(R)1695 5325 y Fp(s)1732 5310 y Fo(u)p Fw(\))27 b(=)h Fo(H)2038 5325 y Fp(B)2098 5310 y Fw(\()p Fo(u)p Fw(\))315 5469 y(for)k(an)m(y)h(con\014guration)f Fo(u)p Fw(,)g(an)m(y)h Fo(B)g Fs(2)28 b(S)40 b Fw(and)33 b(an)m(y)g Fo(s)28 b Fs(2)g Fq(Z)p Fw(.)p eop %%Page: 6 6 6 5 bop 168 251 a Ft(6)1097 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)266 450 y Fz(H5:)41 b Fw(The)j(action)e(of)h(the)g(group)g(on)f(the)i(p)s(oten)m(tial)d(is) h Fr(nontrivial)p Fw(,)j(i.e.,)g(w)m(e)f(supp)s(ose)g(that)482 566 y(if)32 b Fo(g)f Fs(2)d Fo(G)k Fw(is)g(suc)m(h)i(that)f Fo(g)t(B)f Fw(=)27 b Fo(B)38 b Fw(for)32 b(some)g Fo(B)h Fs(2)28 b(S)7 b Fw(,)34 b(then)f Fo(g)j Fw(is)c(the)h(iden)m(tit)m(y)-8 b(.)168 712 y(W)g(e)33 b(also)e(supp)s(ose)j(that)266 857 y Fz(H6:)41 b Fo(H)f Fw(is)32 b Fr(c)-5 b(o)g(er)g(cive)p Fw(,)32 b(i.e.,)g(w)m(e)i(supp)s(ose)g(that)e Fo(H)2078 872 y Fp(B)2139 857 y Fw(\()p Fo(u)p Fw(\))27 b Fs(\025)h Fw(0)k(for)g(an)m(y)h Fo(B)g Fs(2)28 b(S)41 b Fw(and)1321 1036 y Fo(H)1402 1051 y Fp(F)1460 1036 y Fw(\()p Fo(u)p Fw(\))j Fs(\025)h Fo(\024)17 b Fw(max)1832 1098 y Fp(i;j)t Fm(2)p Fp(F)2029 1036 y Fs(j)p Fo(u)2113 1051 y Fp(i)2162 1036 y Fs(\000)23 b Fo(u)2318 1051 y Fp(j)2354 1036 y Fs(j)2382 995 y Fp(\022)2443 1036 y Fs(\000)f Fo(K)i(;)482 1257 y Fw(for)32 b(some)h Fo(\024;)g(K)r(;)g(\022)e(>)d Fw(0.)168 1403 y(W)-8 b(e)24 b(no)m(w)g(mak)m(e)f(an)h(assumption)f(ab) s(out)g(ho)m(w)h(the)g(energy)h(b)s(eha)m(v)m(es)g(with)e(resp)s(ect)i (to)e(the)h(co)s(cycles)168 1519 y(of)32 b(the)h(group.)168 1635 y(That)g(is,)f(w)m(e)h(supp)s(ose)h(that)266 1781 y Fz(H7:)41 b Fw(There)34 b(exist)f Fo(#;)h(S)f(>)28 b Fw(0)k(and)h Fo(&)j Fw(:)27 b Fs(S)36 b(!)27 b Fw([0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\))32 b(in)g(suc)m(h)i(a)e(w)m(a)m(y)i(that) 168 1987 y(\(5\))1109 b Fo(H)1483 2002 y Fp(B)1543 1987 y Fw(\()p Fo(\033)t Fw(\))44 b Fs(\024)h Fo(&)8 b Fw(\()p Fo(B)d Fw(\))2059 1876 y Fn(\020)2118 1987 y Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)2277 1938 y Fp(#)2345 1987 y Fw(+)22 b(1)2492 1876 y Fn(\021)2568 1987 y Fo(;)482 2188 y Fw(for)32 b(an)m(y)i(co)s(cycle)f Fo(\033)j Fw(and)d(an)m(y)g Fo(B)g Fs(2)28 b(S)7 b Fw(,)33 b(and)1621 2284 y Fn(X)1627 2484 y Fk(B)r Fj(2S)1581 2540 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1830 2378 y Fo(&)8 b Fw(\()p Fo(B)d Fw(\))45 b Fs(\024)g Fo(S)22 b(]X)r(;)482 2696 y Fw(for)32 b(an)m(y)i Fo(X)h Fs(\032)28 b Fw(\003.)168 2841 y(W)-8 b(e)38 b(recall)f(that)h(the)h(norm)f Fs(j)-17 b(j)g(j)26 b(\001)f(j)-17 b(j)g(j)38 b Fw(of)g(a)g(co)s(cycle)h(\(whic)m(h)g(has)g (just)f(has)h(b)s(een)g(used)h(in)d(\(5\))h(here)168 2957 y(ab)s(o)m(v)m(e\))33 b(w)m(as)h(de\014ned)g(in)d(\(3\).)168 3074 y(W)-8 b(e)34 b(no)m(w)g(w)m(an)m(t)g(to)f(de\014ne)i Fs(O)1262 3089 y Fp(\033)1342 3074 y Fw(to)f(b)s(e)f(the)h(space)h(of)e (con\014gurations)g(whic)m(h)h(are)f(\\at)g(a)g(b)s(ounded)168 3190 y(distance")f(from)g(a)g(giv)m(en)h(co)s(cycle)g Fo(\033)t Fw(.)43 b(This)33 b(concept)h(is)e(formalized)e(in)i(the)h (follo)m(wing)d(w)m(a)m(y)-8 b(.)168 3306 y(First,)25 b(a)g(co)s(cycle)h Fo(\033)j Fw(de\014nes)d(a)f(con\014guration)f Fo(u)1938 3321 y Fp(\033)2010 3306 y Fw(on)h(\003)f(as)h(follo)m(ws.)40 b(Since)25 b Fo(F)39 b Fw(is)25 b(the)g(fundamen)m(tal)168 3422 y(domain,)40 b(for)g(an)m(y)h Fo(p)g Fs(2)g Fw(\003)f(there)h (exists)g(a)g(unique)f Fo(q)45 b Fs(2)c Fo(F)54 b Fw(suc)m(h)42 b(that)e Fo(p)h Fw(=)g Fo(\015)5 b(q)t Fw(,)42 b(with)e Fo(\015)46 b Fs(2)41 b Fo(G)p Fw(.)168 3539 y(Then,)34 b(w)m(e)f(de\014ne)168 3713 y(\(6\))1409 b Fo(u)1758 3728 y Fp(\033)1804 3713 y Fw(\()p Fo(p)p Fw(\))28 b(=)f Fo(\033)t Fw(\()p Fo(\015)5 b Fw(\))17 b Fo(:)168 3888 y Fw(The)27 b(ab)s(o)m(v)m(e)g(is)f(a)g(b)s(ona-\014de)g(de\014nition,) h(thanks)g(to)f Fz(G3)h Fw(and)f(\(2\).)41 b(Also,)26 b Fo(u)2942 3903 y Fp(\033)2989 3888 y Fw(\()p Fo(g)t(p)p Fw(\))g(=)i Fo(u)3351 3903 y Fp(\033)3397 3888 y Fw(\()p Fo(p)p Fw(\))9 b(+)g Fo(\033)t Fw(\()p Fo(g)t Fw(\),)168 4004 y(that)32 b(is,)g Fo(u)560 4019 y Fp(\033)639 4004 y Fw(\\b)s(eha)m(v)m(es)j(as)d(a)h(linear)e(function".)168 4121 y(Then,)j(if)d Fo(\024;)17 b(K)r(;)g(S;)g(\022)s(;)g(#)32 b Fw(are)h(as)g(in)e Fz(H6)i Fw(and)f Fz(H7)p Fw(,)h(w)m(e)g(set)168 4451 y(\(7\))445 b Fo(K)828 4410 y Fp(?)821 4476 y(\033)912 4451 y Fw(:=)1059 4217 y Fn(8)1059 4306 y(>)1059 4336 y(<)1059 4516 y(>)1059 4546 y(:)1788 4288 y Fw(9)807 b(if)32 b Fo(])p Fw(\(\003)p Fo(=G)p Fw(\))27 b(=)g(1,)1148 4521 y(9)22 b(+)1317 4350 y Fn( )1406 4453 y Fo(S)g Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)1685 4405 y Fp(#)1753 4453 y Fw(+)22 b(1\))17 b Fo(]F)35 b Fw(+)22 b Fo(K)p 1406 4498 874 4 v 1814 4589 a(\024)2289 4350 y Fn(!)2368 4373 y Fl(1)p Fp(=\022)2644 4521 y Fw(if)31 b Fo(])p Fw(\(\003)p Fo(=G)p Fw(\))c Fs(\025)h Fw(2.)168 4794 y(Then,)34 b(giv)m(en)e Fo(K)j Fs(\025)28 b Fw(0,)33 b(w)m(e)g(de\014ne)h Fs(O)1543 4758 y Fp(K)1540 4818 y(\033)1644 4794 y Fw(to)e(b)s(e)h(the)g(space)h(of)e(all)e (con\014gurations)i Fo(u)g Fw(suc)m(h)j(that)168 4968 y(\(8\))1181 b(sup)1482 5050 y Fp(p)p Fm(2)p Fl(\003)1638 4968 y Fs(j)p Fo(u)p Fw(\()p Fo(p)p Fw(\))21 b Fs(\000)i Fo(u)2024 4983 y Fp(\033)2070 4968 y Fw(\()p Fo(p)p Fw(\))p Fs(j)44 b(\024)h Fo(K)23 b(:)168 5237 y Fw(W)-8 b(e)37 b(also)f(set)i Fs(O)776 5252 y Fp(\033)859 5237 y Fw(:=)d Fs(O)1079 5186 y Fp(K)1144 5163 y Fk(?)1139 5203 y(\033)1076 5247 y Fp(\033)1184 5237 y Fw(.)57 b(W)-8 b(e)38 b(will)c(seek)39 b(our)e(solutions)f(in)g(the)i(space)g Fs(O)3068 5252 y Fp(\033)3152 5237 y Fw(\(Lemma)e(29)g(will)168 5353 y(assure)d(an)g(a-priori)d(b)s(ounds)j(for)f(minimizers)e(whic)m(h)j (con\014ne)g(them)g(in)m(to)f(this)g(space\).)168 5469 y(With)g(the)h(ab)s(o)m(v)m(e)g(con)m(v)m(en)m(tions,)h(w)m(e)g(also)e (assume)h(that)p eop %%Page: 7 7 7 6 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)700 b(7)98 450 y Fz(H8:)42 b Fw(F)-8 b(or)33 b(an)m(y)h Fo(K)j Fs(\025)30 b Fw(0,)k(the)g(in)m(teraction)f(p)s(oten)m(tial)f Fo(H)41 b Fw(is)34 b Fo(C)2386 414 y Fl(2)2425 450 y Fw(-b)s(ounded)g(on)g Fs(O)3076 414 y Fp(K)3073 475 y(\033)3144 450 y Fw(,)h(that)e(is,)h(for)315 566 y(an)m(y)f Fo(B)g Fs(2)28 b(S)7 b Fw(,)33 b(the)g(map)f Fo(H)1293 581 y Fp(B)1386 566 y Fw(is)g(t)m(wice)h(di\013eren)m(tiable)e(and)1008 780 y(sup)979 869 y Fp(u)p Fm(2O)1125 850 y Fk(K)1123 886 y(\033)1200 780 y Fw(sup)1207 862 y Fp(p)p Fm(2)p Fl(\003)1367 686 y Fn(X)1373 886 y Fk(B)r Fj(2S)1378 942 y Fk(B)r Fj(3)p Fk(p)1531 636 y Fn(\014)1531 696 y(\014)1531 756 y(\014)1531 815 y(\014)1574 713 y Fo(@)5 b(H)1711 728 y Fp(B)p 1574 757 199 4 v 1598 849 a Fo(@)g(u)1710 864 y Fp(p)1783 780 y Fw(\()p Fo(u)p Fw(\))1915 636 y Fn(\014)1915 696 y(\014)1915 756 y(\014)1915 815 y(\014)984 1146 y Fw(+)51 b(sup)1082 1235 y Fp(u)p Fm(2O)1228 1216 y Fk(K)1226 1251 y(\033)1303 1146 y Fw(sup)1311 1227 y Fp(p)p Fm(2)p Fl(\003)1471 1051 y Fn(X)1477 1251 y Fk(B)r Fj(2S)1481 1308 y Fk(B)r Fj(3)p Fk(p)1635 1051 y Fn(X)1638 1262 y Fp(q)r Fm(2)p Fp(B)1795 1001 y Fn(\014)1795 1061 y(\014)1795 1121 y(\014)1795 1181 y(\014)1871 1078 y Fo(@)1927 1042 y Fl(2)1967 1078 y Fo(H)2048 1093 y Fp(B)p 1839 1123 303 4 v 1839 1214 a Fo(@)5 b(u)1951 1229 y Fp(p)1991 1214 y Fo(@)g(u)2103 1229 y Fp(q)2151 1146 y Fw(\()p Fo(u)p Fw(\))2283 1001 y Fn(\014)2283 1061 y(\014)2283 1121 y(\014)2283 1181 y(\014)2360 1146 y Fo(<)44 b Fw(+)p Fs(1)17 b Fo(:)0 1006 y Fw(\(9\))98 1452 y Fz(H9:)42 b Fo(H)e Fw(satis\014es)33 b(the)g Fr(ferr)-5 b(omagnetic)31 b Fw(\(or)h Fr(twist)p Fw(\))h(condition,)e(i.e.)1517 1606 y Fo(@)1573 1570 y Fl(2)1613 1606 y Fo(H)1694 1621 y Fp(B)p 1476 1651 320 4 v 1476 1742 a Fo(@)5 b(u)1588 1757 y Fp(p)1645 1742 y Fo(@)g(u)1757 1757 y Fp(q)1805 1674 y Fw(\()p Fo(u)p Fw(\))44 b Fs(\024)h Fw(0)17 b Fo(;)315 1902 y Fw(for)32 b(an)m(y)h Fo(u)27 b Fs(2)h(O)907 1865 y Fp(K)904 1926 y(\033)976 1902 y Fw(,)k(an)m(y)h Fo(K)i Fs(\025)28 b Fw(0)33 b(and)f(an)m(y)i Fo(p;)f(q)j Fw(in)c(\003,)g(with)h Fo(p)27 b Fs(6)p Fw(=)h Fo(q)t Fw(.)42 2018 y Fz(H10:)42 b Fw(There)33 b(exist)g Fo(\016)f Fw(:)c Fs(S)35 b(!)27 b Fw([0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\))32 b(in)g(suc)m(h)i(a)e(w)m(a)m(y)i(that)416 2134 y Fz({)41 b Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b(=)f Fo(\016)t Fw(\()p Fo(g)t(B)5 b Fw(\),)32 b(for)g(an)m(y)i Fo(B)e Fs(2)c(S)41 b Fw(and)32 b(an)m(y)i Fo(g)d Fs(2)d Fo(G)p Fw(.)416 2250 y Fz({)41 b Fw(F)-8 b(or)32 b(an)m(y)h Fo(K)i Fs(\025)28 b Fw(0)33 b(and)f(an)m(y)h(\014nite)g(set)g Fo(X)j Fs(\032)28 b Fw(\003,)1240 2425 y(lim)1183 2487 y Fp(R)p Fm(!)p Fl(+)p Fm(1)1478 2425 y Fw(sup)1449 2513 y Fp(u)p Fm(2O)1595 2495 y Fk(K)1593 2530 y(\033)1729 2330 y Fn(X)1735 2530 y Fk(B)r Fj(2S)1690 2581 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1691 2635 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fk(>R)1949 2425 y Fo(H)2030 2440 y Fp(B)2090 2425 y Fw(\()p Fo(u)p Fw(\))44 b(=)g(0)17 b Fo(:)416 2777 y Fz({)41 b Fw(Giv)m(en)32 b(an)m(y)g Fo(R)d(>)f Fw(0)j(and)h(an)m(y)g(\014nite)g(set)g Fo(X)k Fs(\032)28 b Fw(\003,)j(there)i(exists)f(a)g(\014nite)f(set)i Fo(Y)49 b Fs(\032)28 b Fw(\003)513 2893 y(in)k(suc)m(h)i(a)e(w)m(a)m(y) i(that)0 3067 y(\(10\))1583 2973 y Fn([)1573 3173 y Fk(B)r Fj(2S)1528 3223 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1529 3278 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1786 3067 y Fo(B)50 b Fs(\032)45 b Fo(Y)38 b(:)0 3425 y Fw(In)33 b(concrete)h(applications,)c(one)j(ma)m(y)f(tak)m(e)i Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))32 b(in)g Fz(H10)g Fw(to)h(b)s(e)f (diameter)g(of)g(the)h(set)g Fo(B)5 b Fw(.)0 3562 y Fz(Remark)36 b(1.)41 b Fw(The)33 b(ab)s(o)m(v)m(e)g(conditions)e(\(namely)-8 b(,)31 b Fz(G1)p Fw({)p Fz(G4)h Fw(on)g(the)h(group)e Fo(G)h Fw(and)g Fz(H1)p Fw({)p Fz(H10)f Fw(on)0 3679 y(the)38 b(in)m(teraction)f(p)s(oten)m(tial)f Fo(H)8 b Fw(\))37 b(will)e(b)s(e)j(assumed)h(in)e(order)h(to)f(obtain)g(our)g 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(elemen)m(t,)f(w)m(e)i(alw)m(a)m(ys)f(ha)m(v)m(e)h(a)e(co)s(cycle.)0 4635 y(By)j(considering)f(di\013eren)m(t)h(subgroups,)h(w)m(e)f(can)g (obtain)f(a)g(v)-5 b(ariet)m(y)34 b(of)g(di\013eren)m(t)h(ground)f (states)0 4751 y(asso)s(ciated)e(to)h(the)g(co)s(cycles)g(of)f(the)h (subgroups.)0 4888 y Fz(Remark)39 b(3.)k Fw(The)35 b(assumptions)f(in)g Fz(H2)p Fw(,)h Fz(H7)p Fw(,)f Fz(H8)g Fw(and)h Fz(H10)f Fw(are)h(satis\014ed)g(automatically)c(if)0 5005 y(the)f(in)m (teraction)f(has)h(\014nite)g(range.)43 b(So,)30 b(one)g(can)h (consider)f(them)g(as)g(an)g(expression)h(of)e(the)i(fact)0 5121 y(that)36 b(the)h(in)m(teraction)e(deca)m(ys)j(fast)f(enough.)55 b(These)38 b(assumptions)e(are)h(more)f(or)g(less)g(tailored)0 5237 y(to)f(the)i(argumen)m(t)e(that)h(w)m(e)h(will)c(dev)m(elop.)54 b(W)-8 b(e)36 b(write)g(them)f(at)g(this)h(stage,)h(so)f(that)f(they)i (can)0 5353 y(b)s(e)k(v)m(eri\014ed)h(for)f(mo)s(dels)f(of)h(in)m (terest.)70 b(In)41 b(con)m(trast)h(the)g(assumptions)f(statistical)e (mec)m(hanics)0 5469 y(in)h([Rue99])g(whic)m(h)h(in)m(v)m(olv)m(e)g (mostly)e(the)i(size)g(of)f(the)h(in)m(teraction,)g(our)g(assumptions)f (in)m(v)m(olv)m(e)p eop %%Page: 8 8 8 7 bop 168 251 a Ft(8)1097 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(also)28 b(the)i(size)g(of)e(deriv)-5 b(ativ)m(es)30 b(and)f(of)g(second)i (deriv)-5 b(ativ)m(es.)42 b(This)30 b(is)f(natural,)g(since)g(w)m(e)i (consider)168 566 y(the)37 b(existence)h(of)e(critical)e(p)s(oin)m(ts)i (\(the)h(equations)f(for)g(critical)e(p)s(oin)m(ts)i(in)m(v)m(olv)m(e)h (deriv)-5 b(ativ)m(es)37 b(of)168 683 y(the)29 b(in)m(teraction\))f (and)g(their)h(minimization)24 b(prop)s(erties)k(\(whic)m(h)i(is)e (natural)f(in)m(v)m(olv)m(e)j(the)f(second)168 799 y(deriv)-5 b(ativ)m(e\).)168 946 y Fz(Remark)30 b(4.)37 b Fw(W)-8 b(e)27 b(note)f(that)h(the)g(co)s(ercivit)m(y)f(assumption)g Fz(H6)g Fw(could)g(b)s(e)h(somewhat)f(w)m(eak)m(ened.)168 1062 y(Some)33 b(uniformit)m(y)f(of)h(gro)m(wth)h(in)e(the)i(distance)g (is)f(needed.)49 b(The)34 b(c)m(hoice)g(of)f(p)s(o)m(w)m(ers)i(is)e (done)h(to)168 1178 y(mak)m(e)41 b(the)h(other)f(assumptions)g(on)g (deca)m(y)i(explicit.)68 b(W)-8 b(eak)m(er)42 b(conditions)e(on)i(the)f (co)s(ercivit)m(y)168 1294 y(w)m(ould)32 b(require)h(stronger)g (assumptions)f(on)h(the)g(deca)m(y)-8 b(.)168 1441 y Fz(Remark)42 b(5.)j Fw(Note)37 b(that)h(w)m(e)g(ha)m(v)m(e)h(form)m (ulated)d(man)m(y)i(of)f(the)h(assumptions)f(in)g(terms)h(of)f(the)168 1557 y(in)m(teraction)h(functions)h Fo(H)1173 1572 y Fp(B)1234 1557 y Fw(.)63 b(Note)40 b(ho)m(w)m(ev)m(er)i(that)d(there)h (are)f(sev)m(eral)h(c)m(hoices)g(of)f(in)m(teraction)168 1674 y(functions)32 b(that)h(lead)f(to)g(the)h(same)f(in)m(teraction.) 168 1790 y(That)h(is,)f(di\013eren)m(t)h(functions)f Fo(H)1423 1805 y Fp(B)1484 1790 y Fw(,)1568 1765 y(~)1543 1790 y Fo(H)1624 1805 y Fp(B)1717 1790 y Fw(in)g(suc)m(h)i(a)e(w)m(a)m (y)i(that)e(the)h(energy)h(functional)d(is)1499 1889 y Fn(X)1543 2100 y Fp(B)1660 1984 y Fo(H)1741 1999 y Fp(B)1801 1984 y Fw(\()p Fo(u)p Fw(\))c(=)2064 1889 y Fn(X)2108 2100 y Fp(B)2250 1958 y Fw(~)2225 1984 y Fo(H)2306 1999 y Fp(B)2366 1984 y Fw(\()p Fo(u)p Fw(\))168 2249 y(for)34 b(all)e Fo(u)p Fw(,)j(at)f(least)g(formally)-8 b(.)47 b(One)35 b(suc)m(h)h(a)f(w)m(a)m(y)g(of)g(doing)e(this)h(w)m (ould)h(b)s(e,)g(e.g.)50 b(to)34 b(add)h(terms)168 2365 y(to)43 b(the)h(2-b)s(o)s(dy)f(in)m(teractions)g(and)h(to)g(cancel)g (them)f(b)m(y)i(adding)e(carefully)f(c)m(hosen)k(terms)d(to)168 2482 y(the)33 b(3-b)s(o)s(dy)f(and)g(4-b)s(o)s(dy)g(in)m(teractions.) 168 2598 y(In)24 b(the)g(same)f(v)m(ein,)j(w)m(e)f(men)m(tion)e(that)g (it)g(is)g(also)g(customary)h(to)f(add)h(terms)g(to)f(the)h(in)m (teractions)f({)168 2714 y(exact)i(di\013eren)m(tials)e({)i(whic)m(h)g (k)m(eep)h(the)f(same)f(critical)e(p)s(oin)m(ts)i(but)h(that)f(c)m (hange)i(the)f(minimizers.)168 2830 y(The)f(pro)s(cedure)h(of)e(c)m (hanging)g(the)i(in)m(teractions)e(but)h(k)m(eeping)g(the)g(minimizers) d(or)i(critical)e(p)s(oin)m(ts)168 2946 y(is)30 b(one)i(of)f(the)g (main)f(to)s(ols)g(of)h(the)g(renormalization)d(group)j(tec)m(hniques.) 45 b(W)-8 b(e)32 b(will)d(not)i(consider)168 3063 y(those)i(tec)m (hniques)h(here.)168 3278 y(1.2.)48 b Fz(An)27 b(order)g(relation)f (and)i(the)f(Birkho\013)f(prop)s(ert)m(y.)49 b Fw(T)-8 b(o)24 b(state)g(our)f(main)f(theorem,)j(w)m(e)168 3395 y(need)f(no)m(w)g(some)f(further)h(notation)e(concerning)h(the)h(space) g(of)f(con\014gurations.)40 b(First,)25 b(w)m(e)f(endo)m(w)168 3511 y(the)31 b(space)i(of)e(con\014gurations)f(with)h(a)g Fr(p)-5 b(artial)34 b(or)-5 b(dering)30 b Fw(b)m(y)i(sa)m(ying)g(that)f (t)m(w)m(o)h(con\014gurations)f Fo(u)168 3627 y Fw(and)h Fo(v)37 b Fw(are)32 b(suc)m(h)i(that)1862 3764 y Fo(u)44 b Fs(\024)h Fo(v)168 3921 y Fw(if)1737 4058 y Fo(u)p Fw(\()p Fo(p)p Fw(\))f Fs(\024)h Fo(v)t Fw(\()p Fo(p)p Fw(\))168 4215 y(for)32 b(an)m(y)h Fo(p)f 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4711 y Fo(u)d Fs(\024)g(R)2140 4726 y Fp(s)2178 4711 y Fo(u)p Fr(\).)64 b(The)41 b(set)h(of)g(Birkho\013)e(c)-5 b(on\014gur)g(ations)168 4827 y(in)34 b Fs(O)366 4842 y Fp(\033)448 4827 y Fr(wil)5 b(l)35 b(b)-5 b(e)34 b(denote)-5 b(d)34 b(by)h Fs(B)1300 4842 y Fp(\033)1348 4827 y Fr(.)168 4974 y Fz(De\014nition)i(7.)42 b Fr(We)35 b(say)h(that)g(a)f(c)-5 b(on\014gur)g(ation)35 b Fo(u)g Fr(is)g(a)g Fw(solution)d(of)h(our)g(v) -5 b(ariational)29 b(equation)168 5090 y Fr(\(or,)34 b(in)h(the)g(c)-5 b(alculus)34 b(of)h(variations)f(terminolo)-5 b(gy,)34 b(a)h Fw(critical)30 b(p)s(oin)m(t)p Fr(\))k(if,)g(for)h(any)g Fo(p)27 b Fs(2)i Fw(\003)p Fr(,)168 5321 y Fw(\(11\))1615 5226 y Fn(X)1621 5427 y Fk(B)r Fj(2S)1626 5483 y Fk(B)r Fj(3)p Fk(p)1789 5254 y Fo(@)5 b(H)1926 5269 y Fp(B)p 1789 5298 199 4 v 1812 5389 a Fo(@)g(u)1924 5404 y Fp(p)1997 5321 y Fw(\()p Fo(u)p Fw(\))44 b(=)g(0)17 b Fo(:)p eop %%Page: 9 9 9 8 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)700 b(9)0 450 y Fr(We)45 b(also)f(say)h(that)g Fo(u)f Fr(is)g(a)h Fw(ground)e(state)i Fr(\(or,)i(in)d(the)h(terminolo)-5 b(gy)44 b(of)g Fw([Mor24])p Fr(,)j(a)e Fw(class-)p Fo(A)0 566 y Fw(minimizer)p Fr(\))31 b(if)1398 683 y Fo(H)8 b Fw(\()p Fo(u;)17 b(X)8 b Fw(\))42 b Fs(\024)j Fo(H)8 b Fw(\()p Fo(v)t(;)17 b(X)8 b Fw(\))0 820 y Fr(for)34 b(any)g(\014nite)g(subset)g Fo(X)42 b Fr(of)34 b Fw(\003)g Fr(and)f(any)i(c)-5 b(on\014gur)g(ation)33 b Fo(v)38 b Fr(such)c(that)g Fo(u)28 b Fw(=)f Fo(v)38 b Fr(on)c Fw(\003)20 b Fs(\000)h Fo(X)42 b Fr(\(r)-5 b(e)g(c)g(al)5 b(l)0 936 y(the)35 b(de\014nition)f(of)g Fo(H)8 b Fw(\()p Fs(\001)p Fo(;)17 b(X)8 b Fw(\))34 b Fr(given)g(in)42 b Fw(\(4\))o Fr(\).)0 1073 y Fw(Note)33 b(that)f(the)h(sum)g(in)e (\(11\))h(is)g(absolutely)g(con)m(v)m(ergen)m(t,)j(thanks)f(to)e Fz(H8)p Fw(.)0 1248 y(1.3.)49 b Fz(Statemen)m(t)26 b(of)h(the)g (results.)49 b Fw(The)24 b(\014rst)g(result)g(w)m(e)h(pro)m(v)m(e)g 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b(singular\))g(lamination)f(of)i(gr)-5 b(ound)41 b(states)h(induc)-5 b(e)g(d)p eop %%Page: 10 10 10 9 bop 168 251 a Ft(10)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 453 y Fr(by)48 b(The)-5 b(or)g(em)47 b(8.)85 b(Supp)-5 b(ose)47 b(that)i Fo(u)1544 417 y Fl(\(1\))1690 453 y Fs(\025)k Fo(u)1876 417 y Fl(\(0\))2018 453 y Fr(and)48 b Fo(u)2277 417 y Fl(\(1\))2423 453 y Fs(6\021)53 b Fo(u)2609 417 y Fl(\(0\))2703 453 y Fr(.)85 b(Then,)51 b(ther)-5 b(e)48 b(exists)g Fo(u)k Fs(2)168 569 y(B)233 584 y Fp(\033)302 569 y Fs(\000)23 b(f)p Fo(u)508 533 y Fl(\(0\))601 569 y Fo(;)34 b(u)718 533 y Fl(\(1\))812 569 y Fs(g)g Fr(such)h(that)1688 705 y Fo(u)1744 664 y Fl(\(1\))1866 705 y Fs(\025)28 b Fo(u)f Fs(\025)h Fo(u)2215 664 y Fl(\(0\))168 854 y Fr(and)34 b(which)g(is)h(a)f(solution)h(of)f(the)h(variational)f(e)-5 b(quation.)168 997 y Fw(W)d(e)36 b(remark)g(that)g(the)h(h)m(yp)s (othesis)h(that)e Fo(H)43 b Fw(con)m(tains)37 b(only)e(the)i(iden)m 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y(Theorem)k(10)f(ma)m(y)g(b)s(e)h(seen)h(as) e(a)h(generalization)d(of)i(some)g(results)h(in)f(Aubry-Mather)h (theory)168 1926 y(\(see,)48 b(in)43 b(particular,)f([Mat86]\))i(and)g (it)f(\014ts)i(in)m(to)e(the)h(theory)h(of)e(plane-lik)m(e)g (\(quasi\)p)s(erio)s(dic)168 2043 y(structures,)30 b(the)d(recen)m(t)h (literature)e(of)g(whic)m(h)h(is)g(quite)g(ric)m(h)f(\(see,)k(for)c (instance,)h([Mos86)q(,)f(Ban88,)168 2159 y(CdlL01)o(,)40 b(V)-8 b(al01)o(,)40 b(RS03)o(,)h(V)-8 b(al04)n(,)40 b(T)-8 b(or04,)40 b(RS04a,)g(RS04b,)g(RS04c,)g(CdlL05)o(,)h(PV05a,)f (PV05b,)168 2275 y(Bes05]\).)i(With)27 b(resp)s(ect)i(to)e(the)h(ab)s (o)m(v)m(e)g(men)m(tioned)f(theory)-8 b(,)29 b(the)f(notion)e(of)h (\\plane")g(is)g(replaced)168 2391 y(here)33 b(b)m(y)h(the)f(more)e (general)h(concept)i(of)e(\\lev)m(el)g(set)h(of)g(a)f(co)s(cycle".)168 2508 y(The)47 b(pro)s(of)e(of)h(Theorem)h(10)f(is)g(reminiscen)m(t)f (of)h(the)h(one)g(in)e([dlL)-11 b(V05)o(],)50 b(in)c(whic)m(h)h(an)f (ellip-)168 2624 y(tic)39 b(PDE)i(problem)e(is)h(considered.)68 b(As)40 b(a)g(matter)g(of)g(fact,)i(the)f(pro)s(of)e(is)h(somewhat)h (simpler)168 2740 y(here,)g(since)f(the)f(discrete)h(setting)e(a)m(v)m (oids)i(some)e(smo)s(othness)i(and)f(compactness)h(issues)g(that)168 2856 y(complicate)i(the)j(situation)e(in)h([dlL)-11 b(V05)o(].)80 b(On)45 b(the)g(other)f(hand,)k(the)d(case)h(considered)f(here)168 2973 y(b)s(ecomes)32 b(conceptually)f(more)g(di\016cult)g(than)g(the)h (one)g(in)f([dlL)-11 b(V05)o(])32 b(since)g(w)m(e)g(allo)m(w)e(also)h (long-)168 3089 y(range)36 b(in)m(teraction)f(and)i(the)f(problems)g (considered)h(in)f(this)g(pap)s(er)g(ha)m(v)m(e)i(a)e(non-lo)s(cal)e (nature,)168 3205 y(a)e(complication)d(whic)m(h)34 b(is)e(not)g(presen) m(t)i(in)e(the)h(PDE)g(case)g(considered)h(in)e([dlL)-11 b(V05)o(].)168 3403 y(1.4.)48 b Fz(Some)37 b(concrete)f(examples)h(of)g (the)f(mo)s(dels)g(considered.)49 b Fw(The)33 b(abstract)f(form)m(ula-) 168 3520 y(tion)f(dev)m(elop)s(ed)j(in)e(Section)g(1.1)g(mak)m(es)h 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(\003)27 b(=)h Fq(Z)i Fw(and)i Fo(G)c Fw(=)g Fq(Z)p Fw(,)i(acting)h(on) i(\003)f(b)m(y)i(translations.)168 4648 y(The)f(v)-5 b(ariational)29 b(principle)i(of)h(the)h(F)-8 b(renk)m(el-Kon)m(toro)m (v)j(a)32 b(mo)s(del)f(is)h(giv)m(en)h(b)m(y)168 4860 y(\(13\))843 b Fo(S)6 b Fw(\()p Fo(u)p Fw(\))27 b(=)1513 4765 y Fn(X)1573 4975 y Fp(i)1684 4792 y Fw(1)p 1684 4837 49 4 v 1684 4928 a(2)1742 4860 y(\()p Fo(u)1836 4875 y Fp(i)p Fl(+1)1976 4860 y Fs(\000)c Fo(u)2132 4875 y Fp(i)2182 4860 y Fs(\000)f Fo(a)p Fw(\))2370 4819 y Fl(2)2432 4860 y Fw(+)g Fo(V)g Fw(\()p Fo(u)2703 4875 y Fp(i)2730 4860 y Fw(\))17 b Fo(;)168 5121 y Fw(with)23 b Fo(V)f Fw(\()p Fo(x)5 b Fw(+)g(1\))27 b(=)g Fo(V)22 b Fw(\()p Fo(x)p Fw(\).)41 b(The)25 b(energy)g(functional)d(in)h (\(13\))h(pro)m(vides)g(a)g(mo)s(del)e(for)i(a)g(system)g(with)168 5237 y(connecting)e(springs)g(of)g(length)f Fo(a)28 b Fs(2)g Fq(R)33 b Fw(under)23 b(a)f(p)s(erio)s(dic)e(substrate)j(p)s (oten)m(tial)d Fo(V)i Fw(.)40 b(The)23 b(particles)168 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b(the)f(assumption)g Fo(S)6 b Fw(\()p Fo(i)p Fw(;)17 b Fo(x)22 b Fw(+)f(1)p Fo(;)c(y)24 b Fw(+)d(1\))28 b(=)f Fo(S)6 b Fw(\()p Fo(i)p Fw(;)17 b Fo(x;)g(y)t Fw(\),)31 b(these)j(are)e(maps)g(of)g(the)168 5193 y(ann)m(ulus.)168 5309 y(A)g(particularly)f(in)m(teresting)h(case) h(is)f(when)i(the)f(sequence)i(of)e(maps)f(is)g(p)s(erio)s(dic,)f(that) h(is)1438 5469 y Fo(S)6 b Fw(\()p Fo(i)p Fw(;)17 b Fo(x;)g(y)t Fw(\))26 b(=)i Fo(S)6 b Fw(\()p Fo(i)22 b Fw(+)g Fo(N)10 b Fw(;)17 b Fo(x;)g(y)t Fw(\))g Fo(:)p eop %%Page: 13 13 13 12 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(13)0 450 y Fw(This)33 b(corresp)s(onds)h(to)e(a)g(p)s(erio)s(dic)f (sequence)k(of)d(t)m(wist)h(maps.)0 566 y(W)-8 b(e)28 b(note)f(that)g(the)g(comp)s(osition)e(of)i(t)m(wist)g(maps)g(can)h (fail)c(to)j(b)s(e)h(a)f(t)m(wist)g(map.)41 b(The)28 b(v)-5 b(ariational)0 683 y(theory)33 b(for)f(comp)s(ositions)f(of)h(t) 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y(p)s(erturbation)43 b(of)g(the)h(standard)h(map.)76 b(Nev)m(ertheless,)49 b(as)44 b(a)f(dynamical)f(system,)48 b(it)43 b(is)g(v)m(ery)0 5469 y(di\013eren)m(t.)e(Ev)m(en)26 b(the)f(phase)g(space)g(has)g (di\013eren)m(t)g(dimensions)e(and)i(there)g(are)f(large)f(co)s (e\016cien)m(ts.)p eop %%Page: 14 14 14 13 bop 168 251 a Ft(14)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(It)34 b(is)h(also)e(in)m(teresting)h(to)h(note)g(that)f(the)h(v)-5 b(ariational)31 b(metho)s(ds)k(pro)s(duce)g(solutions)e(with)i(just)168 566 y(one)e(frequency)h(ev)m(en)g(if)e(the)h(corresp)s(onding)f (dynamical)f(system)j(has)f(2)f(degrees)i(of)e(freedom.)168 683 y(Mo)s(dels)f(of)g(the)h(t)m(yp)s(e)g(\(21\))f(w)m(ere)i (considered)f(in)f([Ang90)o(],)h(who)g(called)e(them)h Fr(monotone)i(r)-5 b(e)g(cur-)168 799 y(r)g(enc)g(es)p Fw(.)55 b(The)38 b(pap)s(er)f([Ang90])f(used)i(a)f(gradien)m(t)f(\015o) m(w)i(metho)s(d)e(similar)d(to)k(the)g(one)g(w)m(e)h(use)g(in)168 915 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Fw(The)k(standard)g(lattice)e(mo)s(dels)g(as)168 1907 y(in)32 b([Rue99])g(corresp)s(onds)i(to)e(taking)g(\003)27 b(=)h Fq(Z)1858 1871 y Fp(d)1928 1907 y Fw(and)33 b Fo(G)28 b Fw(=)f Fq(Z)2395 1871 y Fp(d)2433 1907 y Fw(.)168 2023 y(W)-8 b(e)44 b(can)h(think)f(of)f(\003)h(as)g(describing)g(a)g (standard)g(crystal,)j(whic)m(h)e(is)f(symmetric)f(under)i Fq(Z)3759 1987 y Fp(d)3797 2023 y Fw(-)168 2139 y(translations.)168 2255 y(In)29 b(our)g(case,)h(eac)m(h)g(site)e(in)g(the)i(crystal)f(has) g(an)f(order)h(parameter)g(c)m(haracterized)g(b)m(y)h(a)f(n)m(um)m(b)s (er.)168 2372 y(The)34 b(assumptions)f(on)g(in)m(v)-5 b(ariance)32 b(of)g(the)i(in)m(teractions)e(under)i(translation)d(are)i (standard)h(and)168 2488 y(can)d(b)s(e)h(found)g(in)f([Rue99].)43 b(The)32 b(more)f(detailed)f(assumptions)i(ab)s(out)f(the)h(regularit)m (y)e(and)i(the)168 2604 y(di\013eren)m(tiabilit)m(y)j(of)i(the)h(in)m (teractions)f(are)h(not)g(considered)g(in)f([Rue99],)i(whic)m(h)g(did)e (not)g(tak)m(e)168 2720 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Fp(i)2514 3440 y Fw(=)28 b(\()p Fo(F)2719 3455 y Fp(i)2747 3440 y Fq(Z)p Fw(\))2854 3404 y Fp(d)2925 3440 y Fw(with)k Fo(F)3210 3455 y Fp(i)3270 3440 y Fw(an)h(increasing)168 3556 y(sequence)i(of)d(p)s(ositiv)m(e)g (in)m(tegers)h(going)e(to)h(+)p Fs(1)p Fw(.)168 3672 y(One)h(can)g(also)f(consider)h(materials)e(with)i(some)f(structure,)j (b)m(y)f(assuming)e(that)h(the)g(symmetry)168 3789 y(group)44 b(of)f(the)i(problem)e(is)g(smaller)g(than)h(the)g(crystal.)78 b(F)-8 b(or)44 b(example,)i(one)f(can)f(tak)m(e)h Fo(G)i Fw(=)168 3836 y Fn(Q)262 3862 y Fp(d)262 3940 y(j)t Fl(=1)389 3911 y Fw(\()p Fo(N)505 3926 y Fp(j)541 3911 y Fq(Z)p Fw(\).)41 b(In)33 b(this)f(case,)i(there)f(is)f(a)g(fundamen)m(tal)g (cell)775 4078 y Fs(f)p Fw(0)p Fo(;)17 b(:)g(:)g(:)e(;)i(N)1170 4093 y Fl(1)1231 4078 y Fs(\000)23 b Fw(1)p Fs(g)f(\002)g(f)p Fw(0)p Fo(;)17 b(:)g(:)g(:)f(;)h(N)1947 4093 y Fl(2)2008 4078 y Fs(\000)23 b Fw(1)p Fs(g)f(\002)g(\001)17 b(\001)g(\001)k(\002)h (f)p Fw(0)p Fo(;)17 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b(further)e(symmetry)h(prop)s (erties.)168 2628 y(It)22 b(also)e(mak)m(es)j(use)f(of)g(the)g(follo)m (wing)d(result,)24 b(whic)m(h)e(can)g(b)s(e)g(found)g(in)f(Prop)s (osition)f(1)h(of)h([CdlL98)o(].)168 2772 y Fz(Prop)s(osition)31 b(11.)39 b Fr(L)-5 b(et)32 b Fo(G)g Fr(b)-5 b(e)31 b(a)h(r)-5 b(esidual)5 b(ly)31 b(\014nite)g(gr)-5 b(oup.)44 b(Then,)31 b(given)g(any)h(c)-5 b(o)g(cycle)31 b Fo(\033)k Fr(we)d(c)-5 b(an)168 2888 y(\014nd)34 b(a)h(se)-5 b(quenc)g(e)34 b Fo(\033)902 2852 y Fl(\()p Fp(n)p Fl(\))1038 2888 y Fr(of)h(r)-5 b(ational)34 b(c)-5 b(o)g(cycles)35 b(such)f(that)1590 3063 y Fw(lim)1538 3123 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1794 3063 y Fs(j)-17 b(j)g(j)o Fo(\033)1902 3022 y Fl(\()p Fp(n)p Fl(\))2026 3063 y Fs(\000)23 b Fo(\033)t Fs(j)-17 b(j)g(j)28 b Fw(=)f(0)17 b Fo(:)168 3273 y Fw(W)-8 b(e)44 b(no)m(w)g(in)m(tro)s(duce)f(the)h(follo)m(wing)d(equiv)-5 b(alence)44 b(relation)e(on)h(subsets)j(of)d(\003,)j(induced)e(b)m(y)g (a)168 3390 y(giv)m(en)35 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b(in\014mal)f(minimizer)40 b Fw(whic)m(h)30 b(enjo)m(ys)h(some)f(extra) 0 5353 y(geometric)40 b(prop)s(ert)m(y)j(\(Birkho\013)d(prop)s(ert)m (y\),)45 b(that)c({)g(b)s(esides)i(in)m(trinsic)d(in)m(terest)i({)f (will)e(allo)m(w)0 5469 y(us)i(to)f(pro)m(v)m(e)i(that)f(the)g(the)g (in\014mal)d(minimizer)f(do)s(es)k(not)g(presen)m(t)h (symmetry-breaking.)67 b(In)p eop %%Page: 18 18 18 17 bop 168 251 a Ft(18)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(fact,)32 b(all)f(minimizers)e(of)k(the)g(p)s(erio)s(di\014ed)e(energy)j({)e(ev)m (en)i(if)e(they)h(do)g(not)f(satisfy)h(the)g(Birkho\013)168 566 y(prop)s(ert)m(y)g(|)f(also)g(satisfy)g(a)h(v)m(ersion)g(of)f(the)h (no-symmetry-breaking)e(prop)s(ert)m(y)-8 b(.)168 770 y(2.1.1.)48 b Fr(The)34 b(r)-5 b(e)g(duc)g(e)g(d)35 b(inter)-5 b(action.)48 b Fw(In)33 b(order)g(to)f(formalize)e(the)j(ab)s(o)m(v)m (e)g(ideas,)g(w)m(e)g(\014x)168 942 y(\(27\))1437 b Fo(R)45 b(>)f(\016)t Fw(\()p Fo(F)14 b Fw(\))168 1114 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Fw(is)k(ob)m(vious.)44 b(Also,)382 4915 y Fs(j)p Fo(H)491 4931 y Fl([)p Fp(B)s Fl(])587 4942 y Fk(H)648 4915 y Fw(\()p Fo(v)737 4874 y Fl(\()p Fp(n)p Fl(\))839 4915 y Fw(\))22 b Fs(\000)h Fo(H)1080 4931 y Fl([)p Fp(B)s Fl(])1176 4942 y Fk(H)1237 4915 y Fw(\()p Fo(v)t Fw(\))p Fs(j)82 b(\024)1635 4821 y Fn(X)1637 5032 y Fp(p)p Fm(2)p Fp(B)1795 4780 y Fn(Z)1895 4806 y Fl(1)1851 5005 y(0)1951 4771 y Fn(\014)1951 4830 y(\014)1951 4890 y(\014)1951 4950 y(\014)1994 4848 y Fo(@)5 b(H)2131 4863 y Fp(B)p 1994 4892 199 4 v 2018 4984 a Fo(@)g(u)2130 4999 y Fp(q)2203 4915 y Fw(\()p Fo(\034)11 b(v)2345 4874 y Fl(\()p Fp(n)p Fl(\))2469 4915 y Fw(+)22 b(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(v)t Fw(\))2955 4771 y Fn(\014)2955 4830 y(\014)2955 4890 y(\014)2955 4950 y(\014)3022 4915 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(v)3277 4874 y Fl(\()p Fp(n)p Fl(\))3273 4940 y Fp(p)3400 4915 y Fs(\000)h Fo(v)3547 4930 y Fp(p)3587 4915 y Fs(j)1474 5197 y(\024)84 b Fo(C)1728 5102 y Fn(X)1731 5314 y Fp(p)p Fm(2)p Fp(B)1889 5197 y Fs(j)p Fo(v)1968 5156 y Fl(\()p Fp(n)p Fl(\))1964 5221 y Fp(p)2091 5197 y Fs(\000)23 b Fo(v)2238 5212 y Fp(p)2278 5197 y Fs(j)17 b Fo(;)168 5469 y Fw(for)32 b(some)g Fo(C)j(>)27 b Fw(0,)33 b(thanks)g(to)g Fz(H8)p Fw(.)p eop %%Page: 19 19 19 18 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(19)0 450 y Fw(Since)33 b Fo(B)k Fw(is)32 b(a)h(\014nite)f(set,)h(w)m (e)h(th)m(us)g(conclude)f(that)1251 623 y(lim)1200 683 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1455 623 y Fo(H)1536 639 y Fl([)p Fp(B)s Fl(])1632 650 y Fk(H)1693 623 y Fw(\()p Fo(v)1782 582 y Fl(\()p Fp(n)p Fl(\))1884 623 y Fw(\))28 b(=)f Fo(H)2134 639 y Fl([)p Fp(B)s Fl(])2230 650 y Fk(H)2292 623 y Fw(\()p Fo(v)t Fw(\))17 b Fo(:)0 832 y Fw(Then)34 b(the)f(claim)d(follo)m(ws)h(from)g(Lemma)h(12.)1844 b Fg(\003)0 1002 y Fw(W)-8 b(e)35 b(no)m(w)h(p)s(oin)m(t)e(out)h(an)g (easy)h(v)-5 b(ariation)33 b(of)h(a)h(classical)f(result)h(in)f (quotien)m(t)h(group)g(theory)-8 b(.)51 b(T)-8 b(o)0 1118 y(this)46 b(exten)m(t,)51 b(w)m(e)c(use)g(the)f(standard)h (notation)e(for)g(the)i(equiv)-5 b(alence)46 b(classes)h(induced)g(b)m (y)g(a)0 1237 y(normal)31 b(subgroup)779 1212 y(~)754 1237 y Fo(H)40 b Fw(of)32 b(\014nite)g(index:)1357 1411 y([)p Fo(p)p Fw(])1478 1421 y Fl(~)1460 1438 y Fp(H)1555 1411 y Fw(:=)27 b Fs(f)p Fo(hp)17 b(;)61 b(h)28 b Fs(2)2148 1386 y Fw(~)2123 1411 y Fo(H)7 b Fs(g)17 b Fo(:)0 1592 y Fz(Lemma)37 b(14.)42 b Fr(L)-5 b(et)775 1567 y Fw(~)750 1592 y Fo(H)35 b Fs(\032)28 b Fo(H)43 b Fr(b)-5 b(e)34 b(normal,)g(\014nite)h(index)f(sub)-5 b(gr)g(oups)34 b(of)h Fo(G)p Fr(.)0 1708 y(Given)53 b Fo(p)62 b Fs(2)g Fw(\003)p Fr(,)c(let)c Fo(f)902 1723 y Fp(p)995 1708 y Fr(b)-5 b(e)53 b(the)g(unique)g(element)g(of)g(the)h(fundamental)e (domain)g Fw(\003)p Fo(=H)61 b Fr(for)0 1824 y(which)34 b Fo(h)330 1839 y Fp(p)370 1824 y Fo(p)27 b Fw(=)h Fo(f)598 1839 y Fp(p)638 1824 y Fr(,)34 b(for)h(some)f Fo(h)1163 1839 y Fp(p)1231 1824 y Fs(2)28 b Fo(H)8 b 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g(is)g(injectiv)m(e.)62 b(F)-8 b(or)38 b(this,)i(supp)s(ose)g(that)f (\()p Fo(f)3209 3618 y Fp(p)3248 3603 y Fo(;)17 b Fw([)p Fo(h)3375 3618 y Fp(p)3415 3603 y Fw(])3460 3613 y Fl(~)3442 3630 y Fp(H)3509 3603 y Fw(\))39 b(=)0 3732 y(\()p Fo(f)86 3747 y Fp(q)124 3732 y Fo(;)17 b Fw([)p Fo(h)251 3747 y Fp(q)289 3732 y Fw(])334 3742 y Fl(~)316 3759 y Fp(H)383 3732 y Fw(\))44 b Fs(2)h Fw(\(\003)p Fo(=H)8 b Fw(\))28 b Fs(\002)h Fw(\()p Fo(H)r(=)1187 3706 y Fw(~)1162 3732 y Fo(H)7 b Fw(\))42 b(for)g(some)g Fo(p;)33 b(q)48 b Fs(2)d Fw(\003.)72 b(Then,)44 b Fo(f)2561 3747 y Fp(p)2645 3732 y Fw(=)g Fo(f)2813 3747 y Fp(q)2893 3732 y Fw(and)e Fo(h)3148 3747 y Fp(q)3231 3732 y Fw(=)3352 3705 y(~)3351 3732 y Fo(hh)3463 3747 y Fp(p)3545 3732 y Fw(for)0 3860 y(some)250 3834 y(~)249 3860 y Fo(h)35 b Fs(2)466 3835 y Fw(~)441 3860 y Fo(H)7 b Fw(.)56 b(Thence,)39 b Fo(h)1041 3875 y Fp(p)1081 3860 y Fo(p)c Fw(=)f Fo(f)1323 3875 y Fp(p)1398 3860 y Fw(=)h Fo(f)1557 3875 y Fp(q)1629 3860 y Fw(=)g Fo(h)1796 3875 y Fp(q)1834 3860 y Fo(q)k Fw(=)2028 3834 y(~)2027 3860 y Fo(hh)2139 3875 y Fp(p)2179 3860 y Fo(q)t Fw(.)55 b(Then,)39 b(since)2864 3835 y(~)2839 3860 y Fo(H)44 b Fw(is)36 b(normal,)f Fo(h)3484 3824 y Fm(\003)3558 3860 y Fw(:=)0 3989 y Fo(h)56 3953 y Fm(\000)p Fl(1)56 4013 y Fp(p)151 3963 y Fw(~)150 3989 y Fo(h)q(h)263 4004 y Fp(p)330 3989 y Fs(2)450 3964 y Fw(~)424 3989 y Fo(H)40 b Fw(and)33 b Fo(p)28 b Fw(=)f Fo(h)971 3953 y Fm(\003)1011 3989 y Fo(q)t Fw(.)43 b(So,)33 b([)p Fo(p)p Fw(])1412 3999 y Fl(~)1394 4016 y Fp(H)1489 3989 y Fw(=)27 b([)p Fo(q)t Fw(])1711 3999 y Fl(~)1693 4016 y Fp(H)1760 3989 y Fw(.)0 4109 y(Accordingly)-8 b(,)32 b(the)h(map)f(in)f(\(30\))h (is)g(a)h(bijection.)1712 b Fg(\003)0 4307 y Fz(Remark)34 b(15.)40 b Fw(The)31 b(result)f(in)f(Lemma)g(14)g(ma)m(y)h(not)g(hold)g (if)e(one)j(drops)f(condition)f Fz(G3)p Fw(.)43 b(As)30 b(an)0 4423 y(example,)36 b(let)f Fo(G)e Fw(=)g Fo(H)43 b Fw(b)s(e)36 b(the)g(group)g(of)f(the)h(p)s(erm)m(utations)f(of)g (three)i(elemen)m(ts)f Fs(f)p Fw(1)p Fo(;)17 b Fw(2)p Fo(;)g Fw(3)p Fs(g)34 b Fw(\(or,)0 4540 y(equiv)-5 b(alen)m(tly)d(,)35 b(the)h(dihedral)e(group)h(of)f(order)i(6,)f(or)g(the)h(symmetric)e (group)h(of)g(order)g(3,)h(in)e(the)0 4656 y(notation)d(of)h([Wik06a,)g (Wik06b]\).)0 4772 y(Let)h Fo(a)28 b Fw(:)g(\(1)p Fo(;)17 b Fw(2)p Fo(;)g Fw(3\))27 b Fs(7!)g Fw(\(2)p Fo(;)17 b Fw(1)p Fo(;)g Fw(3\))31 b(and)i Fo(b)c Fw(:)f(\(1)p Fo(;)17 b Fw(2)p Fo(;)g Fw(3\))26 b Fs(7!)i Fw(\(1)p Fo(;)17 b Fw(3)p Fo(;)g Fw(2\).)42 b(Then,)34 b Fo(G)f Fw(is)f(generated)i(b)m(y)f Fo(a)g Fw(and)g Fo(b)0 4888 y Fw(and)g(it)e(has)i(6)f(elemen)m(ts:)h Fo(a)p Fw(,)g Fo(b)p Fw(,)g Fo(ab)p Fw(,)h Fo(ba)p Fw(,)f Fo(aba)g Fw(and)g(the)g(iden)m(tit)m(y)f(id.)0 5005 y(The)e(fact)e(that)h Fo(G)f Fw(is)h(\014nite)f(implies)e(that)j(it)f(is)g(residually)f (\014nite,)j(b)s(ecause)g(the)f(iden)m(tit)m(y)f(is)h(\014nite)0 5121 y(index.)0 5237 y(The)24 b(group)g Fo(G)f Fw(acts)h(on)f(the)h (lattice)e(\003)27 b(:=)h Fs(f)p Fw(1)p Fo(;)17 b Fw(2)p Fo(;)g Fw(3)p Fs(g)k Fw(in)i(the)h(ob)m(vious)f(w)m(a)m(y)i(\(for)e (instance,)g Fo(a)p Fw(\(1\))28 b(=)g(2,)0 5353 y Fo(a)p Fw(\(2\))j(=)g(1,)k Fo(a)p Fw(\(3\))c(=)g(3\))k(and)f(since)i(an)m(y)f (elemen)m(t)f(of)h(\003)f(is)g(sen)m(t)i(to)e(an)m(y)i(other)e(one)h(b) m(y)h(a)e(suitable)0 5469 y(p)s(erm)m(utation,)d(it)h(follo)m(ws)f (that)h(\003)p Fo(=G)g Fw(con)m(tains)h(only)f(one)h(elemen)m(t.)p eop %%Page: 20 20 20 19 bop 168 251 a Ft(20)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 458 y Fw(Let)370 433 y(~)345 458 y Fo(H)40 b Fw(:=)32 b Fs(f)p Fw(id)o Fo(;)17 b(ab;)g(ba)p Fs(g)p Fw(.)53 b(By)35 b(insp)s(ection,)h(one)f (sees)i(that)2403 433 y(~)2378 458 y Fo(H)43 b Fw(is)35 b(a)g(normal)e(subgroup)j(of)f Fo(G)g Fw(and)168 583 y(that)d Fo(G=)530 558 y Fw(~)505 583 y Fo(H)j Fw(=)28 b Fs(f)p Fw([id)o(])928 593 y Fl(~)910 610 y Fp(H)977 583 y Fo(;)17 b Fw([)p Fo(a)p Fw(])1144 593 y Fl(~)1126 610 y Fp(H)1193 583 y Fs(g)p Fw(.)168 699 y(In)33 b(particular,)168 883 y(\(31\))1342 b Fo(])1722 772 y Fn(\020)1782 883 y Fo(G=)1933 857 y Fw(~)1908 883 y Fo(H)1996 772 y Fn(\021)2100 883 y Fw(=)44 b(2)17 b Fo(:)168 1066 y Fw(F)-8 b(urthermore,)31 b(since)h(eac)m(h)h(p)s(oin)m(t)e(of)g(\003)g(reac)m(hes)j(an)m(y)e (other)g(via)f(either)g Fo(ab)i Fw(or)e Fo(ba)p Fw(,)i(w)m(e)f(ha)m(v)m (e)h(that)168 1185 y(also)e(\003)p Fo(=)505 1160 y Fw(~)480 1185 y Fo(H)40 b Fw(con)m(tains)32 b(only)h(one)f(elemen)m(t.)168 1301 y(Th)m(us,)i(in)e(this)g(case,)1149 1421 y Fo(])p Fw(\(\003)p Fo(=H)8 b Fw(\))21 b Fs(\001)g Fo(])p Fw(\()p Fo(H)r(=)1773 1395 y Fw(~)1747 1421 y Fo(H)8 b Fw(\))27 b(=)h(1)22 b Fs(\001)g Fw(2)27 b Fs(6)p Fw(=)h(1)f(=)h Fo(])p Fw(\(\003)p Fo(=)2703 1395 y Fw(~)2679 1421 y Fo(H)6 b Fw(\))17 b Fo(;)168 1558 y Fw(whic)m(h)33 b(sho)m(ws)h(that)e (Lemma)g(14)g(do)s(es)h(not)f(hold)g(in)g(this)g(case.)168 1733 y(2.1.2.)48 b Fr(Existenc)-5 b(e)36 b(of)h(minimizers)f(for)h(the) g(r)-5 b(e)g(duc)g(e)g(d)36 b(ener)-5 b(gy.)49 b Fw(Note)35 b(that)f Fs(C)3023 1748 y Fp(\033)n(;H)3184 1733 y Fw(is)h(\014nite)f (dimen-)168 1849 y(sional)e(\(that)j(is,)f(an)m(y)h(con\014guration)f Fo(u)c Fs(2)h(C)1839 1864 y Fp(\033)n(;H)2000 1849 y Fw(is)j(determined)g(b)m(y)h(its)f(v)-5 b(alues)34 b(on)h(\003)p Fo(=H)8 b Fw(,)34 b(whic)m(h)168 1967 y(is)41 b(a)g(\014nite)g(set\))g (and)h Fo(S)1086 1931 y Fp(R)1080 1992 y(\033)n(;H)1247 1967 y Fw(is)f(b)s(ounded)h(from)e(b)s(elo)m(w.)69 b(W)-8 b(e)42 b(also)e(note)i(that,)h(b)s(ecause)g(of)d Fz(H4)168 2083 y Fw(the)34 b(problem)f(is)h(in)m(v)-5 b(arian)m(t)33 b(under)i(addition)e(of)g(1)h(to)g(all)e(the)j(sites.)49 b(Hence,)36 b(it)d(su\016ces)j(to)e(con-)168 2199 y(sider)g Fo(u)459 2214 y Fl(0)528 2199 y Fs(2)c Fq(T)687 2163 y Fl(1)730 2199 y Fw(.)47 b(In)35 b(this)e(case,)j(w)m(e)f(see)g(that,) f(b)s(ecause)h(of)f Fz(H6)p Fw(,)g(the)g(energy)h(go)s(es)f(to)g(+)p Fs(1)f Fw(when)168 2316 y(the)41 b(other)h(sites)f(tend)h(to)f(+)p Fs(1)p Fw(.)68 b(Hence,)45 b(w)m(e)d(can)f(conclude)h(that)f Fo(S)2827 2280 y Fp(R)2821 2341 y(\033)n(;H)2988 2316 y Fw(reac)m(hes)i(a)d(minim)m(um)168 2432 y(on)32 b Fs(C)355 2447 y Fp(\033)n(;H)481 2432 y Fw(.)168 2550 y(Of)24 b(course,)j(the)e(minimizers)d(of)i Fo(S)1432 2513 y Fp(R)1426 2575 y(\033)n(;H)1576 2550 y Fw(in)g Fs(C)1734 2565 y Fp(\033)n(;H)1884 2550 y Fw(can)h(b)s(e)g(considered)h(also)d (as)i(con\014gurations)f(in)g Fs(O)3755 2565 y Fp(\033)3802 2550 y Fw(.)168 2666 y(A)m(t)36 b(this)h(stage)f(of)g(the)h(argumen)m (t,)g(w)m(e)h(are)e(not)g(claiming)d(that)k(these)g(minimizers)d(in)i Fs(C)3537 2681 y Fp(\033)n(;H)3699 2666 y Fw(are)168 2782 y(minimizers)29 b(of)k Fo(S)38 b Fw(in)32 b Fs(O)1064 2797 y Fp(\033)1111 2782 y Fw(.)168 2898 y(Unfortunately)-8 b(,)26 b(the)g(minimizers)c(of)j(the)h(p)s(erio)s(dic)e(problem)g(pro)s (duced)i(b)m(y)g(the)g(ab)s(o)m(v)m(e)g(compact-)168 3015 y(ness)32 b(argumen)m(t)f(are)g(not)g(v)m(ery)h(suitable)e(for)h (further)g(dev)m(elopmen)m(ts.)44 b(Indeed,)33 b(at)e(the)g(momen)m(t,) 168 3131 y(w)m(e)43 b(do)g(not)f(kno)m(w)h(whether)h(the)f(minimizers)d (no)m(w)j(pro)s(duced)g(are)g(ground)f(states.)74 b(It)43 b(could)168 3247 y(happ)s(en)30 b(that)f(their)g(energy)i(could)e(b)s (e)h(lo)m(w)m(ered)g(b)m(y)h(p)s(erturbations)e(whic)m(h)h(ha)m(v)m(e)h (p)s(erio)s(ds)e(longer)168 3363 y(than)j(the)h(one)g(no)m(w)g (considered.)168 3539 y(2.1.3.)48 b Fr(The)39 b(in\014mal)f(minimizer.) 47 b Fw(The)38 b(goal)e(of)g(this)h(section)h(is)e(to)h(sho)m(w)i(that) e(among)f(all)f(the)168 3655 y(minimizers)24 b(in)j Fs(C)817 3670 y Fp(\033)n(;H)970 3655 y Fw(there)i(is)d(a)i(sp)s(ecial)e(one)i (whic)m(h)g(is)f(indeed)g(a)h(ground)f(state)h(\(after)f(remo)m(ving) 168 3771 y(the)j(cut-o\013)f Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))27 b Fs(\024)h Fo(R)q Fw(\).)43 b(Moreo)m(v)m(er,)31 b(this)f(minimizer)c(will)h(ha)m(v)m(e)j(sp)s(ecial)f(geometric)f(prop) s(erties)168 3887 y(that)k(will)e(allo)m(w)h(us)j(to)e(tak)m(e)h(the)g (limit)c(of)j(irrational)d(minimizers.)168 4003 y(Note)42 b(that)h(in)f(this)g(construction)h(w)m(e)g(will)e(use)i(hea)m(vily)g (the)g(ferromagnetism)d(prop)s(ert)m(y)j Fz(H9)168 4120 y Fw(whic)m(h)h(w)m(e)g(ha)m(v)m(e)g(not)g(used)g(so)g(far.)75 b(The)44 b(tec)m(hnical)f(result)g(whic)m(h)h(follo)m(ws)e(will)f(allo) m(w)h(us)h(to)168 4236 y(rearrange)32 b(minimizers:)168 4373 y Fz(Lemma)37 b(16.)42 b Fr(L)-5 b(et)35 b Fo(u;)17 b(v)30 b Fs(2)e(C)1241 4388 y Fp(\033)n(;H)1368 4373 y Fr(.)44 b(Denote)1076 4530 y Fo(u)22 b Fs(^)g Fo(v)32 b Fw(=)27 b(max)p Fs(f)p Fo(u;)17 b(v)t Fs(g)98 b Fo(;)116 b(u)22 b Fs(_)h Fo(v)31 b Fw(=)d(min)n Fs(f)p Fo(u;)17 b(v)t Fs(g)34 b Fo(:)168 4688 y Fr(Then,)g Fo(u)21 b Fs(^)i Fo(v)t(;)17 b(u)k Fs(_)i Fo(v)31 b Fs(2)d(C)1103 4703 y Fp(\033)n(;H)1264 4688 y Fr(and)168 4851 y Fw(\(32\))655 b Fo(S)1063 4810 y Fp(R)1057 4876 y(\033)n(;H)1183 4851 y Fw(\()p Fo(u)21 b Fs(^)i Fo(v)t Fw(\))f(+)g Fo(S)1662 4810 y Fp(R)1656 4876 y(\033)n(;H)1781 4851 y Fw(\()p Fo(u)g Fs(_)g Fo(v)t Fw(\))28 b Fs(\024)g Fo(S)2273 4810 y Fp(R)2267 4876 y(\033)n(;H)2392 4851 y Fw(\()p Fo(u)p Fw(\))22 b(+)g Fo(S)2710 4810 y Fp(R)2704 4876 y(\033)n(;H)2830 4851 y Fw(\()p Fo(v)t Fw(\))17 b Fo(:)168 5009 y Fz(Pro)s(of.)43 b Fw(W)-8 b(e)33 b(write)1705 5166 y Fo(u)22 b Fs(^)g Fo(v)32 b Fw(=)27 b Fo(u)22 b Fw(+)g Fo(\013)1705 5312 y(u)g Fs(_)h Fo(v)31 b Fw(=)d Fo(u)21 b Fs(\000)i Fo(\014)168 5469 y Fw(where)34 b Fo(\013)q(;)17 b(\014)37 b Fw(are)c(non-negativ)m (e)f(con\014gurations.)p eop %%Page: 21 21 21 20 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(21)0 450 y Fw(Since)33 b Fo(u)21 b Fs(^)i Fo(v)j Fw(+)c Fo(u)f Fs(_)i Fo(v)31 b Fw(=)d Fo(u)22 b Fw(+)g Fo(v)t Fw(,)32 b(w)m(e)i(ha)m(v)m(e)g(that)e Fo(v)f Fw(=)d Fo(u)21 b Fw(+)i Fo(\013)f Fs(\000)h Fo(\014)6 b Fw(.)0 566 y(W)-8 b(e)26 b(note)g(that)g(\()p Fo(u)9 b Fs(^)g Fo(v)t Fw(\))843 581 y Fp(g)r(i)934 566 y Fw(=)27 b(\()p Fo(u)9 b Fs(^)g Fo(v)t Fw(\))1304 581 y Fp(i)1357 566 y Fw(and)26 b(\()p Fo(u)9 b Fs(_)g Fo(v)t Fw(\))1807 581 y Fp(g)r(i)1897 566 y Fw(=)28 b(\()p Fo(u)9 b Fs(_)g Fo(v)t Fw(\))2268 581 y Fp(i)2321 566 y Fw(for)25 b(an)m(y)i Fo(g)k Fs(2)d Fo(H)8 b Fw(.)41 b(Therefore,)27 b Fo(\013)3494 581 y Fp(g)r(i)3586 566 y Fw(=)0 683 y Fo(\013)62 698 y Fp(i)121 683 y Fw(and)32 b Fo(\014)365 698 y Fp(g)r(i)457 683 y Fw(=)27 b Fo(\014)615 698 y Fp(i)644 683 y Fw(,)k(for)g(an)m(y)h Fo(g)f Fs(2)d Fo(H)8 b Fw(.)43 b(Hence,)32 b Fo(u)19 b Fw(+)h Fo(t\013)g Fw(+)g Fo(s\014)33 b Fs(2)28 b(C)2343 698 y Fp(\033)n(;H)2469 683 y Fw(,)k(for)e(an)m(y)i Fs(\000)p Fw(1)c Fs(\024)g Fo(t;)17 b(s)28 b Fs(\024)g Fw(1,)k(that)0 799 y(yields)g(the)h(\014rst)g(claim)d(in)i(Lemma)f(16.)0 915 y(W)-8 b(e)33 b(also)f(note)g(that,)h(b)m(y)g(construction,)g(w)m (e)h(ha)m(v)m(e)0 1100 y(\(33\))1210 b(supp)17 b Fo(\013)23 b Fs(\\)g Fw(supp)17 b Fo(\014)34 b Fw(=)27 b Fs(;)17 b Fo(:)0 1285 y Fw(Consider)33 b(the)g(function)1193 1429 y Fo(F)14 b Fw(\()p Fo(t;)j(s)p Fw(\))44 b(:=)g Fo(S)1728 1388 y Fp(R)1722 1454 y(\033)n(;H)1848 1429 y Fw(\()p Fo(u)22 b Fw(+)g Fo(t\013)h Fw(+)f Fo(s\014)6 b Fw(\))17 b Fo(:)0 1598 y Fw(W)-8 b(e)32 b(ha)m(v)m(e)h(the)f (calculus)f(iden)m(tit)m(y)g(\(recall)f(indeed)i(that,)f(b)m(y)i Fz(H8)p Fw(,)e Fo(F)45 b Fw(is)31 b(t)m(wice)h(di\013eren)m(tiable)e (and)0 1714 y(Lemma)h(12)h(allo)m(ws)g(to)g(in)m(terc)m(hange)h(the)g (sum)g(with)f(the)h(deriv)-5 b(ativ)m(es\))984 1891 y Fo(F)14 b Fw(\(1)p Fo(;)j Fw(0\))k(+)h Fo(F)14 b Fw(\(0)p Fo(;)j Fs(\000)p Fw(1\))k Fs(\000)i Fo(F)14 b Fw(\(0)p Fo(;)j Fw(0\))k Fs(\000)h Fo(F)14 b Fw(\(1)p Fo(;)j Fs(\000)p Fw(1\))1011 2119 y(=)1115 1984 y Fn(Z)1215 2010 y Fl(1)1170 2209 y(0)1281 2052 y Fo(@)5 b(F)p 1281 2096 134 4 v 1301 2187 a(@)g(t)1424 2119 y Fw(\()p Fo(t;)17 b Fw(0\))g Fo(dt)k Fs(\000)1852 1984 y Fn(Z)1952 2010 y Fl(1)1907 2209 y(0)2018 2052 y Fo(@)5 b(F)p 2018 2096 V 2038 2187 a(@)g(t)2161 2119 y Fw(\()p Fo(t;)17 b Fs(\000)p Fw(1\))g Fo(dt)1011 2399 y Fw(=)1115 2263 y Fn(Z)1215 2289 y Fl(1)1170 2488 y(0)1271 2263 y Fn(Z)1370 2289 y Fl(0)1326 2488 y Fm(\000)p Fl(1)1458 2331 y Fo(@)1514 2295 y Fl(2)1554 2331 y Fo(F)p 1447 2376 196 4 v 1447 2467 a(@)5 b(t@)g(s)1652 2399 y Fw(\()p Fo(t;)17 b(s)p Fw(\))g Fo(ds)g(dt)g(:)0 2180 y Fw(\(34\))0 2642 y(No)m(w)33 b(w)m(e)h(observ)m(e)g(that)0 2865 y(\(35\))840 2797 y Fo(@)896 2761 y Fl(2)936 2797 y Fo(F)p 829 2842 V 829 2933 a(@)5 b(t@)g(s)1034 2865 y Fw(\()p Fo(t;)17 b(s)p Fw(\))44 b(=)1472 2770 y Fn(X)1432 2974 y Fi([)p Fk(B)r Fi(])1516 2990 y Fk(H)1573 2974 y Fj(2S)1434 3035 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1419 3086 y(p)p Fj(2)p Fi(supp)10 b Fk(\013)1421 3137 y(q)r Fj(2)p Fi(supp)g Fk(\014)1747 2797 y Fo(@)1803 2761 y Fl(2)1844 2797 y Fo(H)1925 2812 y Fp(B)p 1715 2842 303 4 v 1715 2933 a Fo(@)5 b(u)1827 2948 y Fp(p)1867 2933 y Fo(@)g(u)1979 2948 y Fp(q)2028 2865 y Fw(\()p Fo(u)21 b Fw(+)h Fo(t\013)h Fw(+)f Fo(s\014)6 b Fw(\))p Fo(\013)2666 2880 y Fp(p)2706 2865 y Fo(\014)2761 2880 y Fp(q)2815 2865 y Fo(:)0 3309 y Fw(T)-8 b(aking)31 b(in)m(to)h(accoun) m(t)g(the)h(ferromagnetic)d(prop)s(ert)m(y)j Fz(H9)p Fw(,)e(the)i(fact)e(that)h Fo(\013)2895 3324 y Fp(i)2923 3309 y Fo(;)17 b(\014)3022 3324 y Fp(j)3086 3309 y Fs(\025)28 b Fw(0)k(and)g(\(33\))o(,)0 3425 y(w)m(e)39 b(obtain)d(that)h(\(35\))g (is)g(non-p)s(ositiv)m(e.)58 b(Hence,)39 b(\(34\))e(establishes)h(the)g (second)h(conclusion)e(of)0 3541 y(Lemma)31 b(16.)3078 b Fg(\003)0 3760 y Fz(Corollary)45 b(17.)g Fr(In)40 b(the)g(hyp)-5 b(othesis)40 b(of)h(L)-5 b(emma)40 b(16,)h(if)g Fo(u)2264 3775 y Fl(1)2343 3760 y Fr(and)f Fo(u)2594 3775 y Fl(2)2674 3760 y Fr(ar)-5 b(e)40 b(minimizers)f(of)i Fo(S)3542 3724 y Fp(R)3536 3785 y(\033)n(;H)0 3877 y Fr(on)34 b Fs(C)191 3892 y Fp(\033)n(;H)318 3877 y Fr(,)g(so)h(ar)-5 b(e)34 b Fo(u)728 3892 y Fl(1)790 3877 y Fs(_)22 b Fo(u)934 3892 y Fl(2)1008 3877 y Fr(and)34 b Fo(u)1253 3892 y Fl(1)1314 3877 y Fs(^)23 b Fo(u)1459 3892 y Fl(2)1498 3877 y Fr(.)0 4027 y Fz(Pro)s(of.)43 b Fw(W)-8 b(e)33 b(argue)g(b)m(y)g(con)m(tradiction.)43 b(Supp)s(ose,)33 b(for)f(instance,)h(that)792 4212 y Fo(S)858 4171 y Fp(R)852 4237 y(\033)n(;H)978 4212 y Fw(\()p Fo(u)1072 4227 y Fl(1)1111 4212 y Fw(\))27 b(=)h Fo(S)1346 4171 y Fp(R)1340 4237 y(\033)n(;H)1465 4212 y Fw(\()p Fo(u)1559 4227 y Fl(2)1598 4212 y Fw(\))g(=)f(min)1775 4274 y Fm(C)1812 4285 y Fk(\033)o(;H)1947 4212 y Fo(S)2013 4171 y Fp(R)2007 4237 y(\033)n(;H)2160 4212 y Fo(<)g(S)2329 4171 y Fp(R)2323 4237 y(\033)n(;H)2449 4212 y Fw(\()p Fo(u)2543 4227 y Fl(1)2604 4212 y Fs(^)c Fo(u)2749 4227 y Fl(2)2788 4212 y Fw(\))17 b Fo(:)0 4451 y Fw(It)33 b(follo)m(ws)e(from)g(Lemma)g(16)i (that)768 4636 y(min)775 4698 y Fm(C)812 4709 y Fk(\033)o(;H)947 4636 y Fo(S)1013 4595 y Fp(R)1007 4660 y(\033)n(;H)1216 4636 y Fs(\024)83 b Fo(S)1442 4595 y Fp(R)1436 4660 y(\033)n(;H)1562 4636 y Fw(\()p Fo(u)1656 4651 y Fl(1)1717 4636 y Fs(_)23 b Fo(u)1862 4651 y Fl(2)1901 4636 y Fw(\))1216 4852 y Fs(\024)83 b Fo(S)1442 4811 y Fp(R)1436 4877 y(\033)n(;H)1562 4852 y Fw(\()p Fo(u)1656 4867 y Fl(1)1695 4852 y Fw(\))22 b(+)g Fo(S)1919 4811 y Fp(R)1913 4877 y(\033)n(;H)2039 4852 y Fw(\()p Fo(u)2133 4867 y Fl(2)2172 4852 y Fw(\))g Fs(\000)g Fo(S)2397 4811 y Fp(R)2391 4877 y(\033)n(;H)2517 4852 y Fw(\()p Fo(u)2611 4867 y Fl(1)2672 4852 y Fs(^)g Fo(u)2816 4867 y Fl(2)2855 4852 y Fw(\))1217 5014 y(=)83 b(2)17 b(min)1449 5076 y Fm(C)1486 5087 y Fk(\033)o(;H)1621 5014 y Fo(S)1687 4973 y Fp(R)1681 5038 y(\033)n(;H)1829 5014 y Fs(\000)22 b Fo(S)1994 4973 y Fp(R)1988 5038 y(\033)n(;H)2114 5014 y Fw(\()p Fo(u)2208 5029 y Fl(1)2269 5014 y Fs(^)h Fo(u)2414 5029 y Fl(2)2453 5014 y Fw(\))1217 5231 y Fo(<)83 b Fw(min)1384 5293 y Fm(C)1421 5304 y Fk(\033)o(;H)1556 5231 y Fo(S)1622 5189 y Fp(R)1616 5255 y(\033)n(;H)1758 5231 y Fo(;)0 5469 y Fw(whic)m(h)33 b(is)f(an)h(ob)m(vious)f(con)m (tradiction.)2130 b Fg(\003)p eop %%Page: 22 22 22 21 bop 168 251 a Ft(22)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(2.2.)48 b Fz(The)36 b(in\014mal)e(minimizer.)46 b Fw(Using)30 b(Corollary)g(17,)h(w)m(e)h(can)f(de\014ne)h(a)f(particularly)e(in)m (ter-)168 566 y(esting)j(minimizer)d(that)k(will)d(ha)m(v)m(e)k(sev)m (eral)f(extra)g(prop)s(erties.)168 683 y(W)-8 b(e)33 b(de\014ne)g(the)g(in\014mal)e(minimizer)e(of)j Fo(S)1743 646 y Fp(R)1737 708 y(\033)n(;H)1895 683 y Fw(on)h Fs(C)2083 698 y Fp(\033)n(;H)2241 683 y Fw(as)168 883 y(\(36\))920 b Fo(u)1318 842 y Fm(\003)p Fp(;\033)n(;H)q(;R)1595 883 y Fw(=)2103 789 y Fn(_)1709 1004 y Fk(u)20 b Fi(is)h(a)g(minimizer)e (of)i Fk(S)2276 983 y Ff(R)2273 1021 y(\033)o(;H)2398 1004 y Fi(in)f Fj(C)2504 1015 y Ff(\033)o(;H)1752 1076 y Fk(u)1789 1091 y(g)r(i)1847 1076 y Fj(\025)p Fk(\033)r Fi(\()p Fk(g)r Fi(\))10 b Fk(;)65 b Fj(8)18 b Fk(i)p Fj(2)p Fi(\003)p Fk(=H)c(;)65 b(g)r Fj(2)p Fk(G)2636 883 y Fo(u)17 b(:)168 1238 y Fw(Note)32 b(that,)h(b)m(y)g Fz(H4)p Fw(,)1557 1364 y Fo(S)1623 1323 y Fp(R)1617 1388 y(\033)n(;H)1743 1364 y Fw(\()p Fs(R)1865 1379 y Fp(`)1898 1364 y Fo(u)p Fw(\))27 b(=)h Fo(S)2189 1323 y Fp(R)2183 1388 y(\033)n(;H)2308 1364 y Fw(\()p Fo(u)p Fw(\))168 1509 y(and)38 b(so,)h(giv)m(en)f(an)m(y)h(minimizer)c Fo(u)p Fw(,)k(w)m(e)g(ha)m(v)m(e)g(that)f Fs(R)2229 1524 y Fp(`)2262 1509 y Fo(u)f Fw(is)h(a)f(minimizer)e(to)s(o,)k(for)e(an)m (y)i Fo(`)d Fs(2)i Fq(Z)p Fw(.)168 1626 y(F)-8 b(urthermore,)32 b(if)f Fo(u)h Fw(satis\014es)i(the)f(constrain)m(t)f(on)h(the)g(righ)m (t)e(hand)i(side)g(of)39 b(\(36\))o(,)33 b(then)1237 1792 y(\()p Fs(R)1359 1807 y Fp(`)1393 1792 y Fo(u)p Fw(\))1487 1807 y Fp(g)r(i)1578 1792 y Fw(=)28 b Fo(u)1738 1807 y Fp(g)r(i)1823 1792 y Fw(+)22 b Fo(`)28 b Fs(\025)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))21 b(+)h Fo(`)28 b Fs(\025)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))168 1958 y(for)41 b(an)m(y)i Fo(`)g Fs(2)h Fq(N)56 b Fw(and)42 b(an)m(y)h Fo(i)h Fs(2)g Fw(\003)p Fo(=H)8 b Fw(.)70 b(Hence,)45 b(the)e(in\014mal)c(minimizer)g (carries)i(information)168 2074 y(ab)s(out)32 b(all)e(minimizers)g(in)i (the)h(righ)m(t)f(hand)g(side)h(of)39 b(\(36\).)168 2190 y(The)31 b(constrain)m(t)f(on)f(the)i(righ)m(t)e(hand)h(side)g(of)37 b(\(36\))29 b(also)g(mak)m(es)i Fo(u)2646 2154 y Fm(\003)p Fp(;\033)n(;H)q(;R)2895 2190 y Fw(\()p Fo(p)p Fw(\))c Fo(>)h Fs(\0001)i Fw(for)f(an)m(y)i Fo(p)d Fs(2)168 2306 y Fw(\003.)168 2423 y(Moreo)m(v)m(er,)j(the)f(constrain)m(t)g(on)f(the) h(righ)m(t)f(hand)g(side)h(of)36 b(\(36\))29 b(do)s(es)h(not)f(giv)m(e) g(rise)h(to)f(the)h(empt)m(y)168 2539 y(set.)44 b(Indeed,)34 b(let)e Fo(u)g Fw(b)s(e)h(an)m(y)g(minimizer)c(of)j Fo(S)1880 2503 y Fp(R)1874 2564 y(\033)n(;H)2032 2539 y Fw(in)g Fs(C)2198 2554 y Fp(\033)n(;H)2357 2539 y Fw(and)h(tak)m(e)g Fo(`)28 b Fs(2)g Fq(N)47 b Fw(suc)m(h)34 b(that)168 2710 y(\(37\))1105 b Fo(`)28 b Fs(\025)52 b Fw(max)1631 2764 y Fk(\015)s Fj(2)p Fk(G=H)1641 2822 y(i)p Fj(2)p Fi(\003)p Fk(=H)1868 2710 y Fw(\()p Fs(j)p Fo(u)1990 2725 y Fp(\015)t(i)2057 2710 y Fs(j)22 b Fw(+)g Fs(j)p Fo(\033)t Fw(\()p Fo(\015)5 b Fw(\))p Fs(j)p Fw(\))32 b Fo(:)168 2978 y Fw(Note)k(that)h Fo(G=H)44 b Fw(and)36 b(\003)p Fo(=H)44 b Fw(are)36 b(\014nite)h(set)g (b)m(y)h(construction,)f(thence)h(the)f(de\014nition)f(in)g(\(37\))168 3094 y(mak)m(es)e(sense.)50 b(Then,)35 b Fs(R)1122 3109 y Fp(`)1156 3094 y Fo(u)e Fw(is)h(a)f(minimizer)e(to)s(o)i(and,)i(if)e Fo(i)d Fs(2)h Fw(\003)p Fo(=H)8 b Fw(,)33 b Fo(g)h Fs(2)c Fo(G)p Fw(,)35 b(w)m(e)g(write)e Fo(g)h Fw(=)c Fo(h\015)5 b Fw(,)168 3210 y(with)32 b Fo(h)c Fs(2)g Fo(H)40 b Fw(and)33 b Fo(\015)f Fs(2)c Fo(G=H)40 b Fw(to)32 b(get)h(that)552 3376 y(\()p Fs(R)674 3391 y Fp(`)707 3376 y Fo(u)p Fw(\))801 3391 y Fp(g)r(i)892 3376 y Fw(=)28 b Fo(u)1052 3391 y Fp(h\015)t(i)1183 3376 y Fw(+)22 b Fo(`)27 b Fw(=)h Fo(u)1509 3391 y Fp(\015)t(i)1599 3376 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(h)p Fw(\))g(+)g Fo(`)28 b Fw(=)f Fo(u)2236 3391 y Fp(\015)t(i)2326 3376 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))g Fs(\000)g Fo(\033)t Fw(\()p Fo(\015)5 b Fw(\))22 b(+)g Fo(`)28 b Fs(\025)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))17 b Fo(;)168 3542 y Fw(thence)34 b(the)f(set)g(in)f(the)h(righ)m(t)e (hand)i(side)g(of)39 b(\(36\))32 b(is)g(not)g(empt)m(y)-8 b(.)168 3658 y(W)g(e)33 b(no)m(w)g(p)s(oin)m(t)f(out)g(that)g(the)h (in\014mal)e(minimizer)e(is)j(indeed)h(a)f(minimizer:)168 3799 y Fz(Lemma)37 b(18.)42 b Fo(u)806 3763 y Fm(\003)p Fp(;\033)n(;H)q(;R)1090 3799 y Fr(is)34 b(a)h(minimizer)f(of)g Fo(S)1924 3763 y Fp(R)1918 3825 y(\033)n(;H)2079 3799 y Fr(in)g Fs(C)2250 3814 y Fp(\033)n(;H)2376 3799 y Fr(.)168 3957 y Fz(Pro)s(of.)40 b Fw(F)-8 b(or)21 b(an)m(y)i Fo(q)32 b Fs(2)c Fw(\003)p Fo(=H)h Fw(let)22 b Fo(u)1432 3921 y Fl(\()p Fp(q)r(;n)p Fl(\))1610 3957 y Fw(b)s(e)g(a)g(sequence)k(of)c (minimizers)d(satisfying)j(the)h(constrain)m(ts)168 4073 y(in)32 b(the)h(righ)m(t)e(hand)i(side)g(of)39 b(\(36\))32 b(in)g(suc)m(h)i(a)e(w)m(a)m(y)i(that)1438 4247 y(lim)1386 4307 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1641 4247 y Fo(u)1697 4206 y Fl(\()p Fp(q)r(;n)p Fl(\))1852 4247 y Fw(\()p Fo(q)t Fw(\))44 b(=)g Fo(u)2195 4206 y Fm(\003)p Fp(;\033)n(;H)q(;R) 2445 4247 y Fw(\()p Fo(q)t Fw(\))17 b Fo(:)168 4451 y Fw(De\014ne)33 b(also)168 4617 y(\(38\))1124 b Fo(u)1522 4576 y Fl(\()p Fp(n)p Fl(\))1623 4617 y Fw(\()p Fo(p)p Fw(\))28 b(:=)1965 4522 y Fn(_)1907 4738 y Fp(q)r Fm(2)p Fl(\003)p Fp(=H)2151 4617 y Fo(u)2207 4576 y Fl(\()p Fp(q)r(;n)p Fl(\))2362 4617 y Fw(\()p Fo(p)p Fw(\))17 b Fo(;)168 4897 y Fw(for)31 b(an)m(y)i Fo(p)28 b Fs(2)g Fw(\003)p Fo(=H)8 b Fw(.)42 b(Note)33 b(that,)f(since)h(\003)p Fo(=H)39 b Fw(is)31 b(a)h(\014nite)g(set,)h(the)g(minimization)27 b(p)s(erformed)32 b(on)168 5017 y(the)f(righ)m(t)g(hand)g(side)g(of)38 b(\(38\))31 b(is)f(done)i(among)e(a)h(\014nite)f(n)m(um)m(b)s(er)i(of)f (candidates,)h(thence)g Fo(u)3631 4980 y Fl(\()p Fp(n)p Fl(\))3764 5017 y Fw(is)168 5133 y(also)f(a)i(minimizer)c(as)k(in)e (the)i(righ)m(t)f(hand)h(side)g(of)39 b(\(36\))o(,)33 b(b)s(ecause)h(of)e(Corollary)f(17.)168 5252 y(Then,)j(b)m(y)g(\(36\),) e Fo(u)875 5216 y Fm(\003)p Fp(;\033)n(;H)q(;R)1152 5252 y Fs(\024)d Fo(u)1314 5216 y Fl(\()p Fp(n)p Fl(\))1415 5252 y Fw(.)43 b(On)33 b(the)g(other)g(hand,)g(b)m(y)g(construction,) 1116 5426 y(lim)1064 5486 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1320 5426 y Fo(u)1376 5385 y Fl(\()p Fp(n)p Fl(\))1477 5426 y Fw(\()p Fo(p)p Fw(\))28 b Fs(\024)79 b Fw(lim)1735 5486 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1990 5426 y Fo(u)2046 5385 y Fl(\()p Fp(p;n)p Fl(\))2203 5426 y Fw(\()p Fo(p)p Fw(\))27 b(=)h Fo(u)2515 5385 y Fm(\003)p Fp(;\033)n(;H)q(;R)2764 5426 y Fw(\()p Fo(p)p Fw(\))17 b Fo(;)p eop %%Page: 23 23 23 22 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(23)0 450 y Fw(for)32 b(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003)p Fo(=H)8 b Fw(.)0 566 y(Consequen)m(tly)-8 b(,)1333 728 y(lim)1281 788 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1537 728 y Fo(u)1593 687 y Fl(\()p Fp(n)p Fl(\))1694 728 y Fw(\()p Fo(p)p Fw(\))28 b(=)f Fo(u)2006 687 y Fm(\003)p Fp(;\033)n(;H)q(;R)2255 728 y Fw(\()p Fo(p)p Fw(\))0 952 y(for)32 b(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003)p Fo(=H)8 b Fw(,)32 b(and)g(so,)h(b)m(y)h (Corollary)d(13,)851 1160 y Fo(S)917 1119 y Fp(R)911 1185 y(\033)n(;H)1037 1160 y Fw(\()p Fo(u)1131 1119 y Fm(\003)p Fp(;\033)n(;H)q(;R)1380 1160 y Fw(\))d(=)79 b(lim)1549 1220 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1805 1160 y Fo(S)1871 1119 y Fp(R)1865 1185 y(\033)n(;H)1990 1160 y Fw(\()p Fo(u)2084 1119 y Fl(\()p Fp(n)p Fl(\))2186 1160 y Fw(\))27 b(=)h Fo(S)2421 1119 y Fp(R)2415 1185 y(\033)n(;H)2541 1160 y Fw(\()p Fo(u)2635 1119 y Fl(\(1\))2728 1160 y Fw(\))17 b Fo(;)0 1415 y Fw(since)29 b(all)e(the)i Fo(u)587 1379 y Fl(\()p Fp(n)p Fl(\))717 1415 y Fw(are)f(minimizers.)40 b(Accordingly)-8 b(,)27 b Fo(u)2018 1379 y Fm(\003)p Fp(;\033)n(;H)q(;R)2296 1415 y Fw(is)h(a)h(minimizer)c(of)j Fo(S)3092 1379 y Fp(R)3086 1440 y(\033)n(;H)3240 1415 y Fw(in)g Fs(C)3402 1430 y Fp(\033)n(;H)3528 1415 y Fw(.)h Fg(\003)0 1876 y Fw(2.3.)49 b Fz(Some)i(group)h(algebra)g (considerations)f(on)i Fo(S)2239 1840 y Fp(R)2233 1902 y(\033)n(;H)2358 1876 y Fz(.)c Fw(Next)d(results)g(will)c(in)m(v)m (estigate)0 1993 y(ho)m(w)d(\\doubling)d(the)j(p)s(erio)s(dicit)m(y)d (induced)j(b)m(y)g(the)f(subgroups")h(re\015ects)h(in)m(to)e (\\doubling)e(the)0 2109 y(energy")d(\(a)g(precise)g(statemen)m(t)g (will)d(b)s(e)j(giv)m(en)g(in)f(Corollary)e(20\).)0 2275 y Fz(Lemma)35 b(19.)40 b Fr(L)-5 b(et)769 2250 y Fw(~)744 2275 y Fo(H)35 b Fs(\032)28 b Fo(H)41 b Fr(b)-5 b(e)32 b(normal,)g(\014nite)h(index)f(sub)-5 b(gr)g(oups)33 b(of)f Fo(G)p Fr(.)44 b(L)-5 b(et)33 b Fo(\015)2992 2290 y Fl(1)3031 2275 y Fo(;)17 b(:)g(:)g(:)f(;)h(\015)3301 2290 y Fp(`)3361 2275 y Fs(2)28 b Fo(H)41 b Fr(in)0 2392 y(such)35 b(a)f(way)h(that)1233 2577 y Fo(H)r(=)1390 2552 y Fw(~)1365 2577 y Fo(H)51 b Fw(=)1617 2467 y Fn(n)1684 2577 y Fw([)p Fo(\015)1762 2592 y Fl(1)1801 2577 y Fw(])1846 2587 y Fl(~)1828 2604 y Fp(H)1895 2577 y Fo(;)17 b(:)g(:)g(:)f(;)h Fw([)p Fo(\015)2192 2592 y Fp(`)2224 2577 y Fw(])2269 2587 y Fl(~)2251 2604 y Fp(H)2319 2467 y Fn(o)2402 2577 y Fo(:)0 2789 y Fr(Then,)29 b(given)f(any)h Fo(B)k Fs(2)28 b(S)37 b Fr(and)28 b(any)h Fo(A)e Fs(2)i Fw([)p Fo(B)5 b Fw(])1698 2804 y Fp(H)1794 2789 y Fr(ther)-5 b(e)29 b(exists)f(a)h(unique)g Fo(j)34 b Fs(2)28 b(f)p Fw(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(`)p Fs(g)28 b Fr(in)h(such)g(a)0 2914 y(way)j(that)g(ther)-5 b(e)32 b(exists)895 2888 y Fw(~)894 2914 y Fo(h)c Fs(2)1097 2889 y Fw(~)1072 2914 y Fo(H)39 b Fr(such)32 b(that)1608 2888 y Fw(~)1607 2914 y Fo(h)q(A)27 b Fw(=)h Fo(\015)1919 2929 y Fp(j)1955 2914 y Fo(B)5 b Fr(.)44 b(A)n(lso,)32 b(if)g Fw([)p Fo(\015)2525 2929 y Fp(i)2552 2914 y Fo(B)5 b Fw(])2677 2924 y Fl(~)2658 2941 y Fp(H)2754 2914 y Fw(=)27 b([)p Fo(\015)2935 2929 y Fp(j)2971 2914 y Fo(B)5 b Fw(])3095 2924 y Fl(~)3077 2941 y Fp(H)3145 2914 y Fr(,)32 b(then)g Fo(i)c Fw(=)g Fo(j)6 b Fr(.)0 3085 y Fz(Pro)s(of.)41 b Fw(Let)27 b Fo(h)h Fs(2)g Fo(H)34 b Fw(suc)m(h)28 b(that)e Fo(A)i Fw(=)g Fo(hB)j Fw(and)c(let)f Fo(\015)1961 3100 y Fp(j)2023 3085 y Fw(b)s(e)h(so)g(that)f([)p Fo(h)p Fw(])2597 3095 y Fl(~)2579 3112 y Fp(H)2674 3085 y Fw(=)i([)p Fo(\015)2856 3100 y Fp(j)2892 3085 y Fw(])2937 3095 y Fl(~)2919 3112 y Fp(H)2987 3085 y Fw(.)41 b(Then)28 b Fo(\015)3355 3100 y Fp(j)3418 3085 y Fw(=)3523 3059 y(~)3522 3085 y Fo(hh)p Fw(,)0 3213 y(for)k(some)394 3187 y(~)393 3213 y Fo(h)c Fs(2)597 3188 y Fw(~)571 3213 y Fo(H)8 b Fw(.)43 b(Then,)1013 3187 y(~)1012 3213 y Fo(hA)28 b Fw(=)g Fo(\015)1324 3228 y Fp(j)1360 3213 y Fo(B)5 b Fw(,)33 b(as)f(desired.)0 3342 y(Supp)s(ose)23 b(no)m(w)f(that)766 3316 y(~)765 3342 y Fo(h)821 3357 y Fp(j)858 3342 y Fo(A)27 b Fw(=)h Fo(\015)1113 3357 y Fp(j)1149 3342 y Fo(B)f Fw(with)1463 3316 y(~)1462 3342 y Fo(h)1518 3357 y Fp(j)1582 3342 y Fs(2)1701 3317 y Fw(~)1676 3342 y Fo(H)8 b Fw(,)24 b(for,)f(sa)m(y)-8 b(,)23 b Fo(j)34 b Fw(=)27 b Fo(j)2377 3357 y Fl(1)2417 3342 y Fo(;)17 b(j)2501 3357 y Fl(2)2540 3342 y Fw(.)40 b(Let)22 b Fo(h)2827 3357 y Fm(\003)2895 3342 y Fw(:=)27 b Fo(\015)3081 3301 y Fm(\000)p Fl(1)3076 3368 y Fp(j)3105 3377 y Fi(1)3177 3316 y Fw(~)3176 3342 y Fo(h)3232 3357 y Fp(j)3261 3366 y Fi(1)3300 3316 y Fw(~)3299 3342 y Fo(h)3355 3301 y Fm(\000)p Fl(1)3355 3368 y Fp(j)3384 3377 y Fi(2)3449 3342 y Fo(\015)3500 3357 y Fp(j)3529 3366 y Fi(2)3595 3342 y Fs(2)0 3464 y Fo(H)8 b Fw(.)40 b(Then,)23 b Fo(h)483 3479 y Fm(\003)523 3464 y Fo(B)33 b Fw(=)27 b Fo(B)g Fw(and)c(so,)i(b)m(y)e Fz(H5)p Fw(,)h(w)m(e)f(see)h(that)e Fo(h)2011 3479 y Fm(\003)2073 3464 y Fw(is)f(the)i(iden)m(tit)m(y)-8 b(.)40 b(Consequen)m(tly)-8 b(,)25 b([)p Fo(\015)3397 3479 y Fp(j)3426 3488 y Fi(1)3464 3464 y Fw(])3508 3474 y Fl(~)3491 3491 y Fp(H)3586 3464 y Fw(=)0 3580 y([)p Fo(\015)78 3595 y Fp(j)107 3604 y Fi(2)145 3580 y Fw(])190 3590 y Fl(~)172 3607 y Fp(H)272 3580 y Fw(and)33 b(so)f Fo(j)621 3595 y Fl(1)689 3580 y Fw(=)27 b Fo(j)832 3595 y Fl(2)872 3580 y Fw(.)0 3708 y(Assume)38 b(no)m(w)f(that)f([)p Fo(\015)866 3723 y Fp(i)894 3708 y Fo(B)5 b Fw(])1018 3718 y Fl(~)1000 3735 y Fp(H)1102 3708 y Fw(=)35 b([)p Fo(\015)1291 3723 y Fp(j)1327 3708 y Fo(B)5 b Fw(])1451 3718 y Fl(~)1433 3735 y Fp(H)1501 3708 y Fw(.)55 b(Then,)39 b Fo(B)h Fw(=)35 b Fo(\015)2151 3667 y Fm(\000)p Fl(1)2146 3734 y Fp(i)2246 3682 y Fw(~)2245 3708 y Fo(h\015)2352 3723 y Fp(j)2388 3708 y Fo(B)5 b Fw(,)38 b(for)e(some)2935 3682 y(~)2934 3708 y Fo(h)f Fs(2)3151 3683 y Fw(~)3126 3708 y Fo(H)44 b Fw(and)37 b(then,)0 3825 y(b)m(y)c Fz(H5)p Fw(,)g(w)m(e)g(ha)m(v)m(e)h(that)f([)p Fo(\015)997 3840 y Fp(i)1025 3825 y Fw(])1069 3835 y Fl(~)1052 3852 y Fp(H)1147 3825 y Fw(=)27 b([)p Fo(\015)1328 3840 y Fp(j)1364 3825 y Fw(])1409 3835 y Fl(~)1391 3852 y Fp(H)1459 3825 y Fw(.)43 b(So)33 b Fo(i)28 b Fw(=)f Fo(j)6 b Fw(.)1649 b Fg(\003)0 4078 y Fz(Corollary)43 b(20.)h Fr(L)-5 b(et)890 4053 y Fw(~)865 4078 y Fo(H)43 b Fs(\032)37 b Fo(H)47 b Fr(b)-5 b(e)39 b(normal,)g(\014nite)g(index)g(sub)-5 b(gr)g(oups)39 b(of)g Fo(G)g Fr(and)g(let)h Fo(u)35 b Fs(2)h(C)3505 4093 y Fp(\033)n(;H)3632 4078 y Fr(.)0 4194 y(Then,)1213 4380 y Fo(S)1279 4339 y Fp(R)1273 4416 y(\033)n(;)1349 4399 y Fl(~)1331 4416 y Fp(H)1398 4380 y Fw(\()p Fo(u)p Fw(\))44 b(=)g Fo(])1732 4269 y Fn(\020)1792 4380 y Fo(H)r(=)1949 4355 y Fw(~)1924 4380 y Fo(H)2012 4269 y Fn(\021)2088 4380 y Fo(S)2154 4339 y Fp(R)2148 4405 y(\033)n(;H)2274 4380 y Fw(\()p Fo(u)p Fw(\))17 b Fo(:)0 4591 y Fz(Pro)s(of.)65 b Fw(W)-8 b(e)41 b(exploit)d(Lemma)h (19)g(and)h(the)h(in)m(v)-5 b(ariance)39 b(of)g Fo(\016)t Fw(\()p Fs(\001)p Fw(\))g(under)i(the)f(action)f(of)h Fo(G)g Fw(\(re-)0 4708 y(call)31 b Fz(H9)p Fw(\))h(to)g(write)138 4829 y Fn(n)205 4940 y Fw([)p Fo(B)5 b Fw(])356 4950 y Fl(~)338 4967 y Fp(H)433 4940 y Fs(2)28 b(S)605 4950 y Fl(~)587 4967 y Fp(H)671 4940 y Fo(;)61 b(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)1169 4829 y Fn(o)1280 4940 y Fw(=)1400 4829 y Fn(n)1466 4940 y Fw([)p Fo(\015)1544 4955 y Fp(j)1581 4940 y Fo(B)5 b Fw(])1705 4950 y Fl(~)1687 4967 y Fp(H)1771 4940 y Fo(;)61 b Fw([)p Fo(B)5 b Fw(])1992 4955 y Fp(H)2087 4940 y Fs(2)28 b(S)2241 4955 y Fp(H)2325 4940 y Fo(;)61 b(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)18 b(;)61 b(j)33 b Fw(=)28 b(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(`)3413 4829 y Fn(o)3496 4940 y Fo(:)0 5188 y Fw(Note)33 b(that,)f(b)m(y)i (Lemma)d(19,)h(the)h([)p Fo(\015)1361 5203 y Fp(j)1397 5188 y Fo(B)5 b Fw(])1521 5198 y Fl(~)1503 5215 y Fp(H)1571 5188 y Fw('s)33 b(do)f(not)h(rep)s(eat)f(themselv)m(es)i(and)f Fo(`)28 b Fw(=)f Fo(])p Fw(\()p Fo(H)r(=)3362 5163 y Fw(~)3337 5188 y Fo(H)7 b Fw(\).)0 5304 y(Also,)32 b(b)m(y)j(\(26\))o (,)905 5464 y Fo(H)986 5479 y Fl([)p Fp(\015)1042 5488 y Fi(1)1076 5479 y Fp(B)s Fl(])1167 5490 y Fi(~)1152 5503 y Fk(H)1214 5464 y Fw(\()p Fo(u)p Fw(\))21 b(+)h Fs(\001)17 b(\001)g(\001)k Fw(+)h Fo(H)1783 5479 y Fl([)p Fp(\015)1839 5491 y Fk(`)1869 5479 y Fp(B)s Fl(])1960 5490 y Fi(~)1945 5503 y Fk(H)2007 5464 y Fw(\()p Fo(u)p Fw(\))43 b(=)h Fo(`H)2424 5479 y Fl([)p Fp(B)s Fl(])2520 5490 y Fk(H)2582 5464 y Fw(\()p Fo(u)p Fw(\))17 b Fo(:)p eop %%Page: 24 24 24 23 bop 168 251 a Ft(24)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Therefore,)1208 647 y Fo(S)1274 606 y Fp(R)1268 683 y(\033)n(;)1344 667 y Fl(~)1326 683 y Fp(H)1394 647 y Fw(\()p Fo(u)p Fw(\))82 b(=)1844 553 y Fn(X)1777 756 y Fi([)p Fk(B)r Fi(])1875 772 y(~)1861 785 y Fk(H)1918 756 y Fj(2S)2012 772 y Fi(~)1998 785 y Fk(H)1806 831 y(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2082 647 y Fo(H)2163 663 y Fl([)p Fp(B)s Fl(])2273 674 y Fi(~)2259 687 y Fk(H)2320 647 y Fw(\()p Fo(u)p Fw(\))1608 1072 y(=)1844 977 y Fn(X)1777 1181 y Fi([)p Fk(B)r Fi(])1861 1197 y Fk(H)1918 1181 y Fj(2S)1998 1197 y Fk(H)1806 1244 y(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2140 947 y Fp(`)2082 977 y Fn(X)2093 1187 y Fp(j)t Fl(=1)2242 1072 y Fo(H)2323 1087 y Fl([)p Fp(\015)2379 1097 y Fk(j)2412 1087 y Fp(B)s Fl(])2503 1098 y Fi(~)2488 1111 y Fk(H)2550 1072 y Fw(\()p Fo(u)p Fw(\))1608 1484 y(=)1844 1389 y Fn(X)1777 1593 y Fi([)p Fk(B)r Fi(])1861 1610 y Fk(H)1918 1593 y Fj(2S)1998 1610 y Fk(H)1806 1656 y(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2140 1360 y Fp(`)2082 1389 y Fn(X)2093 1599 y Fp(j)t Fl(=1)2242 1374 y Fn(\020)2302 1484 y Fo(`)17 b(H)2441 1500 y Fl([)p Fp(B)s Fl(])2537 1511 y Fk(H)2598 1484 y Fw(\()p Fo(u)p Fw(\))2730 1374 y Fn(\021)1608 1802 y Fw(=)83 b Fo(`)17 b(S)1891 1761 y Fp(R)1885 1827 y(\033)n(;H)2010 1802 y Fw(\()p Fo(u)p Fw(\))g Fo(;)168 1983 y Fw(as)32 b(desired.)3107 b Fg(\003)168 2197 y Fz(Remark)37 b(21.)k Fw(The)34 b(result)e(in)g (Corollary)f(20)h(ma)m(y)g(not)h(hold)f(if)f(assumption)h Fz(H5)g Fw(is)g(remo)m(v)m(ed.)168 2316 y(F)-8 b(or)31 b(instance,)i(let)f Fo(G)c Fw(=)g Fo(H)40 b Fw(and)1435 2291 y(~)1409 2316 y Fo(H)g Fw(b)s(e)33 b(as)g(in)f(Remark)g(15.)168 2432 y(Let)39 b(\003)e(:=)h Fs(f)p Fw(1)p Fo(;)17 b Fw(2)p Fo(;)g Fw(3)p Fo(;)g Fw(4)p Fs(g)p Fw(.)59 b(Then,)40 b Fo(G)f Fw(acts)g(on)f(\003)h(in)e(the)i(ob)m(vious)g(w)m(a)m(y)h(b)m (y)g(\014xing)e(the)h(elemen)m(t)f(4)168 2548 y(\(for)31 b(instance,)i(in)e(the)h(notation)f(of)h(Remark)f(15,)h(w)m(e)h(ha)m(v) m(e)g(that)f Fo(a)p Fw(\(1\))c(=)f(2,)32 b Fo(a)p Fw(\(2\))c(=)f(1,)32 b Fo(a)p Fw(\(3\))c(=)f(3)168 2665 y(and)32 b Fo(a)p Fw(\(4\))c(=)f(4\).)168 2781 y(Let)32 b Fs(S)k Fw(:=)28 b Fs(ff)p Fw(4)p Fs(gg)j Fw(and)i(let)f(the)h(in)m(teraction)e(p)s (oten)m(tials)h(b)s(e)g Fo(H)2502 2796 y Fm(f)p Fl(4)p Fm(g)2612 2781 y Fw(\()p Fo(u)p Fw(\))27 b(:=)h(1)22 b Fs(\000)g Fw(cos)c Fo(u)3276 2796 y Fl(4)3315 2781 y Fw(.)168 2897 y(Then,)34 b Fs(S)510 2912 y Fp(H)605 2897 y Fw(=)28 b Fs(S)787 2907 y Fl(~)769 2924 y Fp(H)864 2897 y Fw(=)g Fs(ff)p Fw(4)p Fs(gg)j Fw(and)i(so,)g(b)m(y)h(taking,)e (for)g(instance)h Fo(\016)t Fw(\()p Fs(\001)p Fw(\))f(to)g(b)s(e)h(the) g(cardinalit)m(y)e(of)h(a)168 3013 y(set)h(and)g Fo(R)28 b Fs(\025)h Fw(1,)j(w)m(e)i(ha)m(v)m(e)g(that)1322 3194 y Fo(S)1388 3153 y Fp(R)1382 3219 y(\033)n(;H)1508 3194 y Fw(\()p Fo(u)p Fw(\))27 b(=)g Fo(S)1836 3153 y Fp(R)1830 3230 y(\033)n(;)1906 3214 y Fl(~)1888 3230 y Fp(H)1956 3194 y Fw(\()p Fo(u)p Fw(\))g(=)g(1)22 b Fs(\000)h Fw(cos)17 b Fo(u)2592 3209 y Fl(4)2648 3194 y Fo(:)168 3391 y Fw(This)32 b(and)h(\(31\))f(sho)m(w)i(that)e(Corollary)f(20)h(do)s(es)h(not)g (hold)e(in)h(this)g(case.)168 3614 y(2.4.)48 b Fz(No-symmetry)h (breaking.)g Fw(The)44 b(in\014mal)d(minimizer,)h(for)h(a)g(co)s(cycle) h Fo(\033)j Fw(and)c(a)g(\014xed)168 3730 y(subgroup)k Fo(H)8 b Fw(,)50 b(is)d(ob)m(viously)f(unique,)51 b(due)d(to)e(\(36\).) 86 b(This)47 b(sa)m(ys)i(that)d(it)g(will)f(ha)m(v)m(e)j(all)d(the)168 3846 y(symmetries)36 b(of)g(the)h(b)s(oundary)g(conditions,)g(in)f (particular)f(there)i(is)f(no-symmetry)g(breaking,)168 3962 y(according)c(to)g(the)h(follo)m(wing)d(result:)168 4114 y Fz(Prop)s(osition)e(22.)37 b Fr(Assume)28 b(that)1522 4089 y Fw(~)1497 4114 y Fo(H)35 b Fs(\032)28 b Fo(H)36 b Fr(is)29 b(also)f(a)h(normal)f(sub)-5 b(gr)g(oup)29 b(of)g(\014nite)f(index.)42 b(Then,)168 4239 y Fo(u)224 4203 y Fm(\003)p Fp(;\033)n(;H)q(;R)501 4239 y Fw(=)27 b Fo(u)660 4203 y Fm(\003)p Fp(;\033)n(;)791 4186 y Fl(~)773 4203 y Fp(H)5 b(;R)913 4239 y Fr(.)168 4388 y Fz(Pro)s(of.)43 b Fw(W)-8 b(e)33 b(note)g(that,)f(giv)m(en)h Fo(k)e Fs(2)d Fo(H)8 b Fw(,)1568 4583 y(~)-55 b Fo(u)27 b Fw(=)h Fs(R)1833 4598 y Fm(\000)p Fp(\033)r Fl(\()p Fp(k)r Fl(\))2029 4583 y Fs(T)2083 4598 y Fp(k)2125 4583 y Fo(u)2181 4541 y Fm(\003)p Fp(;\033)n(;)2312 4525 y Fl(~)2294 4541 y Fp(H)5 b(;R)168 4769 y Fw(is)32 b(also)f(a)i(minimizer)c(for)j Fo(S)1213 4733 y Fp(R)1207 4794 y(\033)n(;H)1365 4769 y Fw(in)g Fs(C)1531 4796 y Fp(\033)n(;)1608 4780 y Fl(~)1589 4796 y Fp(H)1657 4769 y Fw(,)h(due)g(to)f Fz(H3)g Fw(and)h Fz(H4)p Fw(.)43 b(Also,)32 b(it)g(satis\014es)942 4991 y(~)-55 b Fo(u)992 5006 y Fp(g)r(i)1083 4991 y Fw(=)28 b Fo(u)1243 4944 y Fm(\003)p Fp(;\033)n(;)1373 4927 y Fl(~)1356 4944 y Fp(H)5 b(;R)1243 5019 y(k)r(g)r(i)1518 4991 y Fs(\000)23 b Fo(\033)t Fw(\()p Fo(k)s Fw(\))k Fs(\025)i Fo(\033)t Fw(\()p Fo(k)s Fw(\))22 b(+)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))f Fs(\000)i Fo(\033)t Fw(\()p Fo(k)s Fw(\))k(=)h Fo(\033)t Fw(\()p Fo(g)t Fw(\))168 5174 y(for)k(an)m(y)h Fo(g)e Fs(2)d Fo(G)p Fw(.)43 b(Hence,)34 b(b)m(y)g(the)f(de\014nition)f (of)g(in\014mal)e(minimizer,)1322 5371 y Fo(u)1378 5330 y Fm(\003)p Fp(;\033)n(;)1509 5313 y Fl(~)1491 5330 y Fp(H)5 b(;R)1659 5371 y Fs(\024)1783 5276 y Fn(_)1764 5488 y Fp(k)r Fm(2)p Fp(H)1929 5371 y Fs(R)2013 5386 y Fm(\000)p Fp(\033)r Fl(\()p Fp(k)r Fl(\))2209 5371 y Fs(T)2263 5386 y Fp(k)2306 5371 y Fo(u)2362 5330 y Fm(\003)p Fp(;\033)n(;)2493 5313 y Fl(~)2475 5330 y Fp(H)g(;R)2648 5371 y Fo(:)p eop %%Page: 25 25 25 24 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(25)0 450 y Fw(Therefore,)34 b(since)f(the)g(con)m(v)m(erse)i (inequalit)m(y)c(is)h(ob)m(vious,)1155 636 y Fo(u)1211 595 y Fm(\003)p Fp(;\033)n(;)1342 578 y Fl(~)1324 595 y Fp(H)5 b(;R)1492 636 y Fw(=)1615 541 y Fn(_)1596 753 y Fp(k)r Fm(2)p Fp(H)1761 636 y Fs(R)1845 651 y Fm(\000)p Fp(\033)r Fl(\()p Fp(k)r Fl(\))2040 636 y Fs(T)2094 651 y Fp(k)2137 636 y Fo(u)2193 595 y Fm(\003)p Fp(;\033)n(;)2324 578 y Fl(~)2306 595 y Fp(H)g(;R)2479 636 y Fo(:)0 899 y Fw(In)33 b(particular,)1245 1042 y Fo(u)1301 1001 y Fm(\003)p Fp(;\033)n(;)1432 984 y Fl(~)1414 1001 y Fp(H)5 b(;R)1582 1042 y Fs(\024)28 b(R)1771 1057 y Fm(\000)p Fp(\033)r Fl(\()p Fp(k)r Fl(\))1967 1042 y Fs(T)2021 1057 y Fp(k)2064 1042 y Fo(u)2120 1001 y Fm(\003)p Fp(;\033)n(;)2250 984 y Fl(~)2233 1001 y Fp(H)t(;R)2389 1042 y Fo(;)0 1191 y Fw(for)k(an)m(y)h Fo(k)e Fs(2)d Fo(H)8 b Fw(.)0 1307 y(By)33 b(applying)e(the)i(ab)s(o)m(v)m(e)h(inequalit)m(y)d(also)h(to)g Fo(k)1817 1271 y Fm(\000)p Fl(1)1939 1307 y Fs(2)c Fo(H)8 b Fw(,)32 b(one)h(gets)g(that)1268 1491 y Fs(R)1352 1507 y Fm(\000)p Fp(\033)r Fl(\()p Fp(k)r Fl(\))1547 1491 y Fs(T)1601 1506 y Fp(k)1644 1491 y Fo(u)1700 1450 y Fm(\003)p Fp(;\033)n(;)1831 1433 y Fl(~)1813 1450 y Fp(H)5 b(;R)1981 1491 y Fw(=)28 b Fo(u)2141 1450 y Fm(\003)p Fp(;\033)n(;)2271 1433 y Fl(~)2254 1450 y Fp(H)t(;R)0 1661 y Fw(for)k(all)f Fo(k)f Fs(2)e Fo(H)8 b Fw(,)32 b(and)h(therefore)1504 1804 y Fo(u)1560 1763 y Fm(\003)p Fp(;\033)n(;)1691 1746 y Fl(~)1673 1763 y Fp(H)5 b(;R)1841 1804 y Fs(2)28 b(C)1987 1819 y Fp(\033)n(;H)2130 1804 y Fo(:)0 1965 y Fw(Since)h(to)g(an)m(y)g(set)h(in)e([)p Fo(B)5 b Fw(])939 1980 y Fp(H)1035 1965 y Fw(corresp)s(ond)30 b Fo(])p Fw(\()p Fo(H)r(=)1761 1939 y Fw(~)1736 1965 y Fo(H)7 b Fw(\))29 b(sets)h(in)f([)p Fo(B)5 b Fw(])2340 1975 y Fl(~)2322 1992 y Fp(H)2389 1965 y Fw(,)30 b(according)e(to)h (Corollary)e(20)h(w)m(e)0 2081 y(ha)m(v)m(e)34 b(that)1366 2210 y Fo(S)1432 2169 y Fp(R)1426 2246 y(\033)n(;)1502 2229 y Fl(~)1484 2246 y Fp(H)1596 2210 y Fw(=)44 b Fo(])p Fw(\()p Fo(H)r(=)1949 2185 y Fw(~)1924 2210 y Fo(H)7 b Fw(\))17 b Fo(S)2133 2169 y Fp(R)2127 2235 y(\033)n(;H)2269 2210 y Fo(:)0 2370 y Fw(Thence,)35 b(w)m(e)e(see)h(that)170 2522 y(\(i\))41 b Fo(u)371 2485 y Fm(\003)p Fp(;\033)n(;)501 2469 y Fl(~)484 2485 y Fp(H)t(;R)657 2522 y Fw(is)32 b(a)h(minimizer)d(for)j Fo(S)1509 2485 y Fp(R)1503 2547 y(\033)n(;H)1661 2522 y Fw(on)h Fs(C)1850 2537 y Fp(\033)n(;H)1976 2522 y Fw(,)f(b)s(ecause,)i(if)d Fo(S)2581 2485 y Fp(R)2575 2563 y(\033)n(;)2651 2546 y Fl(~)2633 2563 y Fp(H)2700 2522 y Fw(\()p Fo(u)2794 2485 y Fm(\003)p Fp(;\033)n(;)2925 2469 y Fl(~)2907 2485 y Fp(H)5 b(;R)3070 2522 y Fw(+)22 b Fo(\014)6 b Fw(\))33 b(do)s(es)h(not)315 2653 y(decrease)g(c)m(ho)s (osing)e Fo(\014)h Fs(2)28 b(C)1330 2680 y Fp(\033)n(;)1407 2663 y Fl(~)1388 2680 y Fp(H)1456 2653 y Fw(,)33 b(a)f(fortiori)e(it)i (do)s(es)h(not)f(decrease)j(c)m(ho)s(osing)d Fo(\014)h Fs(2)28 b(C)3432 2668 y Fp(\033)n(;H)3558 2653 y Fw(.)143 2783 y(\(ii\))40 b Fo(u)371 2747 y Fm(\003)p Fp(;\033)n(;H)q(;R)642 2783 y Fw(is)22 b(a)h(minimizer)c(of)j Fo(S)1414 2747 y Fp(R)1408 2824 y(\033)n(;)1484 2807 y Fl(~)1466 2824 y Fp(H)1556 2783 y Fw(on)g Fs(C)1733 2810 y Fp(\033)n(;)1810 2793 y Fl(~)1791 2810 y Fp(H)1860 2783 y 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3916 y Fm(\003)p Fp(;\033)n(;H)q(;R) 1769 3958 y Fw(=)g Fo(u)1929 3916 y Fm(\003)p Fp(;\033)n(;)2059 3900 y Fl(~)2042 3916 y Fp(H)t(;R)2198 3958 y Fo(;)0 4107 y Fw(as)33 b(claimed..)3083 b Fg(\003)0 4223 y Fw(Thanks)35 b(to)f(Prop)s(osition)e(22,)i(w)m(e)h(can)f(no)m(w)h(suppress)h(the)e (index)g Fo(H)42 b Fw(in)33 b Fo(u)2807 4187 y Fm(\003)p Fp(;\033)n(;H)q(;R)3090 4223 y Fw(and)h(denote)g(it)0 4339 y(just)f(as)g Fo(u)369 4303 y Fm(\003)p Fp(;\033)n(;R)539 4339 y Fw(.)0 4540 y(2.5.)49 b Fz(Ground)d(states)f(for)h(the)f (cut-o\013)h(energy)-9 b(.)49 b Fw(Our)40 b(next)g(goal)e(is)h(to)h (sho)m(w)h(that)e Fo(u)3491 4503 y Fm(\003)p Fp(;\033)n(;R)0 4656 y Fw(is)e(a)g(ground)h(state)g(for)f(the)h(cut-o\013)f(energy)-8 b(.)60 b(This)38 b(will)d(b)s(e)j(obtained)f(b)m(y)i(\014xing)e(a)g (compactly)0 4772 y(supp)s(orted)f(p)s(erturbation,)f(considering)g (all)e(the)j(sets)g(whic)m(h)g(in)m(teract)f(with)g(this)g(supp)s(ort)g (and)0 4888 y(taking)i(a)h(fundamen)m(tal)e(domain)h(whic)m(h)h(con)m (tains)g(them.)59 b(In)38 b(this)f(w)m(a)m(y)-8 b(,)41 b(the)d(compactly)f(sup-)0 5005 y(p)s(orted)29 b(p)s(erturbation)g(in)m (teracts)h(with)f Fo(u)1548 4968 y Fm(\003)p Fp(;\033)n(;R)1748 5005 y Fw(as)g(a)g(p)s(erio)s(dic)f(one,)i(and)g(its)f(energy)h(has)g (to)f(gro)m(w,)0 5121 y(since)41 b Fo(u)303 5085 y Fm(\003)p Fp(;\033)n(;R)513 5121 y Fw(is)f(energy)h(minimizing)35 b(among)k(p)s(erio)s(dic)g(con\014gurations.)65 b(Note)41 b(that)f(Prop)s(osi-)0 5237 y(tion)d(22)h(pla)m(ys)h(a)f(crucial)f (role)h(here,)j(since,)f(when)g(w)m(e)f(put)g(all)d(the)j(in)m (teractions)f(in)m(to)f(a)h(large)0 5353 y(fundamen)m(tal)f(domain,)i (in)e(general,)j(one)e(ma)m(y)h(obtain)e(a)h(di\013eren)m(t)h (minimizer.)57 b(On)39 b(the)g(con-)0 5469 y(trary)-8 b(,)30 b(in)e(our)h(case,)i(Prop)s(osition)d(22)g(assures)j(that)e(the) h(in\014mal)d(minimizers)f(remain)i(the)h(same.)p eop %%Page: 26 26 26 25 bop 168 251 a Ft(26)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Moreo)m(v)m(er,)42 b(the)e(fact)f(that)f(the)i(energy)g(has)g(b)s(een)g(cut)f(o\013)g(is)f (also)h(imp)s(ortan)m(t)e(here,)k(since)f(w)m(e)168 566 y(w)m(an)m(t)k(that)f(all)e(the)i(sets)i(in)m(teracting)d(with)g(a)h (giv)m(en)h(\014nite)f(set)g(ma)m(y)g(b)s(e)h(put)f(inside)g(a)f Fr(\014nite)168 683 y Fw(fundamen)m(tal)31 b(domain)g(\(and,)i(for)f (cut-o\013)g(in)m(teractions,)g(this)g(is)g(assured)i(b)m(y)g Fz(H10)p Fw(\).)168 799 y(The)25 b(concept)g(that)f Fo(u)968 763 y Fm(\003)p Fp(;\033)n(;R)1162 799 y Fw(is)g(a)g(ground)g(state)g (for)g(the)g(cut-o\013)g(energy)h(and)f(the)h(ab)s(o)m(v)m(e)g(argumen) m(ts)168 915 y(are)32 b(formalized)f(in)g(the)i(follo)m(wing)d(w)m(a)m (y:)168 1055 y Fz(Prop)s(osition)38 b(23.)43 b Fo(u)1018 1019 y Fm(\003)p Fp(;\033)n(;R)1226 1055 y Fr(satis\014es)37 b(the)g(fol)5 b(lowing)36 b(lo)-5 b(c)g(al)37 b(minimization)f(pr)-5 b(op)g(erty:)50 b(given)36 b(any)168 1172 y(\014nite)49 b(set)g Fo(X)62 b Fs(\032)55 b Fw(\003)49 b Fr(and)g(any)h(c)-5 b(on\014gur)g(ation)48 b Fo(\014)55 b Fr(such)49 b(that)h Fo(\014)2611 1187 y Fp(p)2705 1172 y Fw(=)55 b(0)49 b Fr(for)g(any)g Fo(p)55 b Fs(2)f Fw(\003)33 b Fs(\000)g Fo(X)168 1289 y Fr(and)h Fo(u)413 1253 y Fm(\003)p Fp(;\033)n(;R)606 1289 y Fw(+)22 b Fo(\014)33 b Fs(2)28 b(O)965 1304 y Fp(\033)1012 1289 y Fr(,)35 b(we)f(have)h(that)1140 1375 y Fn(X)1146 1575 y Fk(B)r Fj(2S)1101 1624 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1101 1680 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1359 1470 y Fo(H)1440 1485 y Fp(B)1501 1470 y Fw(\()p Fo(u)1595 1429 y Fm(\003)p Fp(;\033)n(;R)1765 1470 y Fw(\))44 b Fs(\024)2028 1375 y Fn(X)2034 1575 y Fk(B)r Fj(2S)1989 1624 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1990 1680 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2248 1470 y Fo(H)2329 1485 y Fp(B)2389 1470 y Fw(\()p Fo(u)2483 1429 y Fm(\003)p Fp(;\033)n(;R)2676 1470 y Fw(+)22 b Fo(\014)6 b Fw(\))17 b Fo(:)168 1829 y Fz(Pro)s(of.)42 b Fw(Let)30 b(the)g(con\014guration)f Fo(\014)35 b Fw(b)s(e)30 b(supp)s(orted)h(in)e(the)h(\014nite)f(set)i Fo(X)k Fs(\032)28 b Fw(\003.)43 b(W)-8 b(e)30 b(exploit)e Fz(H10)168 1945 y Fw(and)40 b(w)m(e)i(consider)f Fo(Y)62 b Fw(as)40 b(in)g(\(10\))o(.)68 b(Then,)43 b(w)m(e)f(can)f(alw)m(a)m (ys)g(\014nd)g(a)f(normal)f(\014nite)h(index)h(sub-)168 2069 y(group)464 2044 y(~)438 2069 y Fo(H)35 b Fw(suc)m(h)28 b(that)e Fo(X)18 b Fs([)10 b Fo(Y)50 b Fs(\032)28 b Fw(\003)p Fo(=)1502 2044 y Fw(~)1477 2069 y Fo(H)34 b Fw(\(this)26 b(is)h(a)f(consequence)k(of)c Fz(G4)p Fw(,)i(as)f(discussed)i(on)e (page)f(4\),)168 2200 y(and,)h(b)m(y)g(Prop)s(osition)e(22,)g Fo(u)1233 2164 y Fm(\003)p Fp(;\033)n(;)1364 2147 y Fl(~)1346 2164 y Fp(H)5 b(;R)1514 2200 y Fw(=)28 b Fo(u)1674 2164 y Fm(\003)p Fp(;\033)n(;H)q(;R)1951 2200 y Fw(=)f Fo(u)2110 2164 y Fm(\003)p Fp(;\033)n(;R)2307 2200 y Fw(do)s(es)f(not)g(really)f (dep)s(end)i(on)f(the)h(c)m(hoice)168 2319 y(of)304 2294 y(~)279 2319 y Fo(H)7 b Fw(.)168 2435 y(No)m(w,)41 b(for)e(an)m(y)g Fo(q)k Fs(2)c Fw(\003)p Fo(=)1104 2410 y Fw(~)1079 2435 y Fo(H)7 b Fw(,)41 b(w)m(e)f(de\014ne)g Fo(w)1743 2450 y Fp(q)1820 2435 y Fw(:=)e Fo(u)2017 2399 y Fm(\003)p Fp(;\033)n(;R)2017 2460 y(q)2214 2435 y Fw(+)27 b Fo(\014)2372 2450 y Fp(q)2410 2435 y Fw(.)63 b(Also,)40 b(for)f(an)m(y)g Fo(p)g Fs(2)g Fw(\003,)i(w)m(e)f(de\014ne)168 2573 y Fo(v)215 2588 y Fp(p)284 2573 y Fw(:=)29 b Fo(w)486 2588 y Fp(q)547 2573 y Fw(+)23 b Fo(\033)t Fw(\()p Fo(h)p Fw(\),)34 b(as)g(long)e(as)i Fo(p)29 b Fw(=)h Fo(hq)t Fw(,)k(with)f Fo(q)g Fs(2)d Fw(\003)p Fo(=)2237 2547 y Fw(~)2212 2573 y Fo(H)41 b Fw(and)33 b Fo(h)d Fs(2)2731 2547 y Fw(~)2706 2573 y Fo(H)7 b Fw(.)47 b(Note)34 b(that)f(the)h(ab)s (o)m(v)m(e)h(is)168 2689 y(a)d(w)m(ell-p)s(osed)g(de\014nition,)g(b)m (y)h Fz(G3)p Fw(.)44 b(Moreo)m(v)m(er,)34 b(b)m(y)f(construction,)1012 2853 y(\()p Fs(T)1104 2868 y Fp(g)1144 2853 y Fo(v)t Fw(\))1233 2868 y Fp(p)1300 2853 y Fw(=)28 b Fo(v)1451 2868 y Fp(g)r(hq)1594 2853 y Fw(=)f Fo(w)1767 2868 y Fp(q)1827 2853 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))f(+)h Fo(\033)t Fw(\()p Fo(h)p Fw(\))28 b(=)g(\()p Fs(R)2675 2869 y Fp(\033)r Fl(\()p Fp(g)r Fl(\))2813 2853 y Fo(v)t Fw(\))2902 2868 y Fp(p)2958 2853 y Fo(;)168 3033 y Fw(for)k(an)m(y)h Fo(g)e Fs(2)698 3008 y Fw(~)673 3033 y Fo(H)8 b Fw(,)32 b(so)h Fo(v)e Fs(2)d(C)1165 3060 y Fp(\033)n(;)1242 3044 y Fl(~)1223 3060 y Fp(H)1292 3033 y Fw(.)43 b(Therefore,)168 3232 y(\(39\))1110 b Fo(S)1518 3191 y Fp(R)1512 3269 y(\033)n(;)1588 3252 y Fl(~)1570 3269 y Fp(H)1638 3232 y Fw(\()p Fo(u)1732 3191 y Fm(\003)p Fp(;\033)n(;)1862 3175 y Fl(~)1845 3191 y Fp(H)5 b(;R)1985 3232 y Fw(\))44 b Fs(\024)h Fo(S)2255 3191 y Fp(R)2249 3269 y(\033)n(;)2325 3252 y Fl(~)2307 3269 y Fp(H)2374 3232 y Fw(\()p Fo(v)t Fw(\))17 b Fo(:)168 3407 y Fw(W)-8 b(e)33 b(claim)d(that)1035 3555 y(If)i Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)q Fw(,)33 b Fo(B)27 b Fs(\\)22 b Fo(X)36 b Fw(=)27 b Fs(;)33 b Fw(and)f Fo(B)c Fs(\\)22 b Fw(\(\003)p Fo(=)2653 3530 y Fw(~)2628 3555 y Fo(H)7 b Fw(\))28 b Fs(6)p Fw(=)f Fs(;)p Fw(,)1035 3723 y(then)33 b(\()p Fo(g)t(B)5 b Fw(\))21 b Fs(\\)i Fo(X)35 b Fw(=)28 b Fs(;)k Fw(for)g(an)m(y)h Fo(g)e Fs(2)2406 3698 y Fw(~)2380 3723 y Fo(H)8 b Fw(.)168 3637 y(\(40\))168 3910 y(Indeed,)33 b(if,)d(b)m(y)i(con)m(tradiction,)e Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))27 b Fs(\024)h Fo(R)q Fw(,)k Fo(B)24 b Fs(\\)19 b Fo(X)36 b Fw(=)27 b Fs(;)p Fw(,)k Fo(B)24 b Fs(\\)c Fw(\(\003)p Fo(=)2721 3885 y Fw(~)2697 3910 y Fo(H)7 b Fw(\))27 b Fs(6)p Fw(=)h Fs(;)j Fw(and)g(\()p Fo(g)t(B)5 b Fw(\))18 b Fs(\\)h Fo(X)36 b Fs(6)p Fw(=)27 b Fs(;)p Fw(,)168 4034 y(w)m(e)37 b(w)m(ould)g(ha)m(v) m(e)g(that)g Fo(g)t(B)h Fs(\032)d Fo(Y)21 b Fw(,)38 b(thanks)f(to)f (\(10\),)h(and)g(so)f(\()p Fo(g)t(B)5 b Fw(\))34 b Fs(\032)h Fw(\(\003)p Fo(=)2996 4009 y Fw(~)2972 4034 y Fo(H)7 b Fw(\).)55 b(Then,)38 b(tak)m(e)g Fo(q)g Fs(2)168 4159 y Fo(B)22 b Fs(\\)17 b Fw(\(\003)p Fo(=)526 4134 y Fw(~)502 4159 y Fo(H)7 b Fw(\))27 b Fs(6)p Fw(=)h Fs(;)p Fw(.)42 b(It)30 b(w)m(ould)g(follo)m(w)e(that)i(b)s(oth)g Fo(q)j Fw(and)d Fo(g)t(q)j Fw(lie)c(in)g(\003)p Fo(=)2750 4134 y Fw(~)2725 4159 y Fo(H)7 b Fw(,)30 b(thence)i Fo(g)h Fw(should)d(b)s(e)g(the)168 4275 y(iden)m(tit)m(y)-8 b(.)43 b(But)33 b(then)1388 4399 y Fs(;)28 b(6)p Fw(=)f(\()p Fo(g)t(B)5 b Fw(\))22 b Fs(\\)g Fo(X)36 b Fw(=)27 b Fo(B)h Fs(\\)22 b Fo(X)36 b Fw(=)27 b Fs(;)17 b Fo(;)168 4543 y Fw(and)32 b(this)h(con)m(tradiction)e(pro)m(v)m(es)j(\(40\).)43 b(W)-8 b(e)33 b(no)m(w)g(sho)m(w)h(that)168 4716 y(\(41\))211 b(If)33 b Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)q Fw(,)k Fo(B)c Fs(\\)22 b Fo(X)36 b Fw(=)27 b Fs(;)33 b Fw(and)f Fo(B)c Fs(\\)22 b Fw(\(\003)p Fo(=)2172 4691 y Fw(~)2147 4716 y Fo(H)7 b Fw(\))28 b Fs(6)p Fw(=)f Fs(;)p Fw(,)33 b(then)g Fo(H)2817 4731 y Fp(B)2877 4716 y Fw(\()p Fo(v)t Fw(\))28 b(=)f Fo(H)3216 4731 y Fp(B)3277 4716 y Fw(\()p Fo(u)3371 4680 y Fm(\003)p Fp(;\033)n(;R)3541 4716 y Fw(\).)168 4889 y(T)-8 b(o)37 b(pro)m(v)m(e)i(this,)g(let)d Fo(B)43 b Fw(as)38 b(ab)s(o)m(v)m(e)g(and)g(tak)m(e)g(an)m(y)g Fo(p)e Fs(2)h Fo(B)5 b Fw(.)58 b(Let)38 b Fo(q)i Fs(2)c Fw(\003)p Fo(=)2935 4864 y Fw(~)2910 4889 y Fo(H)45 b Fw(in)36 b(suc)m(h)j(a)e(w)m(a)m(y)i Fo(p)d Fw(=)168 5014 y Fo(hq)t Fw(,)45 b(with)d Fo(h)j Fs(2)813 4988 y Fw(~)787 5014 y Fo(H)8 b Fw(.)74 b(Then,)44 b Fo(q)k Fs(62)d Fo(X)8 b Fw(,)46 b(otherwise)d(\()p Fo(h)2171 4978 y Fm(\000)p Fl(1)2265 5014 y Fo(p)p Fw(\))i Fs(2)g Fw(\()p Fo(h)2602 4978 y Fm(\000)p Fl(1)2697 5014 y Fo(B)5 b Fw(\))29 b Fs(\\)g Fo(X)8 b Fw(,)45 b(con)m(trary)e(to)g(\(40\))o(.) 168 5130 y(Accordingly)-8 b(,)1183 5294 y Fo(v)1230 5309 y Fp(p)1353 5294 y Fw(=)83 b Fo(w)1582 5309 y Fp(q)1642 5294 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(h)p Fw(\))28 b(=)f Fo(u)2118 5253 y Fm(\003)p Fp(;\033)n(;R)2118 5319 y(q)2311 5294 y Fw(+)22 b Fo(\014)2464 5309 y Fp(q)2524 5294 y Fw(+)g Fo(\033)t Fw(\()p Fo(h)p Fw(\))1353 5462 y(=)83 b Fo(u)1568 5421 y Fm(\003)p Fp(;\033)n(;R)1568 5487 y(q)1761 5462 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(h)p Fw(\))28 b(=)f Fo(u)2237 5415 y Fm(\003)p Fp(;\033)n(;R)2237 5490 y(hq)2435 5462 y Fw(=)h Fo(u)2595 5421 y Fm(\003)p Fp(;\033)n(;R)2595 5487 y(p)2782 5462 y Fo(;)p eop %%Page: 27 27 27 26 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(27)0 450 y Fw(since)33 b Fo(u)295 414 y Fm(\003)p Fp(;\033)n(;R)493 450 y Fs(2)28 b(C)639 465 y Fp(\033)n(;H)765 450 y Fw(.)44 b(Then,)34 b(\(41\))e(is)g(a)g(consequence)k(of)c Fz(H1)p Fw(.)0 566 y(No)m(w,)h(giv)m(en)g Fo(B)g Fs(\032)28 b Fw(\003,)k(w)m(e)i(de\014ne)1388 746 y Fo(U)1472 756 y Fl(~)1454 773 y Fp(H)1521 746 y Fw(\()p Fo(B)5 b Fw(\))45 b(:=)1888 652 y Fn([)1868 879 y Fp(h)p Fm(2)1973 862 y Fl(~)1956 879 y Fp(H)2019 746 y Fw(\()p Fo(hB)5 b Fw(\))17 b Fo(:)0 1018 y Fw(Th)m(us,)0 1182 y(\(42\))111 b(If)33 b([)p Fo(B)5 b Fw(])534 1192 y Fl(~)516 1209 y Fp(H)611 1182 y Fs(2)28 b(S)783 1192 y Fl(~)765 1209 y Fp(H)832 1182 y Fw(,)33 b Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)33 b Fw(and)g Fo(U)1608 1192 y Fl(~)1590 1209 y Fp(H)1658 1182 y Fw(\()p Fo(B)5 b Fw(\))22 b Fs(\\)g Fo(X)36 b Fw(=)27 b Fs(;)p Fw(,)33 b(then)g Fo(H)2556 1198 y Fl([)p Fp(B)s Fl(])2666 1209 y Fi(~)2652 1222 y Fk(H)2713 1182 y Fw(\()p Fo(v)t Fw(\))28 b(=)f Fo(H)3052 1198 y Fl([)p Fp(B)s Fl(])3162 1209 y Fi(~)3148 1222 y Fk(H)3210 1182 y Fw(\()p Fo(u)3304 1146 y Fm(\003)p Fp(;\033)n(;R)3474 1182 y Fw(\).)0 1369 y(Indeed,)j(tak)m(e)e Fo(B)k Fw(as)27 b(ab)s(o)m(v)m(e)h(and)g(let)e Fo(B)1436 1332 y Fm(\003)1503 1369 y Fs(2)j Fw([)p Fo(B)5 b Fw(])1749 1379 y Fl(~)1731 1396 y Fp(H)1825 1369 y Fw(suc)m(h)29 b(that)e Fo(B)2325 1332 y Fm(\003)2376 1369 y Fs(\\)11 b Fw(\(\003)p Fo(=)2633 1343 y Fw(~)2608 1369 y Fo(H)c Fw(\))28 b Fs(6)p Fw(=)f Fs(;)p Fw(.)42 b(Then,)28 b Fo(B)3339 1332 y Fm(\003)3390 1369 y Fs(\\)11 b Fo(X)36 b Fs(\032)0 1485 y Fo(U)84 1495 y Fl(~)66 1512 y Fp(H)133 1485 y Fw(\()p Fo(B)5 b Fw(\))23 b Fs(\\)f Fo(X)36 b Fw(=)27 b Fs(;)33 b Fw(and)f(so,)h(b)m(y)i(\(41\))o(,)767 1654 y Fo(H)848 1670 y Fl([)p Fp(B)s Fl(])959 1681 y Fi(~)944 1694 y Fk(H)1006 1654 y Fw(\()p Fo(v)t Fw(\))27 b(=)g Fo(H)1344 1669 y Fp(B)1400 1650 y Fj(\003)1441 1654 y Fw(\()p Fo(v)t Fw(\))g(=)h Fo(H)1780 1669 y Fp(B)1836 1650 y Fj(\003)1876 1654 y Fw(\()p Fo(u)1970 1613 y Fm(\003)p Fp(;\033)n(;R)2141 1654 y Fw(\))f(=)h Fo(H)2391 1670 y Fl([)p Fp(B)s Fl(])2501 1681 y Fi(~)2487 1694 y Fk(H)2548 1654 y Fw(\()p Fo(u)2642 1613 y Fm(\003)p Fp(;\033)n(;R)2813 1654 y Fw(\))17 b Fo(;)0 1824 y Fw(pro)m(ving)34 b(\(42\))o(.)0 1940 y(Moreo)m(v)m(er,)0 2104 y(\(43\))326 b(If)33 b Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)q Fw(,)k Fo(B)c Fs(\\)22 b Fo(X)36 b Fs(6)p Fw(=)27 b Fs(;)33 b Fw(and)f Fo(B)h Fs(6)p Fw(=)28 b Fo(A)f Fs(2)i Fw([)p Fo(B)5 b Fw(])2306 2114 y Fl(~)2288 2131 y Fp(H)2355 2104 y Fw(,)33 b(then)g Fo(A)22 b Fs(\\)h Fo(X)35 b Fw(=)28 b Fs(;)p Fw(.)0 2279 y(Indeed,)42 b(if)i(\(43\))38 b(did)g(not)g(hold,) h(w)m(e)h(w)m(ould)e(ha)m(v)m(e)i(that)e Fo(A)g Fw(=)g Fo(hB)44 b Fw(for)38 b(some)g Fo(h)g Fs(2)3185 2253 y Fw(~)3160 2279 y Fo(H)46 b Fw(and)39 b(that)0 2403 y(b)s(oth)i Fo(A)g Fw(and)g Fo(B)46 b Fw(lie)39 b(in)i Fo(Y)63 b Fs(\032)42 b Fw(\(\003)p Fo(=)1351 2378 y Fw(~)1326 2403 y Fo(H)7 b Fw(\).)69 b(Then,)42 b Fo(h)f Fw(m)m(ust)g(b)s(e)g(the)h (iden)m(tit)m(y)f(and)g(so)g Fo(A)h Fw(=)g Fo(B)5 b Fw(,)43 b(in)0 2519 y(con)m(tradiction)31 b(with)i(the)g(assumption,)f(whic)m (h)h(giv)m(es)g(\(43\))o(.)0 2636 y(As)g(a)f(consequence)k(of)j(\(43\)) o(,)1317 2726 y Fn(X)1250 2930 y Fi([)p Fk(B)r Fi(])1348 2946 y(~)1334 2959 y Fk(H)1391 2930 y Fj(2S)1485 2946 y Fi(~)1471 2959 y Fk(H)1204 3003 y(U)1261 3020 y Fi(~)1246 3033 y Fk(H)1304 3003 y Fi(\()p Fk(B)r Fi(\))p Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1279 3079 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1610 2821 y Fo(H)1691 2837 y Fl([)p Fp(B)s Fl(])1802 2848 y Fi(~)1787 2861 y Fk(H)1893 2821 y Fw(=)2072 2726 y Fn(X)2078 2926 y Fk(B)r Fj(2S)2033 2975 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2034 3031 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2292 2821 y Fo(H)2373 2836 y Fp(B)2450 2821 y Fo(;)0 3227 y Fw(and,)33 b(consequen)m(tly)-8 b(,)993 3407 y Fo(S)1059 3365 y Fp(R)1053 3443 y(\033)n(;)1129 3426 y Fl(~)1111 3443 y Fp(H)1261 3407 y Fw(=)1553 3312 y Fn(X)1486 3515 y Fi([)p Fk(B)r Fi(])1584 3532 y(~)1570 3545 y Fk(H)1627 3515 y Fj(2S)1721 3532 y Fi(~)1707 3545 y Fk(H)1440 3589 y(U)1497 3605 y Fi(~)1482 3618 y Fk(H)1540 3589 y Fi(\()p Fk(B)r Fi(\))p Fj(\\)p Fk(X)t Fi(=)p Fj(;)1515 3664 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1846 3407 y Fo(H)1927 3422 y Fl([)p Fp(B)s Fl(])2038 3433 y Fi(~)2023 3446 y Fk(H)2107 3407 y Fw(+)2264 3312 y Fn(X)2270 3512 y Fk(B)r Fj(2S)2225 3561 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2226 3617 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2484 3407 y Fo(H)2565 3422 y Fp(B)2642 3407 y Fo(:)0 3818 y Fw(W)g(e)33 b(no)m(w)g(mak)m(e)g(use)g (of)g(the)g(ab)s(o)m(v)m(e)g(equalit)m(y)-8 b(,)32 b(\(42\))g(and)h (\(39\))f(to)g(conclude)h(that)618 3908 y Fn(X)625 4109 y Fk(B)r Fj(2S)579 4159 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)580 4214 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)838 4003 y Fo(H)919 4018 y Fp(B)980 4003 y Fw(\()p Fo(u)1074 3962 y Fm(\003)p Fp(;\033)n(;R)1244 4003 y Fw(\))84 b(=)f Fo(S)1591 3962 y Fp(R)1585 4039 y(\033)n(;)1662 4022 y Fl(~)1643 4039 y Fp(H)1711 4003 y Fw(\()p Fo(u)1805 3962 y Fm(\003)p Fp(;\033)n(;R)1975 4003 y Fw(\))23 b Fs(\000)2268 3908 y Fn(X)2201 4112 y Fi([)p Fk(B)r Fi(])2299 4128 y(~)2285 4141 y Fk(H)2342 4112 y Fj(2S)2436 4128 y Fi(~)2422 4141 y Fk(H)2155 4187 y(U)2212 4204 y Fi(~)2197 4217 y Fk(H)2255 4187 y Fi(\()p Fk(B)r Fi(\))p Fj(\\)p Fk(X)t Fi(=)p Fj(;)2230 4261 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2561 4003 y Fo(H)2642 4019 y Fl([)p Fp(B)s Fl(])2753 4030 y Fi(~)2738 4043 y Fk(H)2800 4003 y Fw(\()p Fo(u)2894 3962 y Fm(\003)p Fp(;\033)n(;R)3064 4003 y Fw(\))1366 4425 y(=)83 b Fo(S)1591 4384 y Fp(R)1585 4461 y(\033)n(;)1662 4445 y Fl(~)1643 4461 y Fp(H)1711 4425 y Fw(\()p Fo(u)1805 4384 y Fm(\003)p Fp(;\033)n(;R)1975 4425 y Fw(\))23 b Fs(\000)2268 4331 y Fn(X)2201 4534 y Fi([)p Fk(B)r Fi(])2299 4551 y(~)2285 4564 y Fk(H)2342 4534 y Fj(2S)2436 4551 y Fi(~)2422 4564 y Fk(H)2155 4609 y(U)2212 4626 y Fi(~)2197 4639 y Fk(H)2255 4609 y Fi(\()p Fk(B)r Fi(\))p Fj(\\)p Fk(X)t Fi(=)p Fj(;)2230 4683 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2561 4425 y Fo(H)2642 4441 y Fl([)p Fp(B)s Fl(])2753 4452 y Fi(~)2738 4465 y Fk(H)2800 4425 y Fw(\()p Fo(v)t Fw(\))1365 4848 y Fs(\024)83 b Fo(S)1591 4806 y Fp(R)1585 4884 y(\033)n(;)1662 4867 y Fl(~)1643 4884 y Fp(H)1711 4848 y Fw(\()p Fo(v)t Fw(\))22 b Fs(\000)2092 4753 y Fn(X)2025 4956 y Fi([)p Fk(B)r Fi(])2123 4973 y(~)2109 4986 y Fk(H)2166 4956 y Fj(2S)2260 4973 y Fi(~)2246 4986 y Fk(H)1979 5032 y(U)2036 5048 y Fi(~)2021 5061 y Fk(H)2079 5032 y Fi(\()p Fk(B)r Fi(\))p Fj(\\)p Fk(X)t Fi(=)p Fj(;)2054 5105 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2385 4848 y Fo(H)2466 4863 y Fl([)p Fp(B)s Fl(])2577 4874 y Fi(~)2562 4887 y Fk(H)2624 4848 y Fw(\()p Fo(v)t Fw(\))1366 5270 y(=)1584 5175 y Fn(X)1591 5375 y Fk(B)r Fj(2S)1545 5426 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1546 5480 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1804 5270 y Fo(H)1885 5285 y Fp(B)1946 5270 y Fw(\()p Fo(v)t Fw(\))17 b Fo(;)p eop %%Page: 28 28 28 27 bop 168 251 a Ft(28)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(as)32 b(desired.)3107 b Fg(\003)168 719 y Fw(2.6.)48 b Fz(The)38 b(Birkho\013)e(prop)s(ert)m(y)h(and)h(a-priori)e(b)s(ounds.)51 b Fw(W)-8 b(e)32 b(no)m(w)h(sho)m(w)h(that)e(the)g(in\014mal)168 835 y(minimizer)i(is)i(Birkho\013)h(\(see)h(De\014nition)e(6)h(\))g (and)g(that)g(it)g(b)s(elongs)f(to)h(the)h(space)g Fs(O)3441 850 y Fp(\033)3526 835 y Fw(de\014ned)168 952 y(on)32 b(page)h(6.)168 1097 y Fz(Lemma)k(24.)42 b Fr(The)34 b(in\014mal)g(minimizer)f Fo(u)1798 1061 y Fm(\003)p Fp(;\033)n(;R)2003 1097 y Fr(is)i(Birkho\013.)168 1243 y Fz(Pro)s(of.)82 b Fw(Let)46 b Fo(`)k Fs(2)h Fq(Z)43 b Fw(and)j Fo(g)53 b Fs(2)d Fo(G)c Fw(b)s(e)g(suc)m(h)h(that)f Fo(\033)t Fw(\()p Fo(g)t Fw(\))j Fs(\025)h Fo(`)c Fw(\(resp.,)h Fo(\033)t Fw(\()p Fo(g)t Fw(\))i Fs(\024)h Fo(`)p Fw(\).)83 b(By)46 b Fz(H3)168 1359 y Fw(and)38 b Fz(H4)p Fw(,)h Fs(R)657 1374 y Fm(\000)p Fp(`)745 1359 y Fs(T)799 1374 y Fp(g)839 1359 y Fo(u)895 1323 y Fm(\003)p Fp(;\033)n(;R)1104 1359 y Fw(\(resp.,)g Fs(R)1492 1374 y Fp(`)1525 1359 y Fs(T)1579 1376 y Fp(g)1615 1357 y Fj(\000)p Fi(1)1702 1359 y Fo(u)1758 1323 y Fm(\003)p Fp(;\033)n(;R)1929 1359 y Fw(\))e(is)h(also)f(a)g(minimizer)e(for)i Fo(S)3076 1323 y Fp(R)3070 1384 y(\033)n(;H)3234 1359 y Fw(in)g Fs(C)3405 1374 y Fp(\033)n(;H)3569 1359 y Fw(and)h(it)168 1480 y(satis\014es)32 b(the)h(constrain)m(ts)f(in)g(the)g(righ)m(t)f (hand)h(side)h(of)38 b(\(36\))o(.)44 b(Th)m(us,)33 b(b)m(y)g(the)g (in\014mal)c(minimizer)168 1596 y(prop)s(ert)m(y)-8 b(,)33 b Fo(u)641 1560 y Fm(\003)p Fp(;\033)n(;R)839 1596 y Fs(\024)28 b(R)1028 1611 y Fm(\000)p Fp(`)1117 1596 y Fs(T)1171 1611 y Fp(g)1211 1596 y Fo(u)1267 1560 y Fm(\003)p Fp(;\033)n(;R)1470 1596 y Fw(\(resp.,,)34 b Fo(u)1852 1560 y Fm(\003)p Fp(;\033)n(;R)2050 1596 y Fs(\024)28 b(R)2239 1611 y Fp(`)2273 1596 y Fs(T)2327 1613 y Fp(g)2363 1594 y Fj(\000)p Fi(1)2449 1596 y Fo(u)2505 1560 y Fm(\003)p Fp(;\033)n(;R)2676 1596 y Fw(\).)978 b Fg(\003)168 1771 y Fw(Next)33 b(result)g(p)s(oin)m(ts)f(out)g(a)h(useful)f(prop)s(ert)m (y)i(of)e(the)h(Birkho\013)f(functions:)168 1917 y Fz(Lemma)42 b(25.)i Fr(If)38 b Fo(v)43 b Fr(satis\014es)38 b(the)h(Birkho\013)f(pr) -5 b(op)g(erty,)40 b(then)f Fo(v)2556 1932 y Fp(g)r(q)2655 1917 y Fs(\000)26 b Fo(v)2805 1932 y Fp(q)2878 1917 y Fs(2)36 b Fw([)p Fo(\033)t Fw(\()p Fo(g)t Fw(\))24 b Fs(\000)i Fw(1)p Fo(;)17 b(\033)t Fw(\()p Fo(g)t Fw(\))23 b(+)i(1])p Fr(,)168 2033 y(for)34 b(any)h Fo(q)d Fs(2)c Fw(\003)34 b Fr(and)h(any)f Fo(g)d Fs(2)d Fo(G)p Fr(.)168 2178 y Fz(Pro)s(of.)44 b Fw(Giv)m(en)33 b Fo(g)e Fs(2)e Fo(G)p Fw(,)k(let)f Fo(`)c Fs(2)h Fq(Z)h Fw(b)s(e)j(so)h(that)e Fo(`)c Fs(\024)h Fo(\033)t Fw(\()p Fo(g)t Fw(\))f Fs(\024)g Fo(`)23 b Fw(+)f(1.)44 b(Then,)34 b(since)g Fo(v)i Fw(is)d(Birkho\013,) 168 2295 y Fs(T)222 2310 y Fp(g)262 2295 y Fw(\()p Fo(v)t Fw(\))27 b Fs(\025)i(R)606 2310 y Fp(`)639 2295 y Fw(\()p Fo(v)t Fw(\))j(and)h Fs(T)1042 2310 y Fp(g)1082 2295 y Fw(\()p Fo(v)t Fw(\))27 b Fs(\024)i(R)1426 2310 y Fp(`)p Fl(+1)1549 2295 y Fw(\()p Fo(v)t Fw(\).)43 b(That)33 b(is,)1503 2470 y Fo(v)1550 2485 y Fp(g)r(q)1647 2470 y Fs(\000)22 b Fo(v)1793 2485 y Fp(q)1859 2470 y Fs(\025)28 b Fo(`)g Fs(\025)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))21 b Fs(\000)i Fw(1)168 2644 y(and)1398 2779 y Fo(v)1445 2794 y Fp(g)r(q)1541 2779 y Fs(\000)g Fo(v)1688 2794 y Fp(q)1754 2779 y Fs(\024)28 b Fo(`)22 b Fw(+)g(1)27 b Fs(\024)h Fo(\033)t Fw(\()p Fo(g)t Fw(\))22 b(+)g(1)17 b Fo(;)168 2933 y Fw(for)32 b(an)m(y)h Fo(q)f Fs(2)c Fw(\003,)k(as)h(desired.)2477 b Fg(\003)168 3137 y Fz(Corollary)38 b(26.)k Fr(If)35 b Fo(v)k Fr(satis\014es)c(the)h(Birkho\013)e(pr)-5 b(op)g(erty,)36 b(then)f Fo(v)2627 3152 y Fp(g)2661 3161 y Fi(1)2695 3152 y Fp(p)2758 3137 y Fs(\000)23 b Fo(v)2905 3152 y Fp(g)2939 3161 y Fi(2)2973 3152 y Fp(p)3042 3137 y Fs(2)29 b Fw([)p Fo(\033)t Fw(\()p Fo(g)3308 3152 y Fl(1)3347 3137 y Fw(\))23 b Fs(\000)g Fo(\033)t Fw(\()p Fo(g)3652 3152 y Fl(2)3691 3137 y Fw(\))g Fs(\000)168 3254 y Fw(1)p Fo(;)17 b(\033)t Fw(\()p Fo(g)405 3269 y Fl(1)443 3254 y Fw(\))22 b Fs(\000)h Fo(\033)t Fw(\()p Fo(g)747 3269 y Fl(2)786 3254 y Fw(\))f(+)g(1])p Fr(,)35 b(for)f(any)h Fo(p)28 b Fs(2)g Fw(\003)35 b Fr(and)f(any)h Fo(g)c Fs(2)d Fo(G)p Fr(.)168 3400 y Fz(Pro)s(of.)43 b Fw(Just)33 b(apply)f(Lemma)g(25)g(to)g Fo(g)f Fw(:=)d Fo(g)1849 3415 y Fl(1)1888 3400 y Fo(g)1939 3359 y Fm(\000)p Fl(1)1935 3425 y(2)2065 3400 y Fw(and)33 b Fo(q)e Fw(:=)d Fo(g)2507 3415 y Fl(2)2546 3400 y Fo(p)p Fw(.)1097 b Fg(\003)168 3575 y Fw(W)-8 b(e)30 b(no)m(w)h(p)s(erform)e(an)h(easy)h (energy)g(estimate)e(whic)m(h)h(will)e(imply)g(an)i(a-priori)d (regularit)m(y)i(result)168 3691 y(\(that)j(is,)g(Lemma)f(28\).)168 3837 y Fz(Lemma)45 b(27.)h Fr(L)-5 b(et)42 b Fo(K)r(;)17 b(\024;)g(\022)44 b Fr(b)-5 b(e)42 b(as)f(in)h Fz(H6)p Fr(.)65 b(Supp)-5 b(ose)41 b(that)h Fo(v)k Fr(satis\014es)41 b(the)g(Birkho\013)g(pr)-5 b(op)g(erty)168 3953 y(and)34 b(that)168 4128 y Fw(\(44\))1277 b(max)1578 4195 y Fp(i;j)t Fm(2)p Fl(\003)p Fp(=G)1857 4128 y Fs(j)p Fo(v)1932 4143 y Fp(i)1982 4128 y Fs(\000)23 b Fo(v)2129 4143 y Fp(j)2166 4128 y Fs(j)k(\025)h Fw(2)17 b Fo(:)168 4357 y Fr(Then,)1120 4537 y Fo(H)1201 4552 y Fp(g)r(F)1296 4537 y Fw(\()p Fo(v)t Fw(\))43 b Fs(\025)i Fo(\024)1661 4427 y Fn(\020)1777 4537 y Fw(max)1737 4604 y Fp(i;j)t Fm(2)p Fl(\003)p Fp(=G)2016 4537 y Fs(j)p Fo(v)2091 4552 y Fp(i)2141 4537 y Fs(\000)23 b Fo(v)2288 4552 y Fp(j)2324 4537 y Fs(j)f(\000)h Fw(2)2523 4427 y Fn(\021)2582 4449 y Fp(\022)2643 4537 y Fs(\000)g Fo(K)h(;)168 4746 y Fr(for)34 b(any)h Fo(g)c Fs(2)d Fo(G)p Fr(.)168 4892 y Fz(Pro)s(of.)43 b Fw(Let)33 b Fo(a;)17 b(b)28 b Fs(2)g Fw(\003)p Fo(=G)k Fw(suc)m(h)i(that)1457 5067 y Fo(v)1504 5082 y Fp(a)1568 5067 y Fs(\000)23 b Fo(v)1715 5082 y Fp(b)1777 5067 y Fw(=)68 b(max)1881 5133 y Fp(i;j)t Fm(2)p Fl(\003)p Fp(=G)2160 5067 y Fs(j)p Fo(v)2235 5082 y Fp(i)2285 5067 y Fs(\000)22 b Fo(v)2431 5082 y Fp(j)2468 5067 y Fs(j)17 b Fo(:)168 5295 y Fw(By)33 b(Lemma)e(25,)734 5469 y Fo(v)781 5484 y Fp(g)r(a)881 5469 y Fs(\000)23 b Fo(v)1028 5484 y Fp(g)r(b)1126 5469 y Fs(\025)28 b Fw(\()p Fo(v)1316 5484 y Fp(a)1380 5469 y Fw(+)22 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))f Fs(\000)i Fw(1\))f Fs(\000)g Fw(\()p Fo(v)2078 5484 y Fp(b)2135 5469 y Fw(+)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))f(+)h(1\))28 b(=)f Fo(v)2803 5484 y Fp(a)2867 5469 y Fs(\000)c Fo(v)3014 5484 y Fp(b)3070 5469 y Fs(\000)g Fw(2)17 b Fo(;)p eop %%Page: 29 29 29 28 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(29)0 450 y Fw(whic)m(h)33 b(is)f(greater)h(or)f(equal)g(then)i(zero,) f(b)m(y)i(\(44\))o(.)43 b(Hence,)34 b(from)e Fz(H3)g Fw(and)h Fz(H6)p Fw(,)835 754 y Fo(H)916 769 y Fp(g)r(F)1010 754 y Fw(\()p Fo(v)t Fw(\))27 b(=)h Fo(H)1349 769 y Fp(F)1407 754 y Fw(\()p Fs(T)1499 769 y Fp(g)1540 754 y Fo(v)t Fw(\))f Fs(\025)h Fo(\024)85 b Fw(max)1834 820 y Fp(i;j)t Fm(2)p Fl(\(\003)p Fp(=G)p Fl(\))2168 754 y Fs(j)p Fo(v)2243 769 y Fp(g)r(i)2329 754 y Fs(\000)22 b Fo(v)2475 769 y Fp(g)r(j)2548 754 y Fs(j)2576 713 y Fp(\022)2637 754 y Fs(\000)h Fo(K)1164 971 y Fs(\025)29 b Fo(\024)17 b Fw(\()p Fo(v)1428 986 y Fp(g)r(a)1527 971 y Fs(\000)23 b Fo(v)1674 986 y Fp(g)r(b)1744 971 y Fw(\))1782 930 y Fp(\022)1843 971 y Fs(\000)g Fo(K)1164 1135 y Fs(\025)29 b Fo(\024)17 b Fw(\()p Fo(v)1428 1150 y Fp(a)1491 1135 y Fs(\000)23 b Fo(v)1638 1150 y Fp(b)1694 1135 y Fs(\000)g Fw(2\))1881 1094 y Fp(\022)1942 1135 y Fs(\000)g Fo(K)g(;)0 1435 y Fw(as)33 b(desired.)3106 b Fg(\003)0 1734 y Fw(W)-8 b(e)33 b(are)g(no)m(w)g(in)f(the)h(p)s(osition)d(of)i(estimating)f(the) i(oscillation)c(of)j(our)h(minimizers:)0 1942 y Fz(Lemma)27 b(28.)36 b Fr(L)-5 b(et)27 b Fo(\024;)17 b(K)r(;)g(S;)g(\022)s(;)g(#)27 b Fr(b)-5 b(e)27 b(as)g(in)g Fz(H6)g Fr(and)f Fz(H7)p Fr(.)42 b(L)-5 b(et)27 b Fo(v)32 b Fs(2)c(C)2529 1957 y Fp(\033)n(;H)2682 1942 y Fr(b)-5 b(e)27 b(a)g(minimizer)f(for)h Fo(S)3542 1906 y Fp(R)3536 1967 y(\033)n(;H)0 2058 y Fr(on)34 b Fs(C)191 2073 y Fp(\033)n(;H)318 2058 y Fr(.)44 b(Then,)786 2476 y Fw(max)745 2542 y Fp(i;j)t Fm(2)p Fl(\003)p Fp(=G)1024 2476 y Fs(j)p Fo(v)1099 2491 y Fp(i)1149 2476 y Fs(\000)23 b Fo(v)1296 2491 y Fp(j)1333 2476 y Fs(j)43 b(\024)i Fw(2)22 b(+)1695 2305 y Fn( )1784 2408 y Fo(S)g Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)2063 2360 y Fp(#)2131 2408 y Fw(+)22 b(1\))17 b Fo(]F)35 b Fw(+)22 b Fo(K)p 1784 2453 874 4 v 2193 2544 a(\024)2667 2305 y Fn(!)2746 2327 y Fl(1)p Fp(=\022)2889 2476 y Fo(:)0 2866 y Fz(Pro)s(of.)54 b Fw(Without)36 b(loss)g(of)g(generalit)m(y)-8 b(,)36 b(w)m(e)i(ma)m(y)e(assume)g(that)h(\(44\))e(holds,)i(otherwise)g (w)m(e)g(are)0 2983 y(done.)44 b(Note)33 b(also)e(that)0 3282 y(\(45\))897 b([)p Fo(iF)14 b Fw(])1235 3297 y Fp(H)1330 3282 y Fs(6)p Fw(=)28 b([)p Fo(j)6 b(F)14 b Fw(])1611 3297 y Fp(H)1710 3282 y Fw(for)32 b(an)m(y)i Fo(i)28 b Fs(6)p Fw(=)f Fo(j)34 b Fs(2)28 b Fo(G=H)0 3581 y Fw(Indeed,)33 b(if)c Fo(F)42 b Fw(=)27 b Fo(g)t(F)14 b Fw(,)30 b(with)g Fo(g)h Fw(=)d Fo(i)1258 3545 y Fm(\000)p Fl(1)1353 3581 y Fo(hj)36 b Fw(and)31 b Fo(h)d Fs(2)g Fo(H)8 b Fw(,)31 b(then,)g(since)h Fo(F)44 b Fw(is)30 b(\014nite)h(\(and)f(not)h(empt)m (y\),)0 3698 y(\014xed)39 b Fo(f)46 b Fs(2)37 b Fo(F)14 b Fw(,)38 b(w)m(e)h(ha)m(v)m(e)g(that)e(there)h(exist)g Fo(n)1723 3713 y Fl(1)1799 3698 y Fo(>)e(n)1969 3713 y Fl(2)2044 3698 y Fs(2)h Fq(N)52 b Fw(in)37 b(suc)m(h)i(a)e(w)m(a)m(y) i(that)e Fo(g)3156 3662 y Fp(n)3199 3671 y Fi(1)3237 3698 y Fo(f)47 b Fw(=)36 b Fo(g)3495 3662 y Fp(n)3538 3671 y Fi(2)3576 3698 y Fo(f)11 b Fw(.)0 3814 y(That)37 b(is,)h(if)e Fo(n)f Fw(:=)h Fo(n)763 3829 y Fl(1)827 3814 y Fs(\000)26 b Fo(n)988 3829 y Fl(2)1063 3814 y Fo(>)35 b Fw(0,)j Fo(g)1339 3778 y Fp(n)1385 3814 y Fo(f)46 b Fw(=)35 b Fo(f)48 b Fw(and)37 b(so,)i(b)m(y)e Fz(G3)p Fw(,)h Fo(g)g Fs(2)e Fo(H)44 b Fw(and)37 b(so)h Fo(i)f Fw(and)g Fo(j)43 b Fw(are)37 b(the)0 3930 y(same)32 b(in)g Fo(G=H)8 b Fw(,)32 b(yielding)g(\(45\).)0 4046 y(Moreo)m(v)m(er,)38 b(b)m(y)f Fz(H10)f Fw(and)g(\(27\))o(,)h Fo(\016)t Fw(\()p Fo(iF)14 b Fw(\))33 b Fs(\024)h Fo(R)q Fw(,)j(for)e(an)m(y)h Fo(i)e Fs(2)g Fo(G=H)8 b Fw(.)52 b(Also,)37 b(since)f Fo(F)47 b Fw(=)33 b(\003)p Fo(=G)p Fw(,)j(w)m(e)0 4163 y(ha)m(v)m(e)31 b(that,)f(for)f(an)m(y)i Fo(i)d Fs(2)g Fo(G=H)8 b Fw(,)29 b(there)i(exists)f Fo(h)e Fs(2)g Fo(H)37 b Fw(in)29 b(suc)m(h)j(a)d(w)m(a)m(y)i(that)e(\()p Fo(hiF)14 b Fw(\))i Fs(\\)g Fw(\(\003)p Fo(=H)8 b Fw(\))28 b Fs(6)p Fw(=)g Fs(;)p Fw(.)0 4279 y(Th)m(us,)34 b(w)m(e)g(deduce)g(from)d Fz(H6)i Fw(that)1304 4594 y Fo(S)1370 4553 y Fp(R)1364 4619 y(\033)n(;H)1490 4594 y Fw(\()p Fo(v)t Fw(\))44 b Fs(\025)1823 4500 y Fn(X)1782 4716 y Fp(i)p Fm(2)p Fp(G=H)2023 4594 y Fo(H)2104 4609 y Fp(iF)2187 4594 y Fw(\()p Fo(v)t Fw(\))17 b Fo(:)0 5003 y Fw(Moreo)m(v)m(er,)34 b(b)m(y)g Fz(H7)p Fw(,)532 5318 y Fo(S)598 5277 y Fp(R)592 5343 y(\033)n(;H)718 5318 y Fw(\()p Fo(\033)t Fw(\))27 b Fs(\024)1095 5224 y Fn(X)1101 5424 y Fk(B)r Fj(2S)996 5480 y Fk(B)r Fj(\\)p Fi(\(\003)p Fk(=H)t Fi(\))p Fj(6)p Fi(=)p Fj(;)1365 5318 y Fo(&)8 b Fw(\()p Fo(B)d Fw(\))17 b(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)1777 5270 y Fp(#)1845 5318 y Fw(+)22 b(1\))28 b Fs(\024)g Fo(S)22 b Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)2442 5270 y Fp(#)2510 5318 y Fw(+)22 b(1\))g Fs(\001)g Fo(])p Fw(\(\003)p Fo(=H)8 b Fw(\))17 b Fo(:)p eop %%Page: 30 30 30 29 bop 168 251 a Ft(30)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Hence,)34 b(w)m(e)f(exploit)f(Lemma)f(27)h(to)g(conclude)h(that)1210 523 y Fn(\022)1283 664 y Fo(\024)1356 553 y Fn(\020)1479 664 y Fw(max)1432 730 y Fp(j;h)p Fm(2)p Fl(\003)p Fp(=G)1724 664 y Fs(j)p Fo(v)1799 679 y Fp(j)1857 664 y Fs(\000)23 b Fo(v)2004 679 y Fp(h)2049 664 y Fs(j)e(\000)i Fw(2)2247 553 y Fn(\021)2306 575 y Fp(\022)2368 664 y Fs(\000)f Fo(K)2557 523 y Fn(\023)2653 664 y Fs(\001)g Fo(])p Fw(\()p Fo(G=H)8 b Fw(\))1049 940 y Fs(\024)1250 846 y Fn(X)1210 1061 y Fp(i)p Fm(2)p Fp(G=H)1451 800 y Fn(\022)1524 940 y Fo(\024)1597 830 y Fn(\020)1720 940 y Fw(max)1673 1007 y Fp(j;h)p Fm(2)p Fl(\003)p Fp(=G)1965 940 y Fs(j)p Fo(v)2040 955 y Fp(j)2098 940 y Fs(\000)23 b Fo(v)2245 955 y Fp(h)2290 940 y Fs(j)e(\000)i Fw(2)2488 830 y Fn(\021)2547 852 y Fp(\022)2609 940 y Fs(\000)f Fo(K)2798 800 y Fn(\023)1049 1230 y Fs(\024)1250 1135 y Fn(X)1210 1351 y Fp(i)p Fm(2)p Fp(G=H)1451 1230 y Fo(H)1532 1245 y Fp(iF)1614 1230 y Fw(\()p Fo(v)t Fw(\))1049 1502 y Fs(\024)84 b Fo(S)1276 1460 y Fp(R)1270 1526 y(\033)n(;H)1395 1502 y Fw(\()p Fo(v)t Fw(\))1049 1663 y Fs(\024)g Fo(S)1276 1622 y Fp(R)1270 1688 y(\033)n(;H)1395 1663 y Fw(\()p Fo(\033)t Fw(\))1049 1832 y Fs(\024)g Fo(S)22 b Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)1489 1784 y Fp(#)1557 1832 y Fw(+)22 b(1\))g Fs(\001)g Fo(])p Fw(\(\003)p Fo(=H)8 b Fw(\))1050 1989 y(=)84 b Fo(S)22 b Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)1489 1941 y Fp(#)1557 1989 y Fw(+)22 b(1\))g Fs(\001)g Fo(])p Fw(\(\003)p Fo(=G)p Fw(\))f Fs(\001)h Fo(])p Fw(\()p Fo(G=H)8 b Fw(\))17 b Fo(;)168 2147 y Fw(whic)m(h)33 b(yields)f(the)h(desired)g(result.)2233 b Fg(\003)168 2326 y Fz(Lemma)37 b(29.)42 b Fr(L)-5 b(et)35 b Fo(K)1008 2289 y Fp(?)1001 2350 y(\033)1083 2326 y Fr(b)-5 b(e)34 b(as)h(in)42 b Fw(\(7\))o Fr(.)j(Then,)1466 2493 y Fs(j)p Fo(u)1550 2446 y Fm(\003)p Fp(;\033)n(;R)1550 2518 y(g)r(i)1742 2493 y Fs(\000)23 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))p Fs(j)j(\024)i Fo(K)2277 2452 y Fp(?)2270 2517 y(\033)2339 2493 y Fs(\000)23 b Fw(1)17 b Fo(;)168 2656 y Fr(for)34 b(al)5 b(l)35 b Fo(i)28 b Fs(2)g Fw(\003)p Fo(=G)p Fr(.)168 2802 y Fz(Pro)s(of.)42 b Fw(First,)29 b(note)g(that,)g(b)m(y)h(the)g(constrain)m(t)f(in)f(the)h(righ)m(t)f (hand)i(side)f(of)35 b(\(36\))o(,)30 b Fo(u)3314 2755 y Fm(\003)p Fp(;\033)n(;R)3314 2828 y(g)r(i)3512 2802 y Fs(\025)e Fo(\033)t Fw(\()p Fo(g)t Fw(\),)168 2924 y(for)k(an)m(y)h Fo(g)e Fs(2)d Fo(G)33 b Fw(and)f(an)m(y)h Fo(i)28 b Fs(2)h Fw(\003)p Fo(=G)p Fw(.)168 3040 y(In)k(fact,)f(w)m(e)i (also)d(ha)m(v)m(e)j(that)f(there)g(exist)g Fo(i)1798 3004 y Fm(\003)1865 3040 y Fs(2)28 b Fw(\003)p Fo(=G)k Fw(and)h Fo(g)2426 3004 y Fm(\003)2492 3040 y Fs(2)28 b Fo(G)33 b Fw(in)f(suc)m(h)i(a)e(w)m(a)m(y)i(that)168 3207 y(\(46\))1190 b Fo(u)1588 3160 y Fm(\003)p Fp(;\033)n(;R)1588 3233 y(g)1624 3214 y Fj(\003)1659 3233 y Fp(i)1683 3214 y Fj(\003)1780 3207 y Fs(\000)23 b Fo(\033)t Fw(\()p Fo(g)2028 3166 y Fm(\003)2066 3207 y Fw(\))28 b Fs(2)g Fw([0)p Fo(;)17 b Fw(1])g Fo(:)168 3388 y Fw(T)-8 b(o)35 b(pro)m(v)m(e)h(this,)g(w)m(e)g(argue)f(b)m(y)h(con)m(tradiction,)f (supp)s(osing)g(that)g Fo(u)2702 3341 y Fm(\003)p Fp(;\033)n(;R)2702 3414 y(g)r(i)2896 3388 y Fs(\000)24 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))32 b Fs(\025)g Fw(1)j(for)f(all)f Fo(i)g Fs(2)168 3510 y Fw(\003)p Fo(=G)f Fw(and)g Fo(g)f Fs(2)d Fo(G)p Fw(.)44 b(Then,)1281 3677 y(\()p Fs(R)1403 3692 y Fm(\000)p Fl(1)1497 3677 y Fo(u)1553 3636 y Fm(\003)p Fp(;\033)n(;R)1724 3677 y Fw(\))1762 3692 y Fp(g)r(i)1854 3677 y Fw(=)27 b Fo(u)2013 3630 y Fm(\003)p Fp(;\033)n(;R)2013 3703 y(g)r(i)2206 3677 y Fs(\000)c Fw(1)k Fs(\025)h Fo(\033)t Fw(\()p Fo(g)t Fw(\))17 b Fo(;)168 3847 y Fw(that)38 b(is,)h Fs(R)601 3862 y Fm(\000)p Fl(1)695 3847 y Fo(u)751 3811 y Fm(\003)p Fp(;\033)n(;R)960 3847 y Fw(satis\014es)h(the)f (constrain)m(ts)g(in)f(the)h(righ)m(t)f(hand)h(side)g(of)45 b(\(36\))38 b(and)h(so,)h(due)168 3963 y(to)33 b Fz(H3)h Fw(and)g Fz(H4)p Fw(,)g(it)e(is)i(also)f(a)g(minimizer.)44 b(Since)34 b Fs(R)2176 3978 y Fm(\000)p Fl(1)2271 3963 y Fo(u)2327 3927 y Fm(\003)p Fp(;\033)n(;R)2527 3963 y Fo(<)c(u)2689 3927 y Fm(\003)p Fp(;\033)n(;R)2860 3963 y Fw(,)k(this)f(w)m(ould)h(con)m(tradict)168 4079 y(the)f(fact)f(that)g Fo(u)795 4043 y Fm(\003)p Fp(;\033)n(;R)998 4079 y Fw(is)h(the)g (in\014mal)d(minimizer.)40 b(This)33 b(yields)f(the)h(pro)s(of)f(of)39 b(\(46\))o(.)168 4196 y(No)m(w,)33 b(if)e Fo(])p Fw(\(\003)p Fo(=G)p Fw(\))c(=)h(1,)k(that)h(is,)f(if)f(\003)p Fo(=G)c Fw(=)h Fs(f)p Fo(i)1887 4159 y Fm(\003)1926 4196 y Fs(g)p Fw(,)33 b(then)g(\(46\))f(and)h(Corollary)d(26)j(imply)d(that)425 4363 y Fs(j)p Fo(u)509 4316 y Fm(\003)p Fp(;\033)n(;R)509 4388 y(g)r(i)569 4369 y Fj(\003)702 4363 y Fs(\000)22 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))p Fs(j)27 b(\024)h(j)p Fo(u)1231 4316 y Fm(\003)p Fp(;\033)n(;R)1231 4388 y(g)r(i)1291 4369 y Fj(\003)1423 4363 y Fs(\000)23 b Fo(u)1579 4316 y Fm(\003)p Fp(;\033)n(;R)1579 4388 y(g)1615 4369 y Fj(\003)1650 4388 y Fp(i)1674 4369 y Fj(\003)1772 4363 y Fs(\000)f Fo(\033)t Fw(\()p Fo(g)t Fw(\))g(+)g Fo(\033)t Fw(\()p Fo(g)2325 4322 y Fm(\003)2363 4363 y Fw(\))p Fs(j)g Fw(+)g Fs(j)p Fo(u)2633 4316 y Fm(\003)p Fp(;\033)n(;R)2633 4388 y(g)2669 4369 y Fj(\003)2704 4388 y Fp(i)2728 4369 y Fj(\003)2825 4363 y Fs(\000)h Fo(\033)t Fw(\()p Fo(g)3073 4322 y Fm(\003)3112 4363 y Fw(\))p Fs(j)k(\024)h Fw(2)22 b(+)g(1)17 b Fo(;)168 4521 y Fw(whic)m(h)33 b(is)f(the)h(desired)g (result)g(for)f(this)g(case.)168 4637 y(If,)44 b(on)e(the)h(other)f (hand,)h Fo(])p Fw(\(\003)p Fo(=G)p Fw(\))g Fs(\025)i Fw(2)c(w)m(e)j(exploit)d(\(46\))o(,)k(Corollary)40 b(26)i(and)g(Lemma)f (28)g(to)168 4753 y(obtain)31 b(that)192 4911 y Fs(j)p Fo(u)276 4864 y Fm(\003)p Fp(;\033)n(;R)276 4936 y(g)r(i)468 4911 y Fs(\000)23 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))p Fs(j)82 b(\024)h(j)p Fo(u)1108 4864 y Fm(\003)p Fp(;\033)n(;R)1108 4936 y(g)r(i)1301 4911 y Fs(\000)22 b Fo(u)1456 4864 y Fm(\003)p Fp(;\033)n(;R)1456 4936 y(g)r(i)1516 4917 y Fj(\003)1627 4911 y Fs(j)g Fw(+)g Fs(j)p Fo(u)1859 4864 y Fm(\003)p Fp(;\033)n(;R)1859 4936 y(g)r(i)1919 4917 y Fj(\003)2051 4911 y Fs(\000)h Fo(u)2207 4864 y Fm(\003)p Fp(;\033)n(;R)2207 4936 y(g)2243 4917 y Fj(\003)2278 4936 y Fp(i)2302 4917 y Fj(\003)2399 4911 y Fs(\000)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))e(+)h Fo(\033)t Fw(\()p Fo(g)2952 4870 y Fm(\003)2991 4911 y Fw(\))p Fs(j)g Fw(+)g Fs(j)p Fo(u)3261 4864 y Fm(\003)p Fp(;\033)n(;R)3261 4936 y(g)3297 4917 y Fj(\003)3332 4936 y Fp(i)3356 4917 y Fj(\003)3453 4911 y Fs(\000)h Fo(\033)t Fw(\()p Fo(g)3701 4870 y Fm(\003)3739 4911 y Fw(\))p Fs(j)864 5184 y(\024)83 b Fw(2)22 b(+)1193 5014 y Fn( )1282 5117 y Fo(S)g Fw(\()p Fs(j)-17 b(j)g(j)p Fo(\033)t Fs(j)g(j)g(j)1561 5069 y Fp(#)1629 5117 y Fw(+)22 b(1\))17 b Fo(]F)35 b Fw(+)22 b Fo(K)p 1282 5162 874 4 v 1691 5253 a(\024)2166 5014 y Fn(!)2244 5036 y Fl(1)p Fp(=\022)2376 5184 y Fw(+)g(2)g(+)g(1)17 b Fo(;)168 5428 y Fw(as)32 b(desired.)3107 b Fg(\003)p eop %%Page: 31 31 31 30 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(31)0 450 y Fw(2.7.)49 b Fz(Appro)m(ximating)29 b(irrational)g(co)s (cycles)i(with)f(rational)g(ones.)50 b Fw(W)-8 b(e)28 b(no)m(w)g(remo)m(v)m(e)g(the)0 566 y(assumption)f(\(made)g(on)h(page)g (16\))f(that)h(the)g(co)s(cycle)g Fo(\033)k Fw(is)27 b(rational.)39 b(W)-8 b(e)28 b(k)m(eep,)j(for)c(the)h(momen)m(t)0 683 y(the)38 b(assumption)f(that)h(the)g(in)m(teraction)f(is)g (\014nite)h(range.)59 b(The)39 b(\014nite)f(range)f(assumption)h(will)0 799 y(b)s(e)33 b(remo)m(v)m(ed)g(later.)0 952 y Fz(Lemma)40 b(30.)j Fr(Fix)36 b Fo(R)d(>)f Fw(0)p Fr(.)51 b(L)-5 b(et)38 b Fo(K)1367 915 y Fp(?)1360 976 y(\033)1444 952 y Fr(b)-5 b(e)36 b(as)h(in)44 b Fw(\(7\))p Fr(.)51 b(Then,)37 b(\014xe)-5 b(d)37 b(any)g(c)-5 b(o)g(cycle)36 b Fo(\033)t Fr(,)i(ther)-5 b(e)37 b(exists)0 1068 y(a)e(Birkho\013)e(c)-5 b(on\014gur)g(ation)34 b Fo(u)1103 1032 y Fm(\003)p Fp(;\033)n(;R)1309 1068 y Fr(such)g(that)0 1261 y Fw(\(47\))1124 b Fs(j)p Fo(u)1382 1214 y Fm(\003)p Fp(;\033)n(;R)1382 1287 y(g)r(i)1574 1261 y Fs(\000)23 b Fo(\033)t Fw(\()p Fo(g)t Fw(\))p Fs(j)j(\024)i Fo(K)2109 1220 y Fp(?)2102 1286 y(\033)2171 1261 y Fs(\000)23 b Fw(1)17 b Fo(;)0 1456 y Fr(for)35 b(al)5 b(l)34 b Fo(i)28 b Fs(2)g Fw(\003)p Fo(=G)p Fr(.)0 1572 y(A)n(lso,)33 b(given)f(any)h(\014nite)f(set)h Fo(X)j Fs(\032)28 b Fw(\003)33 b Fr(and)f(any)h(c)-5 b(on\014gur)g(ation)32 b Fo(\014)39 b Fr(such)33 b(that)g Fo(\014)2927 1587 y Fp(p)2994 1572 y Fw(=)28 b(0)33 b Fr(for)f(any)h Fo(p)28 b Fs(2)0 1688 y Fw(\003)22 b Fs(\000)g Fo(X)8 b Fr(,)35 b(one)f(has)h(that)0 1894 y Fw(\(48\))972 1799 y Fn(X)978 1999 y Fk(B)r Fj(2S)933 2048 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)934 2104 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1192 1894 y Fo(H)1273 1909 y Fp(B)1333 1894 y Fw(\()p Fo(u)1427 1852 y Fm(\003)p Fp(;\033)n(;R)1598 1894 y Fw(\))44 b Fs(\024)1861 1799 y Fn(X)1867 1999 y Fk(B)r Fj(2S)1821 2048 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1822 2104 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2080 1894 y Fo(H)2161 1909 y Fp(B)2222 1894 y Fw(\()p Fo(u)2316 1852 y Fm(\003)p Fp(;\033)n(;R)2508 1894 y Fw(+)22 b Fo(\014)6 b Fw(\))17 b Fo(:)0 2289 y Fw(The)34 b(fact)f(that)f Fo(u)661 2253 y Fm(\003)p Fp(;\033)n(;R)865 2289 y Fw(satis\014es)i(\(48\))e(means)h(that)g(it)f(is)g(a)h(ground)g (state)g(for)g(the)g(cut{o\013)g(in)m(ter-)0 2406 y(action.)0 2522 y Fz(Pro)s(of.)43 b Fw(Prop)s(osition)31 b(23)h(and)h(Lemma)e(29)h (yields)g(the)h(result)g(for)f(rational)e(co)s(cycle.)0 2638 y(Th)m(us,)j(let)c Fo(\033)35 b Fw(b)s(e)c(an)f(irrational)d(co)s (cycle,)32 b(and)e(let)g Fo(\033)1938 2602 y Fl(\()p Fp(n)p Fl(\))2071 2638 y Fw(b)s(e)g(as)h(in)f(Prop)s(osition)f(11.)42 b(Consider)31 b(the)0 2763 y(in\014mal)21 b(minimizer)g Fo(u)819 2727 y Fm(\003)p Fp(;\033)916 2703 y Fi(\()p Fk(n)p Fi(\))1006 2727 y Fp(;R)1107 2763 y Fw(in)i(the)i(ligh)m(t)d(of) h(Prop)s(osition)f(23)i(and)f(Lemma)g(29.)40 b(Since)24 b Fo(u)3315 2727 y Fm(\003)p Fp(;\033)3412 2703 y Fi(\()p Fk(n)p Fi(\))3502 2727 y Fp(;R)3584 2763 y Fs(\000)0 2879 y Fo(u)56 2902 y Fp(\033)98 2882 y Fi(\()p Fk(n)p Fi(\))231 2879 y Fw(b)s(elongs)38 b(to)g(the)h(top)s(ological)c(pro)s (duct)k(\005)1833 2894 y Fp(p)p Fm(2)p Fl(\003)1969 2879 y Fw([)p Fs(\000)p Fo(K)2163 2843 y Fp(?)2156 2916 y(\033)2198 2896 y Fi(\()p Fk(n)p Fi(\))2293 2879 y Fo(;)34 b(K)2444 2843 y Fp(?)2437 2916 y(\033)2479 2896 y Fi(\()p Fk(n)p Fi(\))2574 2879 y Fw(],)40 b(and)f(so)f(to)h(\005)3188 2894 y Fp(p)p Fm(2)p Fl(\003)3323 2879 y Fw([)p Fs(\000)p Fo(K)3517 2843 y Fp(?)3510 2904 y(\033)3584 2879 y Fs(\000)0 2999 y Fw(1)p Fo(;)33 b(K)199 2963 y Fp(?)192 3023 y(\033)243 2999 y Fw(+)t(1])23 b(as)h(long)f(as)h Fo(n)g Fw(is)f(large)f(enough,) 27 b(and)c(suc)m(h)j(a)d(set)h(is)g(compact)f(b)m(y)i(T)m(yc)m (hono\013)g(Theorem,)0 3129 y(w)m(e)39 b(deduce)g(that,)g(up)g(to)e (subsequences,)42 b Fo(u)1673 3093 y Fm(\003)p Fp(;\033)1770 3069 y Fi(\()p Fk(n)p Fi(\))1861 3093 y Fp(;R)1976 3129 y Fw(con)m(v)m(erges)e(to)e(a)f(suitable)g Fo(u)3054 3093 y Fm(\003)p Fp(;\033)n(;R)3263 3129 y Fw(p)s(oin)m(t)m(wise)0 3254 y(in)32 b(\003.)43 b(Of)32 b(course,)i Fo(u)772 3217 y Fm(\003)p Fp(;\033)n(;R)975 3254 y Fw(satis\014es)f(\(47\),)f (since)h(so)g(do)s(es)g Fo(u)2202 3217 y Fm(\003)p Fp(;\033)2299 3194 y Fi(\()p Fk(n)p Fi(\))2390 3217 y Fp(;R)2467 3254 y Fw(.)0 3370 y(Also,)f(b)m(y)i Fz(H8)p Fw(,)e(there)h(exists)h Fo(C)g(>)28 b Fw(0)k(in)g(suc)m(h)i(a)e(w)m(a)m(y)i(that,)e(for)g(an)m (y)i Fo(B)e Fs(2)d(S)7 b Fw(,)431 3572 y Fs(j)p Fo(H)540 3587 y Fp(B)600 3572 y Fw(\()p Fo(u)694 3531 y Fm(\003)p Fp(;\033)791 3508 y Fi(\()p Fk(n)p Fi(\))881 3531 y Fp(;R)981 3572 y Fw(+)22 b Fo(\014)6 b Fw(\))21 b Fs(\000)i Fo(H)1380 3587 y Fp(B)1441 3572 y Fw(\()p Fo(u)1535 3531 y Fm(\003)p Fp(;\033)n(;R)1727 3572 y Fw(+)f Fo(\014)6 b Fw(\))p Fs(j)22 b Fw(+)g Fs(j)p Fo(H)2181 3587 y Fp(B)2241 3572 y Fw(\()p Fo(u)2335 3531 y Fm(\003)p Fp(;\033)2432 3508 y Fi(\()p Fk(n)p Fi(\))2522 3531 y Fp(;R)2600 3572 y Fw(\))g Fs(\000)g Fo(H)2840 3587 y Fp(B)2901 3572 y Fw(\()p Fo(u)2995 3531 y Fm(\003)p Fp(;\033)n(;R)3165 3572 y Fw(\))p Fs(j)986 3753 y(\024)28 b Fo(C)1185 3659 y Fn(X)1187 3870 y Fp(p)p Fm(2)p Fp(B)1346 3753 y Fs(j)p Fo(u)1430 3712 y Fm(\003)p Fp(;\033)1527 3689 y Fi(\()p Fk(n)p Fi(\))1617 3712 y Fp(;R)1430 3778 y(p)1716 3753 y Fs(\000)23 b Fo(u)1872 3712 y Fm(\003)p Fp(;\033)n(;R)1872 3778 y(p)2042 3753 y Fs(j)0 4044 y Fw(and)33 b(so,)g(since)g Fo(B)k Fw(is)32 b(a)h(\014nite)f(set,)1179 4246 y(lim)1127 4306 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1382 4246 y Fo(H)1463 4261 y Fp(B)1524 4246 y Fw(\()p Fo(u)1618 4205 y Fm(\003)p Fp(;\033)1715 4182 y Fi(\()p Fk(n)p Fi(\))1805 4205 y Fp(;R)1905 4246 y Fw(+)22 b Fo(\014)6 b Fw(\))44 b(=)g Fo(H)2347 4261 y Fp(B)2407 4246 y Fw(\()p Fo(u)2501 4205 y Fm(\003)p Fp(;\033)n(;R)2694 4246 y Fw(+)22 b Fo(\014)6 b Fw(\))771 4462 y(and)251 b(lim)1127 4522 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1382 4462 y Fo(H)1463 4477 y Fp(B)1524 4462 y Fw(\()p Fo(u)1618 4421 y Fm(\003)p Fp(;\033)1715 4398 y Fi(\()p Fk(n)p Fi(\))1805 4421 y Fp(;R)1883 4462 y Fw(\))44 b(=)g Fo(H)2166 4477 y Fp(B)2226 4462 y Fw(\()p Fo(u)2320 4421 y Fm(\003)p Fp(;\033)n(;R)2491 4462 y Fw(\))17 b Fo(:)0 4690 y Fw(Therefore,)900 4895 y(lim)848 4955 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1163 4800 y Fn(X)1169 5000 y Fk(B)r Fj(2S)1124 5049 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1124 5105 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1382 4895 y Fo(H)1463 4910 y Fp(B)1524 4895 y Fw(\()p Fo(u)1618 4854 y Fm(\003)p Fp(;\033)1715 4830 y Fi(\()p Fk(n)p Fi(\))1805 4854 y Fp(;R)1905 4895 y Fw(+)22 b Fo(\014)6 b Fw(\))44 b(=)2325 4800 y Fn(X)2331 5000 y Fk(B)r Fj(2S)2286 5049 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2286 5105 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2544 4895 y Fo(H)2625 4910 y Fp(B)2686 4895 y Fw(\()p Fo(u)2780 4854 y Fm(\003)p Fp(;\033)n(;R)2972 4895 y Fw(+)22 b Fo(\014)6 b Fw(\))492 5270 y(and)251 b(lim)848 5330 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1163 5175 y Fn(X)1169 5375 y Fk(B)r Fj(2S)1124 5424 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1124 5480 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1382 5270 y Fo(H)1463 5285 y Fp(B)1524 5270 y Fw(\()p Fo(u)1618 5229 y Fm(\003)p Fp(;\033)1715 5205 y Fi(\()p Fk(n)p Fi(\))1805 5229 y Fp(;R)1883 5270 y Fw(\))44 b(=)2144 5175 y Fn(X)2150 5375 y Fk(B)r Fj(2S)2105 5424 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2106 5480 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2364 5270 y Fo(H)2445 5285 y Fp(B)2505 5270 y Fw(\()p Fo(u)2599 5229 y Fm(\003)p Fp(;\033)n(;R)2770 5270 y Fw(\))17 b Fo(;)p eop %%Page: 32 32 32 31 bop 168 251 a Ft(32)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(since,)36 b(b)m(y)g(Lemma)d(12,)i(the)h(ab)s(o)m(v)m(e)f(series)h(are)f(just)g (\014nite)g(sums.)51 b(It)35 b(follo)m(ws)f(that)g Fo(u)3400 414 y Fm(\003)p Fp(;\033)n(;R)3606 450 y Fw(satis-)168 575 y(\014es)f(\(48\),)f(since,)h(b)m(y)h(Prop)s(osition)d(23,)h(so)h (do)s(es)g Fo(u)2049 539 y Fm(\003)p Fp(;\033)2146 515 y Fi(\()p Fk(n)p Fi(\))2236 539 y Fp(;R)2314 575 y Fw(.)1411 b Fg(\003)168 1144 y Fw(2.8.)48 b Fz(Remo)m(ving)33 b(the)h(cut-o\013.) 49 b Fw(W)-8 b(e)30 b(no)m(w)g(complete)f(the)h(pro)s(of)f(of)g (Theorem)h(8)f(b)m(y)i(taking)e(the)168 1260 y(limit)k(as)j Fo(R)g Fs(!)e Fw(+)p Fs(1)p Fw(,)j(that)g(is,)g(w)m(e)g(will)e(remo)m (v)m(e)i(the)g(conditions)e(that)i(the)g(in)m(teraction)e(sets)j Fo(B)168 1376 y Fw(ha)m(v)m(e)c(to)e(satisfy)g Fo(\016)t Fw(\()p Fo(B)5 b Fw(\))28 b Fs(\024)g Fo(R)q Fw(.)168 1493 y(T)-8 b(o)38 b(this)h(exten)m(t,)i(giv)m(en)e(a)f(\(p)s(ossibly)g (irrational\))e(co)s(cycle)j Fo(\033)t Fw(,)h(let)e Fo(u)2757 1456 y Fm(\003)p Fp(;\033)n(;R)2966 1493 y Fw(b)s(e)h(as)f(in)g(Lemma)f (30.)168 1609 y(By)i(\(47\))e(and)h(T)m(yc)m(hono\013)h(Theorem,)h(w)m (e)e(ma)m(y)g(supp)s(ose)h(that)e Fo(u)2648 1573 y Fm(\003)p Fp(;\033)n(;R)2856 1609 y Fw(con)m(v)m(erges)j(to)d(some)h Fo(u)3728 1573 y Fm(\003)p Fp(;\033)168 1725 y Fw(p)s(oin)m(t)m(wise)32 b(in)g(\003,)g(p)s(ossibly)g(up)h(to)f(a)h(subsequence.)168 1841 y(W)-8 b(e)36 b(will)d(sho)m(w)k(that)e Fo(u)1041 1805 y Fm(\003)p Fp(;\033)1178 1841 y Fw(is)h(the)g(con\014guration)e (claimed)g(in)h(Theorem)h(8.)53 b(Indeed,)37 b Fo(u)3469 1805 y Fm(\003)p Fp(;\033)3606 1841 y Fw(satis-)168 1957 y(\014es)c(\(12\),)f(just)h(b)m(y)h(taking)d(the)i(limit)d(as)i Fo(R)d Fs(!)e Fw(+)p Fs(1)32 b Fw(in)g(\(47\).)168 2074 y(Then,)i(the)f(pro)s(of)e(of)h(Theorem)h(8)g(is)f(completed)g(once)h (w)m(e)g(sho)m(w)h(that)f Fo(u)2916 2038 y Fm(\003)p Fp(;\033)3049 2074 y Fw(is)f(a)h(ground)f(state.)168 2190 y(Pro)m(ving)26 b(that)g Fo(u)787 2154 y Fm(\003)p Fp(;\033)914 2190 y Fw(is)g(a)g(minimizer)d(of)i(the)i(energy)g (without)f(the)h(cut{o\013)f(is)f(somewhat)i(delicate.)168 2306 y(Of)e(course,)j(the)e(energies)g(considered)g(are)g(di\013eren)m (t.)41 b(the)26 b(reduced)h(energy)g(energy)g Fo(S)3381 2270 y Fp(R)3375 2331 y(\033)n(;H)3526 2306 y Fw(de\014ned)168 2427 y(in)d(\(28\))g(is)h(di\013eren)m(t.)41 b(in)24 b(\(1\).)41 b(W)-8 b(e)25 b(also)g(note)g(that)g(w)m(e)h(ha)m(v)m(e)g (to)f(deal)f(with)h(the)g(fact)g(that)g(when)h(w)m(e)168 2543 y(de\014ne)34 b(the)g(p)s(erio)s(di\014ed)e(energy)-8 b(,)35 b(w)m(e)f(are)g(considering)e(only)h(p)s(erio)s(di\014ed)f (con\014gurations.)46 b(If)33 b(w)m(e)168 2660 y(consider)e(increasing) f(the)i(range)f(of)f(the)h(in)m(teraction,)g(then)g(the)h(in)m (teracting)e(sets)i(will)c(b)s(e)k(m)m(uc)m(h)168 2776 y(larger)42 b(than)i(the)g(fundamen)m(tal)f(domain)f(of)h(the)h(p)s (erio)s(di\014cation.)74 b(Hence,)48 b(the)c(in)m(teractions)168 2892 y(could)32 b(b)s(e)h(coun)m(ted)g(man)m(y)g(times.)168 3008 y(As)38 b(w)m(e)h(will)c(see,)40 b(this)d(di\016cult)m(y)h(is)f (solv)m(ed)h(thanks)h(to)e(the)h(no-symmetry)g(breaking)f(result)h(of) 168 3125 y(Prop)s(osition)h(22)i(whic)m(h)h(mak)m(es)g(p)s(ossible)f (to)g(consider)g(large)f(fundamen)m(tal)h(domains)f(for)h(the)168 3241 y(p)s(erio)s(di\014cation)34 b(without)j(c)m(hanging)f(the)i (minimizers.)54 b(Then,)39 b(w)m(e)f(can)g(increase)f(the)h(range)f(of) 168 3357 y(the)c(p)s(erturbation)e(without)i(problems)e(of)i(o)m(v)m (ercoun)m(ting.)168 3473 y(W)-8 b(e)34 b(no)m(w)g(pro)s(ceed)h(to)e (the)h(formal)e(pro)s(of)h(that)g Fo(u)2024 3437 y Fm(\003)p Fp(;\033)2159 3473 y Fw(is)h(a)f(ground)h(state,)g(and)g(this)g(will)d (end)k(the)168 3590 y(pro)s(of)c(of)i(Theorem)f(8.)168 3706 y(F)-8 b(or)32 b(this,)h(\014x)h(a)f(\014nite)g(set)g Fo(X)k Fs(\032)29 b Fw(\003)k(and)g(consider)h(a)e(con\014guration)h Fo(v)k Fw(in)32 b(suc)m(h)j(a)d(w)m(a)m(y)j Fo(v)3499 3721 y Fp(p)3567 3706 y Fw(=)29 b Fo(u)3728 3670 y Fm(\003)p Fp(;\033)3728 3730 y(p)168 3826 y Fw(for)k(an)m(y)i Fo(p)30 b Fs(2)g Fw(\003)23 b Fs(\000)h Fo(X)8 b Fw(,)34 b(and)g(let)f Fo(\014)j Fw(:=)30 b Fo(v)d Fs(\000)c Fo(u)1808 3790 y Fm(\003)p Fp(;\033)1910 3826 y Fw(.)47 b(F)-8 b(or)34 b(a)f(\014xed)i Fo(R)d(>)d Fw(0,)35 b(let)e Fo(Y)55 b Fw(b)s(e)34 b(the)h(set)f(in)g(\(10\))o(;)168 3943 y(then,)f(for)f(an)m (y)h Fo(r)e Fs(\025)d Fo(R)q Fw(,)908 4100 y Fn(X)914 4300 y Fk(B)r Fj(2S)868 4351 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)869 4405 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1127 4195 y Fs(j)p Fo(H)1236 4210 y Fp(B)1296 4195 y Fw(\()p Fo(u)1390 4154 y Fm(\003)p Fp(;\033)1491 4195 y Fw(\))23 b Fs(\000)f Fo(H)1732 4210 y Fp(B)1793 4195 y Fw(\()p Fo(u)1887 4154 y Fm(\003)p Fp(;\033)n(;r)2037 4195 y Fw(\))p Fs(j)688 4617 y(\024)908 4523 y Fn(X)914 4723 y Fk(B)r Fj(2S)868 4773 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)869 4828 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1127 4523 y Fn(X)1130 4734 y Fp(p)p Fm(2)p Fp(B)1288 4482 y Fn(Z)1387 4508 y Fl(1)1343 4707 y(0)1443 4473 y Fn(\014)1443 4532 y(\014)1443 4592 y(\014)1443 4652 y(\014)1486 4550 y Fo(@)5 b(H)1623 4565 y Fp(B)p 1486 4594 199 4 v 1510 4686 a Fo(@)g(u)1622 4701 y Fp(p)1695 4617 y Fw(\()p Fo(\034)11 b(u)1842 4576 y Fm(\003)p Fp(;\033)1966 4617 y Fw(+)22 b(\(1)f Fs(\000)i Fo(\034)11 b Fw(\))p Fo(u)2419 4576 y Fm(\003)p Fp(;\033)n(;r)2570 4617 y Fw(\))2608 4473 y Fn(\014)2608 4532 y(\014)2608 4592 y(\014)2608 4652 y(\014)2675 4617 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)2935 4576 y Fm(\003)p Fp(;\033)2935 4642 y(p)3058 4617 y Fs(\000)h Fo(u)3214 4576 y Fm(\003)p Fp(;\033)n(;r)3214 4642 y(p)3364 4617 y Fs(j)688 5040 y(\024)848 4945 y Fn(X)851 5156 y Fp(p)p Fm(2)p Fp(Y)1013 4945 y Fn(X)1019 5145 y Fk(B)r Fj(2S)1024 5201 y Fk(B)r Fj(3)p Fk(p)1177 4904 y Fn(Z)1277 4930 y Fl(1)1232 5129 y(0)1333 4895 y Fn(\014)1333 4955 y(\014)1333 5015 y(\014)1333 5074 y(\014)1376 4972 y Fo(@)5 b(H)1513 4987 y Fp(B)p 1376 5017 V 1399 5108 a Fo(@)g(u)1511 5123 y Fp(p)1584 5040 y Fw(\()p Fo(\034)11 b(u)1731 4998 y Fm(\003)p Fp(;\033)1855 5040 y Fw(+)22 b(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2309 4998 y Fm(\003)p Fp(;\033)n(;r)2460 5040 y Fw(\))2498 4895 y Fn(\014)2498 4955 y(\014)2498 5015 y(\014)2498 5074 y(\014)2564 5040 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)2824 4998 y Fm(\003)p Fp(;\033)2824 5064 y(p)2947 5040 y Fs(\000)h Fo(u)3103 4998 y Fm(\003)p Fp(;\033)n(;r)3103 5064 y(p)3254 5040 y Fs(j)688 5363 y(\024)83 b Fo(C)942 5268 y Fn(X)944 5480 y Fp(p)p Fm(2)p Fp(Y)1103 5363 y Fs(j)p Fo(u)1187 5322 y Fm(\003)p Fp(;\033)1187 5388 y(p)1310 5363 y Fs(\000)22 b Fo(u)1465 5322 y Fm(\003)p Fp(;\033)n(;r)1465 5388 y(p)1616 5363 y Fs(j)17 b Fo(;)p eop %%Page: 33 33 33 32 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(33)0 450 y Fw(where)34 b Fz(H8)e Fw(has)h(b)s(een)g(used.)45 b(Then,)34 b(since,)f(b)m(y)g Fz(H10)p Fw(,)g Fo(Y)53 b Fw(is)33 b(a)f(\014nite)g(set,)999 642 y(lim)952 702 y Fp(r)r Fm(!)p Fl(+)p Fm(1)1257 547 y Fn(X)1263 747 y Fk(B)r Fj(2S)1218 798 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1219 852 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1477 642 y Fs(j)p Fo(H)1586 657 y Fp(B)1646 642 y Fw(\()p Fo(u)1740 601 y Fm(\003)p Fp(;\033)1841 642 y Fw(\))22 b Fs(\000)h Fo(H)2082 657 y Fp(B)2142 642 y Fw(\()p Fo(u)2236 601 y Fm(\003)p Fp(;\033)n(;r)2387 642 y Fw(\))p Fs(j)44 b Fw(=)g(0)17 b Fo(;)0 1012 y Fw(for)32 b(an)m(y)h(\014xed)h Fo(R)29 b(>)e Fw(0.)0 1128 y(Analogously)-8 b(,)31 b(if)h(w)m(e)h(set)h Fo(v)1009 1092 y Fl(\()p Fp(r)r Fl(\))1129 1128 y Fw(:=)27 b Fo(u)1315 1092 y Fm(\003)p Fp(;\033)n(;r)1488 1128 y Fw(+)22 b Fo(\014)6 b Fw(,)1084 1320 y(lim)1037 1380 y Fp(r)r Fm(!)p Fl(+)p Fm(1)1342 1225 y Fn(X)1348 1425 y Fk(B)r Fj(2S)1303 1476 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1304 1530 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1562 1320 y Fs(j)p Fo(H)1671 1335 y Fp(B)1731 1320 y Fw(\()p Fo(v)t Fw(\))22 b Fs(\000)g Fo(H)2060 1335 y Fp(B)2121 1320 y Fw(\()p Fo(v)2210 1279 y Fl(\()p Fp(r)r Fl(\))2302 1320 y Fw(\))p Fs(j)44 b Fw(=)g(0)17 b Fo(;)0 1690 y Fw(for)32 b(an)m(y)h(\014xed)h Fo(R)29 b(>)e Fw(0.)0 1806 y(Fix)32 b(no)m(w)h Fo(\017)28 b(>)g Fw(0.)43 b(From)31 b Fz(H10)p Fw(,)h(there)i(exists)f Fo(R)1784 1821 y Fp(\017)1845 1806 y Fo(>)27 b Fw(0)32 b(in)g(suc)m(h)i(a)f(w)m(a)m(y)1349 1998 y(sup)1329 2079 y Fp(u)p Fm(2O)1473 2087 y Fk(\033)1590 1903 y Fn(X)1597 2103 y Fk(B)r Fj(2S)1551 2154 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1552 2208 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fk(>R)1810 1998 y Fo(H)1891 2013 y Fp(B)1952 1998 y Fw(\()p Fo(u)p Fw(\))43 b Fs(\024)i Fo(\017)17 b(;)0 2367 y Fw(pro)m(vided)33 b(that)f Fo(R)d Fs(\025)f Fo(R)894 2382 y Fp(\017)927 2367 y Fw(.)0 2484 y(Therefore,)34 b(for)e(an)m(y)h Fo(R)c Fs(\025)f Fo(R)1084 2499 y Fp(\017)1117 2484 y Fw(,)898 2581 y Fn(X)905 2781 y Fk(B)r Fj(2S)859 2837 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1108 2675 y Fo(H)1189 2690 y Fp(B)1249 2675 y Fw(\()p Fo(u)1343 2634 y Fm(\003)p Fp(;\033)1445 2675 y Fw(\))22 b Fs(\000)1653 2581 y Fn(X)1660 2781 y Fk(B)r Fj(2S)1614 2837 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1863 2675 y Fo(H)1944 2690 y Fp(B)2005 2675 y Fw(\()p Fo(v)t Fw(\))689 2999 y Fs(\024)83 b Fw(2)p Fo(\017)23 b Fw(+)1117 2904 y Fn(X)1123 3104 y Fk(B)r Fj(2S)1077 3153 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1078 3209 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1336 2999 y Fo(H)1417 3014 y Fp(B)1478 2999 y Fw(\()p Fo(u)1572 2958 y Fm(\003)p Fp(;\033)1673 2999 y Fw(\))f Fs(\000)1892 2904 y Fn(X)1898 3104 y Fk(B)r Fj(2S)1853 3153 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1853 3209 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2111 2999 y Fo(H)2192 3014 y Fp(B)2253 2999 y Fw(\()p Fo(v)t Fw(\))690 3374 y(=)83 b(2)p Fo(\017)23 b Fw(+)69 b(lim)1058 3434 y Fp(r)r Fm(!)p Fl(+)p Fm(1)1363 3279 y Fn(X)1369 3479 y Fk(B)r Fj(2S)1324 3528 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1325 3584 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1583 3374 y Fo(H)1664 3389 y Fp(B)1724 3374 y Fw(\()p Fo(u)1818 3333 y Fm(\003)p Fp(;\033)n(;r)1969 3374 y Fw(\))22 b Fs(\000)2188 3279 y Fn(X)2194 3479 y Fk(B)r Fj(2S)2148 3528 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2149 3584 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2407 3374 y Fo(H)2488 3389 y Fp(B)2549 3374 y Fw(\()p Fo(v)2638 3333 y Fl(\()p Fp(r)r Fl(\))2730 3374 y Fw(\))690 3749 y(=)83 b(2)p Fo(\017)23 b Fw(+)69 b(lim)1058 3808 y Fp(r)r Fm(!)p Fl(+)p Fm(1)1363 3654 y Fn(X)1369 3854 y Fk(B)r Fj(2S)1324 3903 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1325 3959 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1583 3749 y Fo(H)1664 3764 y Fp(B)1724 3749 y Fw(\()p Fo(u)1818 3707 y Fm(\003)p Fp(;\033)n(;r)1969 3749 y Fw(\))22 b Fs(\000)2188 3654 y Fn(X)2194 3854 y Fk(B)r Fj(2S)2148 3903 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2149 3959 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2407 3749 y Fo(H)2488 3764 y Fp(B)2549 3749 y Fw(\()p Fo(u)2643 3707 y Fm(\003)p Fp(;\033)n(;r)2815 3749 y Fw(+)g Fo(\014)6 b Fw(\))17 b Fo(:)0 4124 y Fw(Then,)34 b(b)m(y)h(\(48\))o(,)1090 4186 y Fn(X)1096 4386 y Fk(B)r Fj(2S)1051 4443 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1300 4281 y Fo(H)1381 4296 y Fp(B)1441 4281 y Fw(\()p Fo(u)1535 4240 y Fm(\003)p Fp(;\033)1636 4281 y Fw(\))22 b Fs(\000)1845 4186 y Fn(X)1851 4386 y Fk(B)r Fj(2S)1806 4443 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2055 4281 y Fo(H)2136 4296 y Fp(B)2196 4281 y Fw(\()p Fo(v)t Fw(\))44 b Fs(\024)h Fw(2)p Fo(\017)17 b(:)0 4579 y Fw(Since)30 b Fo(\017)f Fw(ma)m(y)h(b)s(e)f(tak)m(en)i(arbitrarily)26 b(small,)i(w)m(e)j(ha)m(v)m(e)g(just)e(pro)m(v)m(ed)i(that)f Fo(u)2797 4543 y Fm(\003)p Fp(;\033)2927 4579 y Fw(is)f(a)g(ground)h (state,)0 4695 y(th)m(us)k(completing)c(the)j(pro)s(of)f(of)g(Theorem)h (8.)1009 4946 y(3.)48 b Fv(Lamina)-7 b(tions)39 b(of)f(gr)n(ound)g(st) -7 b(a)g(tes)0 5121 y Fw(W)f(e)32 b(w)m(ould)g(lik)m(e)g(no)m(w)g(to)g (stress)i(the)e(fact)g(that)g(the)g(ground)g(states)h(constructed)h(in) d(Theorem)i(8)0 5237 y(induce)g(a)f(\(p)s(ossibly)g(singular\))f (lamination)e(of)j Fo(F)k Fs(\002)22 b Fq(T)2070 5201 y Fl(1)2113 5237 y Fw(.)44 b(See)33 b(De\014nition)e(9.)0 5353 y(First)k(of)h(all,)g(w)m(e)h(pro)m(v)m(e)h(that)e(the)h(ground)f (states)i(are)e(a)g(closed)h(set)g(under)g(p)s(oin)m(t)m(wise)f(con)m (v)m(er-)0 5469 y(gence:)p eop %%Page: 34 34 34 33 bop 168 251 a Ft(34)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 453 y Fz(Lemma)37 b(31.)42 b Fr(L)-5 b(et)35 b Fo(u)974 417 y Fl(\()p Fp(n)p Fl(\))1103 453 y Fs(2)28 b(O)1276 468 y Fp(\033)1358 453 y Fr(b)-5 b(e)35 b(such)f(that)1692 626 y Fw(lim)1640 685 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1896 626 y Fo(u)1952 584 y Fl(\()p Fp(n)p Fl(\))1952 650 y Fp(p)2097 626 y Fw(=)45 b Fo(u)2274 641 y Fp(p)2329 626 y Fo(;)168 853 y Fr(for)34 b(any)h Fo(p)28 b Fs(2)g Fw(\003)p Fr(.)44 b(Supp)-5 b(ose)34 b(that)i(any)f Fo(u)1637 817 y Fl(\()p Fp(n)p Fl(\))1773 853 y Fr(is)f(a)h(gr)-5 b(ound)35 b(state.)45 b(Then)34 b(so)g(is)h Fo(u)p Fr(.)168 997 y Fz(Pro)s(of.)43 b Fw(Let)33 b Fo(\014)38 b Fw(b)s(e)33 b(a)f(con\014guration)g(whic)m (h)h(v)-5 b(anishes)33 b(outside)f(a)h(\014nite)f(set)h Fo(X)8 b Fw(.)44 b(Then,)33 b(if)1525 1170 y Fo(K)i Fw(:=)28 b Fo(K)1864 1129 y Fp(?)1857 1195 y(\033)1926 1170 y Fw(+)22 b(sup)2024 1251 y Fp(p)p Fm(2)p Fp(X)2187 1170 y Fs(j)p Fo(\014)6 b Fw(\()p Fo(p)p Fw(\))p Fs(j)17 b Fo(;)168 1408 y Fw(w)m(e)33 b(ha)m(v)m(e)h(that)1509 1540 y Fo(u)1565 1499 y Fl(\()p Fp(n)p Fl(\))1689 1540 y Fw(+)22 b Fo(\014)g(;)44 b(u)22 b Fw(+)g Fo(\014)33 b Fs(2)28 b(O)2375 1499 y Fp(K)2372 1564 y(\033)2460 1540 y Fo(:)168 1692 y Fw(Th)m(us,)34 b(\014x)f Fo(\017)28 b(>)g Fw(0)k(and)h(use)g Fz(H10)f Fw(to)h(\014nd)g Fo(R)c(>)e Fw(0)32 b(in)g(suc)m(h)i(a)f(w)m(a)m(y)g(that)1516 1881 y(sup)1488 1970 y Fp(u)p Fm(2O)1634 1951 y Fk(K)1632 1986 y(\033)1768 1786 y Fn(X)1774 1986 y Fk(B)r Fj(2S)1729 2037 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1729 2091 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fk(>R)1987 1881 y Fo(H)2068 1896 y Fp(B)2129 1881 y Fw(\()p Fo(u)p Fw(\))43 b Fs(\024)i Fo(\017)17 b(:)168 2247 y Fw(Observ)m(e)34 b(that)1119 2420 y Fs(j)p Fo(H)1228 2435 y Fp(B)1288 2420 y Fw(\()p Fo(u)1382 2379 y Fl(\()p Fp(n)p Fl(\))1484 2420 y Fw(\))22 b Fs(\000)g Fo(H)1724 2435 y Fp(B)1785 2420 y Fw(\()p Fo(u)p Fw(\))p Fs(j)959 2634 y(\024)1119 2540 y Fn(X)1121 2751 y Fp(p)p Fm(2)p Fp(B)1280 2499 y Fn(Z)1379 2525 y Fl(1)1335 2724 y(0)1435 2490 y Fn(\014)1435 2550 y(\014)1435 2610 y(\014)1435 2669 y(\014)1478 2567 y Fo(@)5 b(H)1615 2582 y Fp(B)p 1478 2611 199 4 v 1502 2703 a Fo(@)g(u)1614 2718 y Fp(p)1687 2634 y Fw(\()p Fo(\034)11 b(u)1834 2593 y Fl(\()p Fp(n)p Fl(\))1958 2634 y Fw(+)22 b(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(u)p Fw(\))2449 2490 y Fn(\014)2449 2550 y(\014)2449 2610 y(\014)2449 2669 y(\014)2516 2634 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)2776 2593 y Fl(\()p Fp(n)p Fl(\))2776 2659 y Fp(p)2899 2634 y Fs(\000)g Fo(u)3054 2649 y Fp(p)3094 2634 y Fs(j)959 2916 y(\024)83 b Fo(C)1229 2821 y Fn(X)1232 3033 y Fp(p)p Fm(2)p Fp(B)1390 2916 y Fs(j)p Fo(u)1474 2875 y Fl(\()p Fp(n)p Fl(\))1474 2941 y Fp(p)1597 2916 y Fs(\000)23 b Fo(u)1753 2931 y Fp(p)1792 2916 y Fs(j)p Fo(;)17 b(;)168 3190 y Fw(for)32 b(some)g Fo(C)j(>)27 b Fw(0,)33 b(thanks)g(to)g Fz(H8)p Fw(.)168 3306 y(Th)m(us,)h(since)f Fo(B)38 b Fw(is)32 b(a)g(\014nite)g(set,)1495 3481 y(lim)1443 3541 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1698 3481 y Fo(H)1779 3496 y Fp(B)1840 3481 y Fw(\()p Fo(u)1934 3440 y Fl(\()p Fp(n)p Fl(\))2035 3481 y Fw(\))44 b(=)g Fo(H)2318 3496 y Fp(B)2379 3481 y Fw(\()p Fo(u)p Fw(\))17 b Fo(:)168 3691 y Fw(Therefore,)1216 3839 y(lim)1164 3899 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1479 3745 y Fn(X)1485 3945 y Fk(B)r Fj(2S)1440 3994 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1440 4050 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1698 3839 y Fo(H)1779 3854 y Fp(B)1840 3839 y Fw(\()p Fo(u)1934 3798 y Fl(\()p Fp(n)p Fl(\))2035 3839 y Fw(\))44 b(=)2297 3745 y Fn(X)2303 3945 y Fk(B)r Fj(2S)2257 3994 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2258 4050 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2516 3839 y Fo(H)2597 3854 y Fp(B)2658 3839 y Fw(\()p Fo(u)p Fw(\))17 b Fo(;)168 4186 y Fw(since)33 b(the)g(ab)s(o)m(v)m(e)g(series) g(are)g(\014nite)f(sums,)h(due)h(to)e Fz(H10)p Fw(.)43 b(Analogously)-8 b(,)1035 4374 y(lim)983 4434 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1298 4280 y Fn(X)1304 4480 y Fk(B)r Fj(2S)1259 4529 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1260 4585 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1518 4374 y Fo(H)1599 4389 y Fp(B)1659 4374 y Fw(\()p Fo(u)1753 4333 y Fl(\()p Fp(n)p Fl(\))1876 4374 y Fw(+)22 b Fo(\014)6 b Fw(\))44 b(=)2297 4280 y Fn(X)2303 4480 y Fk(B)r Fj(2S)2257 4529 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2258 4585 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2516 4374 y Fo(H)2597 4389 y Fp(B)2658 4374 y Fw(\()p Fo(u)21 b Fw(+)h Fo(\014)6 b Fw(\))17 b Fo(:)168 4758 y Fw(Also,)32 b(since)h Fo(u)710 4722 y Fl(\()p Fp(n)p Fl(\))844 4758 y Fw(is)f(a)g(ground)h(state,)943 4946 y(0)83 b Fs(\024)1285 4852 y Fn(X)1291 5052 y Fk(B)r Fj(2S)1246 5108 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1495 4946 y Fo(H)1576 4961 y Fp(B)1636 4946 y Fw(\()p Fo(u)1730 4905 y Fl(\()p Fp(n)p Fl(\))1853 4946 y Fw(+)22 b Fo(\014)6 b Fw(\))22 b Fs(\000)2221 4852 y Fn(X)2227 5052 y Fk(B)r Fj(2S)2182 5108 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2430 4946 y Fo(H)2511 4961 y Fp(B)2572 4946 y Fw(\()p Fo(u)2666 4905 y Fl(\()p Fp(n)p Fl(\))2767 4946 y Fw(\))1075 5270 y Fs(\024)84 b Fw(2)p Fo(\017)22 b Fw(+)1503 5175 y Fn(X)1509 5375 y Fk(B)r Fj(2S)1464 5424 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1465 5480 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1723 5270 y Fo(H)1804 5285 y Fp(B)1864 5270 y Fw(\()p Fo(u)1958 5229 y Fl(\()p Fp(n)p Fl(\))2082 5270 y Fw(+)g Fo(\014)6 b Fw(\))21 b Fs(\000)2459 5175 y Fn(X)2465 5375 y Fk(B)r Fj(2S)2420 5424 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2421 5480 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2679 5270 y Fo(H)2760 5285 y Fp(B)2820 5270 y Fw(\()p Fo(u)2914 5229 y Fl(\()p Fp(n)p Fl(\))3015 5270 y Fw(\))p eop %%Page: 35 35 35 34 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(35)0 450 y Fw(and)33 b(so,)g(b)m(y)g(passing)g(to)f(the)h(limit,)876 654 y(0)82 b Fs(\024)i Fw(2)p Fo(\017)22 b Fw(+)1435 560 y Fn(X)1442 760 y Fk(B)r Fj(2S)1396 809 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1397 865 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)1655 654 y Fo(H)1736 669 y Fp(B)1796 654 y Fw(\()p Fo(u)g Fw(+)g Fo(\014)6 b Fw(\))22 b Fs(\000)2289 560 y Fn(X)2296 760 y Fk(B)r Fj(2S)2250 809 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2251 865 y Fk(\016)r Fi(\()p Fk(B)r Fi(\))p Fj(\024)p Fk(R)2509 654 y Fo(H)2590 669 y Fp(B)2650 654 y Fw(\()p Fo(u)p Fw(\))1007 1029 y Fs(\024)84 b Fw(4)p Fo(\017)22 b Fw(+)1425 935 y Fn(X)1432 1135 y Fk(B)r Fj(2S)1386 1191 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1635 1029 y Fo(H)1716 1044 y Fp(B)1777 1029 y Fw(\()p Fo(u)f Fw(+)h Fo(\014)6 b Fw(\))22 b Fs(\000)2260 935 y Fn(X)2266 1135 y Fk(B)r Fj(2S)2220 1191 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)2469 1029 y Fo(H)2550 1044 y Fp(B)2611 1029 y Fw(\()p Fo(u)p Fw(\))17 b Fo(;)0 1360 y Fw(whic)m(h)33 b(giv)m(es)g(the)g(desired)g(result)g(b)m(y)g(taking)f Fo(\017)h Fw(as)g(small)d(as)j(w)m(e)g(wish.)899 b Fg(\003)0 1548 y Fw(W)-8 b(e)33 b(no)m(w)g(p)s(oin)m(t)f(out)g(a)h(useful)f(prop) s(ert)m(y)i(of)e(Birkho\013)g(functions:)0 1701 y Fz(Lemma)42 b(32.)i Fr(If)39 b Fo(u)c Fs(2)h(B)954 1716 y Fp(\033)1001 1701 y Fr(,)k Fo(g)1118 1716 y Fl(1)1157 1701 y Fo(;)17 b(g)1248 1716 y Fl(2)1322 1701 y Fs(2)36 b Fo(G)j Fr(and)g Fo(`)1775 1716 y Fl(1)1814 1701 y Fo(;)17 b(`)1899 1716 y Fl(2)1974 1701 y Fs(2)36 b Fq(Z)p Fr(,)h(then)i(either)g Fs(R)2799 1716 y Fp(`)2828 1725 y Fi(1)2867 1701 y Fs(T)2921 1716 y Fp(g)2955 1725 y Fi(1)2993 1701 y Fo(u)c Fs(\025)h(R)3281 1716 y Fp(`)3310 1725 y Fi(2)3349 1701 y Fs(T)3403 1716 y Fp(g)3437 1725 y Fi(2)3476 1701 y Fo(u)j Fr(or)0 1817 y Fs(R)84 1832 y Fp(`)113 1841 y Fi(1)152 1817 y Fs(T)206 1832 y Fp(g)240 1841 y Fi(1)279 1817 y Fo(u)27 b Fs(\024)h(R)551 1832 y Fp(`)580 1841 y Fi(2)619 1817 y Fs(T)673 1832 y Fp(g)707 1841 y Fi(2)746 1817 y Fo(u)p Fr(.)0 1969 y Fz(Pro)s(of.)49 b Fw(Let)34 b Fo(u)c Fs(2)i(B)777 1984 y Fp(\033)824 1969 y Fw(.)49 b(Since)34 b Fo(\033)39 b Fw(is)33 b(a)i(co)s(cycle,)g(b)m(y)g(p)s(ossibly)f(exc)m(hanging)h (the)g(indices,)f(w)m(e)i(ma)m(y)0 2085 y(supp)s(ose)h(that)f Fo(\033)t Fw(\()p Fo(g)729 2100 y Fl(1)768 2085 y Fo(g)819 2044 y Fm(\000)p Fl(1)815 2110 y(2)913 2085 y Fw(\))e Fs(\025)g Fo(`)1137 2100 y Fl(2)1201 2085 y Fs(\000)25 b Fo(`)1344 2100 y Fl(1)1383 2085 y Fw(.)54 b(Then,)38 b(b)m(y)f(the)g(Birkho\013)e(prop)s(ert)m(y)-8 b(,)37 b Fs(T)2917 2113 y Fp(g)2951 2122 y Fi(1)2985 2113 y Fp(g)3021 2083 y Fj(\000)p Fi(1)3019 2134 y(2)3108 2085 y Fo(u)c Fs(\025)h(R)3392 2100 y Fp(`)3421 2109 y Fi(2)3456 2100 y Fm(\000)p Fp(`)3540 2109 y Fi(1)3579 2085 y Fo(u)p Fw(,)0 2211 y(and)f(so)f Fs(R)393 2226 y Fp(`)422 2235 y Fi(1)461 2211 y Fs(T)515 2226 y Fp(g)549 2235 y Fi(1)588 2211 y Fo(u)27 b Fs(\025)h(R)860 2226 y Fp(`)889 2235 y Fi(2)928 2211 y Fs(T)982 2226 y Fp(g)1016 2235 y Fi(2)1055 2211 y Fo(u)p Fw(.)2413 b Fg(\003)0 2436 y Fz(Corollary)37 b(33.)42 b Fr(L)-5 b(et)35 b Fo(u)g Fr(b)-5 b(e)34 b(a)h(Birkho\013)f (function.)44 b(F)-7 b(or)34 b(any)h Fo(g)c Fs(2)d Fo(G)35 b Fr(and)f Fo(`)28 b Fs(2)g Fq(Z)p Fr(,)k(let)1119 2650 y Fs(L)1188 2665 y Fp(`;g)1304 2650 y Fw(:=)1435 2539 y Fn(n\020)1561 2650 y Fo(p;)17 b Fw(\()p Fs(R)1776 2665 y Fp(`)1809 2650 y Fs(T)1863 2665 y Fp(g)1903 2650 y Fo(u)p Fw(\))1997 2665 y Fp(p)2036 2539 y Fn(\021)2124 2535 y(\014)2124 2595 y(\014)2124 2655 y(\014)2184 2650 y Fo(p)28 b Fs(2)g Fo(F)2432 2539 y Fn(o)2515 2650 y Fo(:)0 2864 y Fr(L)-5 b(et)35 b Fs(L)g Fr(b)-5 b(e)34 b(the)h(closur)-5 b(e)35 b(of)1667 2933 y Fn([)1661 3133 y Fk(g)r Fj(2)p Fk(G)1672 3189 y(`)p Fj(2)p Fe(Z)1809 3028 y Fs(L)1878 3043 y Fp(`;g)1983 3028 y Fo(:)0 3338 y Fr(Then,)f Fs(L)h Fr(is)f(a)h(\(p)-5 b(ossibly)34 b(singular\))g (lamination)g(on)g Fo(F)i Fs(\002)23 b Fq(R)5 b Fr(.)0 3455 y(A)n(lso,)39 b(if)g Fw(\()p Fo(p;)17 b(r)s Fw(\))34 b Fs(2)h(L)p Fr(,)k(then)g Fw(\()p Fo(p;)17 b(r)27 b Fw(+)e Fo(s)p Fw(\))35 b Fs(2)g(L)j Fr(for)h(any)f Fo(s)d Fs(2)h Fq(Z)p Fr(,)h(thus)i Fs(L)f Fr(may)h(b)-5 b(e)38 b(quotiente)-5 b(d)39 b(to)g(a)0 3571 y(lamination)34 b(of)g Fo(F)i Fs(\002)23 b Fq(T)867 3535 y Fl(1)910 3571 y Fr(.)0 3723 y Fz(Pro)s(of.)45 b Fw(The)34 b(\014rst)f(claim)e(follo)m (ws)h(from)g(Lemma)f(32.)45 b(The)34 b(second)g(one)g(from)d(the)j (observ)-5 b(ation)0 3839 y(that)32 b(if)g(\()p Fo(p;)17 b(r)s Fw(\))27 b Fs(2)h(L)707 3854 y Fp(`;g)795 3839 y Fw(,)33 b(then)g(\()p Fo(p;)17 b(r)24 b Fw(+)e Fo(s)p Fw(\))28 b Fs(2)g(L)1649 3854 y Fp(`)p Fl(+)p Fp(s;g)1825 3839 y Fw(.)1699 b Fg(\003)0 4028 y Fw(As)33 b(a)f(consequence)k(of)c (Corollary)f(33)h(and)h(Lemma)e(31,)h(w)m(e)i(obtain:)0 4180 y Fz(Corollary)j(34.)k Fr(L)-5 b(et)34 b Fo(u)g Fr(b)-5 b(e)34 b(a)g(Birkho\013)f(gr)-5 b(ound)34 b(state.)44 b(Then)34 b(the)g(\(p)-5 b(ossibly)33 b(singular\))h(lamina-)0 4296 y(tion)h(of)f(Cor)-5 b(ol)5 b(lary)35 b(33)f(is)h(made)f(of)g(gr) -5 b(ound)35 b(states.)0 4448 y Fw(Let)0 4636 y(\(49\))1076 b Fo(H)52 b Fw(:=)44 b Fs(f)p Fo(g)31 b Fs(2)d Fo(G)44 b Fs(j)g Fo(\033)t Fw(\()p Fo(g)t Fw(\))27 b Fs(2)h Fq(Z)p Fs(g)17 b Fo(:)0 4825 y Fw(It)33 b(is)f(easily)g(seen)h(that,)g(if)e Fo(u)d Fs(2)g(B)1255 4840 y Fp(\033)1302 4825 y Fw(,)33 b(then)0 5013 y(\(50\))1356 b Fs(T)1584 5028 y Fp(h)1629 5013 y Fo(u)44 b Fw(=)g Fs(R)1933 5028 y Fp(\033)r Fl(\()p Fp(h)p Fl(\))2076 5013 y Fo(u)0 5201 y Fw(for)32 b(an)m(y)i Fo(h)28 b Fs(2)g Fo(H)8 b Fw(,)33 b(so)g(all)d(the)k(v)-5 b(alues)32 b(of)h Fo(u)f Fw(are)h(determined)f(b)m(y)i(the)f(v)-5 b(alues)33 b(in)f(\003)p Fo(=H)8 b Fw(.)43 b(This)33 b(giv)m(es)0 5317 y(another)g(c)m(haracterization)e(of)h(the)h(group)g Fo(H)8 b Fw(:)0 5469 y Fz(Lemma)37 b(35.)42 b Fo(H)35 b Fw(=)28 b Fs(f)p Fo(g)i Fs(2)e Fo(G)35 b Fr(s.t.)g Fs(9)p Fo(s)28 b Fs(2)g Fq(Z)k Fr(s.t.)45 b Fs(T)1857 5484 y Fp(g)1898 5469 y Fo(u)27 b Fw(=)g Fs(R)2168 5484 y Fp(s)2206 5469 y Fo(u)34 b Fs(8)h Fo(u)27 b Fs(2)i(B)2629 5484 y Fp(\033)2676 5469 y Fs(g)p Fr(.)p eop %%Page: 36 36 36 35 bop 168 251 a Ft(36)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fz(Pro)s(of.)41 b Fw(Let)27 b Fo(g)i Fw(b)s(e)e(in)e(the)i(set)g(describ)s(ed)h(b)m(y)f (Lemma)e(35.)41 b(Then,)29 b(since)e Fo(u)2928 465 y Fp(\033)3002 450 y Fs(2)h(B)3161 465 y Fp(\033)3208 450 y Fw(,)g(for)e(an)m(y)h Fo(q)k Fs(2)d Fo(F)782 616 y(\033)t Fw(\()p Fo(g)t Fw(\))e(=)i Fo(u)1154 631 y Fp(\033)1200 616 y Fw(\()p Fo(g)t(q)t Fw(\))f(=)g(\()p Fs(T)1596 631 y Fp(g)1637 616 y Fo(u)1693 631 y Fp(\033)1739 616 y Fw(\))1777 631 y Fp(q)1843 616 y Fw(=)g(\()p Fs(R)2068 631 y Fp(s)2106 616 y Fo(u)2162 631 y Fp(\033)2208 616 y Fw(\))2246 631 y Fp(q)2312 616 y Fw(=)g Fo(u)2471 631 y Fp(\033)2517 616 y Fw(\()p Fo(q)t Fw(\))22 b(+)g Fo(s)28 b Fw(=)g Fo(s)f Fs(2)h Fq(Z)17 b Fo(;)168 782 y Fw(due)33 b(to)f(\(6\),)g(so)h Fo(g)j Fw(b)s(elongs)c(to)g(the)h(set)h(in)d (\(49\).)168 898 y(Vicev)m(ersa,)j(let)d Fo(g)36 b Fw(b)s(e)d(in)f(set) h(describ)s(ed)g(b)m(y)i(\(49\))o(.)44 b(Then,)34 b(it)d(also)h(b)s (elongs)g(to)g(the)h(set)g(describ)s(ed)168 1014 y(b)m(y)g(Lemma)e(35,) i(thanks)g(to)f(\(50\).)2273 b Fg(\003)168 1180 y Fw(When)36 b(the)f(co)s(cycle)g(is)g(rational,)e Fo(H)42 b Fw(happ)s(ens)36 b(to)f(b)s(e)g(\014nite)g(index.)51 b(Th)m(us,)37 b(the)e(assumption)g (in)168 1297 y(Theorem)e(10)f(that)g Fo(H)40 b Fw(is)32 b(just)h(the)g(iden)m(tit)m(y)f(ma)m(y)h(b)s(e)g(seen)h(as)e(an)h (irrationalit)m(y)c(condition.)168 1413 y(With)j(this)h(notation,)f(w)m (e)i(see)g(that)f(the)g(gap)g(is)g(\\summable")e(o)m(v)m(er)j(the)f (quotien)m(t)h(b)s(et)m(w)m(een)h(the)168 1529 y(crystal)d(and)h(the)g (group)f Fo(H)8 b Fw(:)168 1670 y Fz(Lemma)31 b(36.)38 b Fr(L)-5 b(et)31 b Fo(u)960 1634 y Fl(\(0\))1084 1670 y Fr(and)f Fo(u)1325 1634 y Fl(\(1\))1449 1670 y Fr(b)-5 b(e)30 b(at)h(the)f(e)-5 b(dge)30 b(of)g(a)g(gap)g(in)g(the)g(minimal)f (\(p)-5 b(ossibly)30 b(singular\))168 1786 y(lamination)j(of)i(Cor)-5 b(ol)5 b(lary)34 b(34.)45 b(Then,)34 b(the)h(fol)5 b(lowing)33 b(estimate)i(holds:)1393 1874 y Fn(X)1350 2090 y Fp(p)p Fm(2)p Fl(\003)p Fp(=H)1596 1968 y Fs(j)p Fo(u)1680 1927 y Fl(\(1\))1774 1968 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)g Fo(u)2076 1927 y Fl(\(0\))2170 1968 y Fw(\()p Fo(p)p Fw(\))p Fs(j)44 b(\024)h Fo(]F)30 b(:)168 2249 y Fz(Pro)s(of.)43 b Fw(Giv)m(en)32 b Fo(g)f Fs(2)e Fo(G=H)39 b Fw(let)870 2423 y(\002)946 2438 y Fp(g)1031 2423 y Fw(:=)44 b Fs(f)p Fw(\()p Fo(p;)17 b Fw([)p Fo(t)p Fw(]\))g Fo(;)130 b(p)28 b Fs(2)g Fo(F)i(;)131 b(u)2138 2382 y Fl(\(0\))2232 2423 y Fw(\()p Fo(g)t(p)p Fw(\))27 b Fs(\024)h Fo(t)g Fs(\024)g Fo(u)2764 2382 y Fl(\(1\))2858 2423 y Fw(\()p Fo(g)t(p)p Fw(\))p Fs(g)17 b Fo(;)168 2589 y Fw(where)38 b([)p Fs(\001)p Fw(])f(here)h(ab)s(o)m(v)m(e)g(denotes)g(the)g(in)m(teger)f(part.)57 b(Then,)39 b(\002)2553 2604 y Fp(g)2629 2589 y Fs(\032)d Fo(F)j Fs(\002)25 b Fq(T)3009 2553 y Fl(1)3089 2589 y Fw(and)38 b(\002)3360 2604 y Fp(g)3425 2589 y Fs(\\)26 b Fw(\002)3595 2604 y Fl(~)-37 b Fp(g)3668 2589 y Fw(=)35 b Fs(;)168 2711 y Fw(if)28 b Fo(g)j Fs(6)p Fw(=)g(~)-53 b Fo(g)t Fw(,)30 b(thanks)g(to)f(the)g(Birkho\013)g(prop)s(ert)m(y)h (and)f(the)h(fact)f(that)g Fo(u)2711 2675 y Fl(\(0\))2834 2711 y Fw(and)h Fo(u)3077 2675 y Fl(\(1\))3200 2711 y Fw(are)f(at)g(the)g(edge)168 2827 y(of)j(a)g(gap)g(in)g(the)h(minimal)c (\(p)s(ossibly)j(singular\))f(lamination.)168 2943 y(Let)k Fo(\026)g Fw(b)s(e)g(the)g(pro)s(duct)h(measure)f(b)s(et)m(w)m(een)i (the)e(coun)m(ting)g(measure)h(on)f Fo(F)48 b Fw(and)35 b(the)h(Leb)s(esgue)168 3060 y(measure)d Fs(L)f Fw(on)g Fq(T)847 3024 y Fl(1)890 3060 y Fw(.)44 b(Then,)33 b(the)g(ab)s(o)m(v)m (e)h(argumen)m(ts)f(and)f(F)-8 b(ubini's)32 b(Theorem)h(yield)e(that) 955 3341 y Fo(]F)41 b Fw(=)28 b Fo(\026)p Fw(\()p Fo(F)35 b Fs(\002)23 b Fq(T)1559 3300 y Fl(1)1601 3341 y Fw(\))28 b Fs(\025)g Fo(\026)1848 3141 y Fn(0)1848 3320 y(@)1998 3247 y([)1935 3463 y Fp(g)r Fm(2)p 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b(w)m(e)g(ma)m(y)f(supp)s(ose)h(that)f Fo(u)3735 4614 y Fl(\(0\))168 4766 y Fw(and)k Fo(u)426 4730 y Fl(\(1\))564 4766 y Fw(are)g(at)g(the)g(edge)h(of)e(a)h(gap)f(in)h(the)g(\(p)s (ossibly)f(singular\))g(lamination)d(induced)k(b)m(y)168 4882 y(the)c(minimizers)c(giv)m(en)k(in)f(Theorem)h(8)f(\(see)i(the)f (discussion)g(in)f(Section)g(3)g(and,)j(in)d(particu-)168 5002 y(lar,)d(Corollaries)e(33)i(and)g(34)g(there\).)58 b(Indeed,)40 b(if)c(the)h(lamination)d(has)j(no)h(gaps)f(b)s(et)m(w)m (een)i Fo(u)3735 4965 y Fl(\(0\))168 5121 y Fw(and)31 b Fo(u)412 5085 y Fl(\(1\))506 5121 y Fw(,)h(then)g(in\014nitely)f(man) m(y)g(lea)m(v)m(es)i(lie)d(b)s(et)m(w)m(een)k Fo(u)2299 5085 y Fl(\(0\))2424 5121 y Fw(and)e Fo(u)2669 5085 y Fl(\(1\))2763 5121 y Fw(,)g(th)m(us)g(giving)e(the)i(claim)e(of)168 5237 y(Theorem)j(10.)168 5353 y(Th)m(us,)42 b(w)m(e)e(are)e(no)m(w)i (going)d(to)i(construct)h(a)e(new)i(solution)d(inside)h(the)i(gap)e(b)s (ordered)i(b)m(y)f Fo(u)3735 5317 y Fl(\(0\))168 5469 y Fw(and)32 b Fo(u)413 5433 y Fl(\(1\))507 5469 y Fw(.)p eop %%Page: 37 37 37 36 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(37)0 450 y Fw(Notice)30 b(that,)h(due)h(to)e(Lemma)g(36)g(and)h(the)g (fact)g(that)g Fo(H)38 b Fw(only)30 b(con)m(tains)h(the)g(iden)m(tit)m (y)-8 b(,)31 b(w)m(e)h(ha)m(v)m(e)0 566 y(that)0 739 y(\(51\))1225 644 y Fn(X)1231 856 y Fp(p)p Fm(2)p Fl(\003)1386 739 y Fs(j)p Fo(u)1470 698 y Fl(\(1\))1563 739 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)1866 698 y Fl(\(0\))1960 739 y Fw(\()p Fo(p)p Fw(\))p Fs(j)43 b(\024)i Fo(]F)30 b(:)0 1012 y Fw(This)37 b(estimate)f(will)e(pla)m(y)i(a)g(crucial)g (role)f(in)h(the)h(sequel)h(since)f(it)e(will)g(pro)m(vide)i(some)f (amoun)m(t)0 1129 y(of)c(compactness.)0 1245 y(The)44 b(a-priori)c(b)s(ound)j(\(51\))f(giv)m(es)h(us)g(an)g Fo(`)1666 1209 y Fl(1)1748 1245 y Fw(a)g(priori)d(b)s(ound)j(for)g(the) g(di\013erence)g(of)f(the)i(t)m(w)m(o)0 1361 y(solutions.)58 b(Note)37 b(that)h(it)e(is)h(a)h(purely)f(geometric)g(b)s(ound)h(and)f (that)h(it)e(only)h(dep)s(ends)i(on)f(the)0 1477 y(fact)33 b(that)g(they)i(are)e(at)g(the)h(edge)g(of)f(the)h(gaps)g(of)f(a)g(\(p) s(ossibly)g(singular\))f(lamination.)42 b(Indeed,)0 1594 y(the)e(term)f(in)g(the)h(left)f(is)h(just)g(the)g(total)e(v)m(olume)h (of)g(the)i(gap.)64 b(Since)40 b(w)m(e)h(kno)m(w)g(that)e(all)f(the)0 1710 y(translates)33 b(of)f(the)h(gap)g(cannot)f(in)m(tersect,)i(w)m(e) g(just)f(divide)f(the)h(pieces)h(of)e(the)h(in)m(tegral)f(so)h(that)0 1826 y(they)g(\014t)g(in)f(the)h(fundamen)m(tal)f(domain.)0 2032 y(4.1.)49 b Fz(Set-up)41 b(and)g(compactness)f(features.)50 b Fw(During)34 b(this)h(pro)s(of,)g(w)m(e)h(will)c(\014x)k(a)f(co)s (cycle)g Fo(\033)0 2148 y Fw(as)e(in)f(the)h(statemen)m(t)g(of)f (Theorem)h(10.)0 2264 y(W)-8 b(e)33 b(in)m(tro)s(duce)g(the)g(sup-norm) 1436 2396 y Fs(k)p Fo(u)p Fs(k)43 b Fw(:=)i(sup)1790 2477 y Fp(p)p Fm(2)p Fl(\003)1946 2396 y Fs(j)p Fo(u)p Fw(\()p Fo(p)p 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y(from)d(\(52\))h(that)h Fo(u)27 b Fs(2)h(C)6 b Fw(.)2641 b Fg(\003)0 5469 y Fw(P)m(oin)m(t)m(wise)33 b(and)g(uniform)e(con)m(v)m(ergence)k(th)m(us)e(agree)g(on)g Fs(C)6 b Fw(,)33 b(as)f(next)i(result)e(p)s(oin)m(ts)h(out:)p eop %%Page: 38 38 38 37 bop 168 251 a Ft(38)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fz(Lemma)37 b(38.)42 b Fr(If)34 b Fo(u)908 465 y Fp(n)982 450 y Fs(2)28 b(C)41 b Fr(is)35 b(a)g(se)-5 b(quenc)g(e)34 b(such)g(that)1631 631 y Fw(lim)1579 691 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1835 631 y Fo(u)1891 646 y Fp(n)1937 631 y Fw(\()p Fo(p)p Fw(\))28 b(=)f Fo(u)p Fw(\()p Fo(p)p Fw(\))17 b Fo(;)168 856 y Fr(for)34 b(any)h Fo(p)28 b Fs(2)g Fw(\003)p Fr(,)34 b(then)h Fo(u)1086 871 y Fp(n)1167 856 y Fr(c)-5 b(onver)g(ges)34 b(to)h Fo(u)g Fr(in)f(the)h Fw(sup)q Fr(-norm)f Fs(k)22 b(\001)g(k)p Fr(.)168 1005 y Fz(Pro)s(of.)43 b Fw(Fix)32 b Fo(\017)c(>)f Fw(0.)44 b(By)34 b(\(51\))o(,)f(there)g(exists)h(a)e (\014nite)g(set)1515 1186 y Fo(X)1596 1201 y Fp(\017)1656 1186 y Fw(=)c Fs(f)p Fo(p)1859 1201 y Fl(1)1898 1186 y Fo(;)17 b(:)g(:)g(:)f(;)h(p)2166 1201 y Fp(m)2232 1186 y Fs(g)27 b(\032)h Fw(\003)168 1367 y(in)k(suc)m(h)i(a)e(w)m(a)m(y)i (that)1534 1510 y Fs(j)p Fo(u)1618 1525 y Fp(n)1664 1510 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)1967 1469 y Fl(\(0\))2061 1510 y Fw(\()p Fo(p)p Fw(\))p Fs(j)44 b(\024)g Fo(\017)17 b(;)168 1670 y Fw(for)33 b(an)m(y)h Fo(n)c Fs(2)g Fq(N)49 b Fw(and)34 b(an)m(y)g Fo(p)c Fs(2)g Fw(\003)23 b Fs(\000)g Fo(X)1616 1685 y Fp(\017)1649 1670 y Fw(.)47 b(Since)34 b Fs(C)40 b Fw(is)33 b(ob)m(viously)g(closed) h(under)h(p)s(oin)m(t)m(wise)f(limit,)168 1787 y Fo(u)27 b Fs(2)h(C)39 b Fw(and)1558 1927 y Fs(j)p Fo(u)p Fw(\()p Fo(p)p Fw(\))21 b Fs(\000)i Fo(u)1944 1886 y Fl(\(0\))2037 1927 y Fw(\()p Fo(p)p Fw(\))p Fs(j)44 b(\024)h Fo(\017)17 b(;)168 2088 y Fw(for)32 b(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003)22 b Fs(\000)g Fo(X)942 2103 y Fp(\017)975 2088 y Fw(.)168 2204 y(Also,)29 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Fs(\025)g Fo(n)p Fw(\()p Fo(\017)p Fw(\))33 b(and)g(an)m(y)g Fo(p)28 b Fs(2)g Fo(X)1524 3361 y Fp(\017)1556 3346 y Fw(.)168 3463 y(Th)m(us,)34 b(if)d Fo(n)d Fs(\025)g Fo(n)p Fw(\()p Fo(\017)p Fw(\),)720 3644 y Fs(k)p Fo(u)826 3659 y Fp(n)894 3644 y Fs(\000)23 b Fo(u)p Fs(k)82 b(\024)96 b Fw(sup)1343 3725 y Fp(p)p Fm(2)p Fp(X)1484 3733 y Fk(\017)1530 3644 y Fs(j)p Fo(u)1614 3659 y Fp(n)1660 3644 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)p Fw(\()p Fo(p)p Fw(\))p Fs(j)1343 3900 y Fw(+)80 b(sup)1436 3981 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1681 3989 y Fk(\017)1727 3789 y Fn(\020)1786 3900 y Fs(j)p Fo(u)1870 3915 y Fp(n)1916 3900 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)2219 3859 y Fl(\(0\))2313 3900 y Fw(\()p Fo(p)p Fw(\))p Fs(j)e Fw(+)h Fs(j)p Fo(u)2669 3859 y Fl(\(0\))2763 3900 y Fw(\()p Fo(p)p Fw(\))g Fs(\000)g Fo(u)p Fw(\()p Fo(p)p Fw(\))p Fs(j)3218 3789 y Fn(\021)1182 4106 y Fs(\024)84 b Fw(3)p Fo(\017)17 b(;)168 4287 y Fw(pro)m(ving)32 b(the)h(desired)g(claim.)2443 b Fg(\003)168 4468 y Fw(As)35 b(an)g(easy)h(consequence)i(of)d(Lemmata)e(37)h(and)h (38,)g(w)m(e)h(th)m(us)g(deduce)h(that)e Fs(C)41 b Fw(is)34 b(compact)h(in)168 4584 y(the)e(sup)q(-norm:)168 4733 y Fz(Corollary)k(39.)k Fr(L)-5 b(et)35 b Fo(u)1075 4748 y Fp(n)1150 4733 y Fs(2)28 b(C)41 b Fr(for)34 b(any)h Fo(n)27 b Fs(2)i Fq(N)8 b Fr(.)51 b(Then,)34 b(ther)-5 b(e)35 b(exists)f(a)g(subse)-5 b(quenc)g(e)34 b Fo(u)3471 4748 y Fp(n)3514 4760 y Fk(k)3590 4733 y Fr(and)g(a)168 4849 y(c)-5 b(on\014gur)g(ation)34 b Fo(u)27 b Fs(2)h(C)41 b Fr(so)35 b Fo(u)1203 4864 y Fp(n)1246 4876 y Fk(k)1322 4849 y Fr(c)-5 b(onver)g(ges)34 b(to)h Fo(u)f Fr(in)h(the)g Fw(sup)q Fr(-norm)f Fs(k)22 b(\001)g(k)p Fr(.)168 4998 y Fw(As)45 b(in)f([dlL)-11 b(V05)o(],)48 b(it)c(is)g(also)g(con)m(v)m (enien)m(t)i(to)e(in)m(tro)s(duce)h(a)f Fr(r)-5 b(enormalize)g(d)45 b(ener)-5 b(gy)46 b(functional)p Fw(.)168 5114 y(Namely)-8 b(,)31 b(w)m(e)j(set)168 5321 y(\(53\))856 b Fo(S)1258 5336 y Fp(\033)1305 5321 y Fw(\()p Fo(u)p Fw(\))44 b(:=)1671 5226 y Fn(X)1677 5427 y Fk(B)r Fj(2S)1638 5483 y Fk(B)r Fj(\\)p Fi(\003)p Fj(6)p Fi(=)p Fj(;)1874 5211 y Fn(\020)1934 5321 y Fo(H)2015 5336 y Fp(B)2075 5321 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)g Fo(H)2409 5336 y Fp(B)2470 5321 y Fw(\()p Fo(u)2564 5280 y Fl(\(0\))2658 5321 y Fw(\))2696 5211 y Fn(\021)2772 5321 y Fo(:)p eop %%Page: 39 39 39 38 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(39)0 450 y Fw(The)30 b(pap)s(er)e([dlL)-11 b(V05)o(])29 b(con)m(tains)g(a)f(similar)d(result)k(in)f(the)h(case)g(of)f(partial)f (di\013eren)m(tial)g(equation.)0 566 y(In)37 b(partial)d(di\013eren)m (tial)h(equations)i(renormalized)e(energies)i(for)f(the)h(study)h(of)e (homo)s(clinic)e(so-)0 683 y(lutions)e(to)h(p)s(erio)s(dic)f(solutions) g(w)m(ere)i(also)f(used)h(in)f([RS03)o(].)46 b(W)-8 b(e)34 b(also)e(note)h(that)h(some)f(similar)0 799 y(expressions)h(w)m(ere)g (considered)g(in)e([Mat86)o(])h(in)f(the)h(con)m(text)h(of)e(the)h(P)m (eierls-Nabarro)e(barrier.)0 915 y(The)37 b(ab)s(o)m(v)m(e)f (renormalized)e(energy)39 b(\(53\))c(pro)m(vides)h(a)g(\\nice")f (functional)f(in)h Fs(C)6 b Fw(,)37 b(in)e(view)i(of)e(the)0 1031 y(follo)m(wing)30 b(result:)0 1225 y Fz(Lemma)37 b(40.)42 b Fr(F)-7 b(or)34 b(any)h Fo(u)27 b Fs(2)h(C)6 b Fr(,)35 b(the)g(series)f(in)42 b Fw(\(53\))34 b Fr(is)h(absolutely)g (c)-5 b(onver)g(gent,)34 b(and)0 1497 y Fw(\(54\))1238 b(0)44 b Fs(\024)h Fo(S)1687 1512 y Fp(\033)1734 1497 y Fw(\()p Fo(u)p Fw(\))f Fo(<)g Fw(+)p Fs(1)17 b Fo(:)0 1770 y Fr(Mor)-5 b(e)g(over,)45 b(if)f Fo(S)633 1785 y Fp(\033)680 1770 y Fw(\()p Fo(u)p Fw(\))f(=)g(0)h Fr(for)f(some)g Fo(u)g Fs(2)h(C)6 b Fr(,)46 b(then)d Fo(u)h Fr(is)f(a)g(gr)-5 b(ound)43 b(state,)j(and,)f(c)-5 b(onversely,)0 1886 y(if)35 b Fo(u)27 b Fs(2)h(C)41 b Fr(is)35 b(a)f(gr)-5 b(ound)35 b(state,)g(then)g Fo(S)1418 1901 y Fp(\033)1464 1886 y Fw(\()p Fo(u)p Fw(\))28 b(=)f(0)p Fr(.)0 2080 y Fz(Pro)s(of.)43 b Fw(F)-8 b(or)32 b(an)m(y)h Fo(u)27 b Fs(2)i(C)6 b Fw(,)868 2273 y Fn(X)865 2485 y Fp(B)s Fm(2S)1032 2368 y Fs(j)p Fo(H)1141 2383 y Fp(B)1202 2368 y Fw(\()p Fo(u)p Fw(\))21 b Fs(\000)i Fo(H)1536 2383 y Fp(B)1596 2368 y Fw(\()p Fo(u)1690 2327 y Fl(\(0\))1784 2368 y Fw(\))p Fs(j)704 2688 y(\024)868 2594 y Fn(X)865 2805 y Fp(B)s Fm(2S)1032 2594 y Fn(X)1035 2805 y Fp(p)p Fm(2)p Fp(B)1193 2544 y Fn(\014)1193 2604 y(\014)1193 2664 y(\014)1193 2723 y(\014)1226 2553 y(Z)1326 2579 y Fl(1)1282 2778 y(0)1392 2621 y Fo(@)5 b(H)1529 2636 y Fp(B)p 1392 2666 199 4 v 1415 2757 a Fo(@)g(u)1527 2772 y Fp(p)1600 2688 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2223 2647 y Fl(\(0\))2317 2688 y Fw(\))17 b Fo(d\034)33 b Fs(\001)22 b Fw(\()p Fo(u)2642 2703 y Fp(p)2703 2688 y Fs(\000)h Fo(u)2859 2647 y Fl(\(0\))2859 2713 y Fp(p)2952 2688 y Fw(\))2990 2544 y Fn(\014)2990 2604 y(\014)2990 2664 y(\014)2990 2723 y(\014)704 3018 y Fs(\024)865 2923 y Fn(X)871 3134 y Fp(p)p Fm(2)p Fl(\003)1029 2923 y Fn(X)1035 3123 y Fk(B)r Fj(2S)1040 3179 y Fk(B)r Fj(3)p Fk(p)1193 2882 y Fn(Z)1293 2908 y Fl(1)1249 3108 y(0)1349 2873 y Fn(\014)1349 2933 y(\014)1349 2993 y(\014)1349 3052 y(\014)1392 2950 y Fo(@)5 b(H)1529 2965 y Fp(B)p 1392 2995 V 1415 3086 a Fo(@)g(u)1527 3101 y Fp(p)1600 3018 y Fw(\()p Fo(\034)11 b(u)23 b Fw(+)f(\(1)f Fs(\000)i Fo(\034)11 b Fw(\))p Fo(u)2223 2976 y Fl(\(0\))2317 3018 y Fw(\))2355 2873 y Fn(\014)2355 2933 y(\014)2355 2993 y(\014)2355 3052 y(\014)2422 3018 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)2682 3033 y Fp(p)2743 3018 y Fs(\000)g Fo(u)2898 2976 y Fl(\(0\))2898 3042 y Fp(p)2992 3018 y Fs(j)704 3341 y(\024)84 b Fo(C)975 3246 y Fn(X)981 3458 y Fp(p)p Fm(2)p Fl(\003)1135 3341 y Fs(j)p Fo(u)22 b Fs(\000)g Fo(u)1396 3300 y Fl(\(0\))1490 3341 y Fs(j)17 b Fo(;)0 3720 y Fw(thanks)49 b(to)f Fz(H8)p Fw(.)91 b(Th)m(us,)54 b(b)m(y)c(\(51\),)i(the)d(series)g(in)e(\(53\))h (is)g(absolutely)f(con)m(v)m(ergen)m(t,)55 b(and)48 b(so)0 3836 y Fo(S)60 3851 y Fp(\033)107 3836 y Fw(\()p Fo(u)p Fw(\))27 b Fo(<)g Fw(+)p Fs(1)p Fw(,)33 b(as)f(desired.)0 3952 y(Then,)g(let)e Fo(u)d Fs(2)h(C)37 b Fw(and)31 b(\014x)g Fo(\017)d(>)g Fw(0.)42 b(W)-8 b(e)31 b(exploit)f(again)f(\(51\))h(to)g (construct)i(a)e(set)h Fo(X)3101 3967 y Fp(\017)3161 3952 y Fs(\032)e Fw(\003)h(so)h(that)0 4224 y(\(55\))1428 b Fo(]X)1721 4239 y Fp(\017)1781 4224 y Fo(<)27 b Fw(+)p Fs(1)0 4496 y Fw(and)0 4768 y(\(56\))1387 4674 y Fn(X)1322 4885 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1567 4893 y Fk(\017)1613 4768 y Fs(j)p Fo(u)1697 4727 y Fl(\(1\))1813 4768 y Fs(\000)c Fo(u)1969 4727 y Fl(\(0\))2062 4768 y Fs(j)44 b(\024)h Fo(\017)17 b(:)0 5140 y Fw(Let)33 b(no)m(w)1131 5408 y Fo(v)t Fw(\()p Fo(p)p Fw(\))27 b(:=)1465 5268 y Fn(\032)1587 5346 y Fo(u)p Fw(\()p Fo(p)p Fw(\))257 b(if)31 b Fo(p)d Fs(2)g Fo(X)2366 5361 y Fp(\017)2398 5346 y Fw(,)1540 5467 y Fo(u)1596 5431 y Fl(\(0\))1690 5467 y Fw(\()p Fo(p)p Fw(\))115 b(if)31 b Fo(p)d Fs(2)g Fw(\003)22 b Fs(\000)h Fo(X)2461 5482 y Fp(\017)2493 5467 y Fw(.)p eop %%Page: 40 40 40 39 bop 168 251 a Ft(40)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Since)31 b(the)g(series)g(in)f(\(53\))g(is)g(absolutely)g(con)m(v)m(ergen)m(t)j (\(as)e(w)m(e)h(ha)m(v)m(e)g(just)f(pro)m(v)m(ed\),)h(b)m(y)g(using)e Fz(H8)168 566 y Fw(and)i(\(56\),)g(w)m(e)i(see)g(that)603 899 y Fs(j)p Fo(S)691 914 y Fp(\033)737 899 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)h Fo(S)1051 914 y Fp(\033)1097 899 y Fw(\()p Fo(v)t Fw(\))p Fs(j)44 b Fw(=)1400 724 y Fn(\014)1400 784 y(\014)1400 844 y(\014)1400 904 y(\014)1400 964 y(\014)1436 804 y(X)1433 1016 y Fp(B)s Fm(2S)1600 899 y Fo(H)1681 914 y Fp(B)1742 899 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)g Fo(H)2076 914 y Fp(B)2137 899 y Fw(\()p Fo(v)t Fw(\))2264 724 y Fn(\014)2264 784 y(\014)2264 844 y(\014)2264 904 y(\014)2264 964 y(\014)1296 1219 y Fs(\024)1405 1125 y Fn(X)1401 1336 y Fp(B)s Fm(2S)1569 1125 y Fn(X)1571 1336 y Fp(p)p Fm(2)p Fp(B)1729 1084 y Fn(Z)1829 1110 y Fl(1)1785 1309 y(0)1885 1075 y Fn(\014)1885 1134 y(\014)1885 1194 y(\014)1885 1254 y(\014)1928 1152 y Fo(@)5 b(H)2065 1167 y Fp(B)p 1928 1196 199 4 v 1951 1288 a Fo(@)g(u)2063 1303 y Fp(p)2137 1219 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(v)t Fw(\))2792 1075 y Fn(\014)2792 1134 y(\014)2792 1194 y(\014)2792 1254 y(\014)2859 1219 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)3119 1234 y Fp(p)3180 1219 y Fs(\000)g Fo(v)3326 1234 y Fp(p)3366 1219 y Fs(j)1296 1548 y Fw(=)1400 1454 y Fn(X)1406 1665 y Fp(p)p Fm(2)p Fl(\003)1564 1454 y Fn(X)1570 1654 y Fk(B)r Fj(2S)1575 1710 y Fk(B)r Fj(3)p Fk(p)1728 1413 y Fn(Z)1828 1439 y Fl(1)1783 1638 y(0)1884 1404 y Fn(\014)1884 1464 y(\014)1884 1523 y(\014)1884 1583 y(\014)1927 1481 y Fo(@)5 b(H)2064 1496 y Fp(B)p 1927 1525 V 1950 1617 a Fo(@)g(u)2062 1632 y Fp(p)2135 1548 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(v)t Fw(\))2791 1404 y Fn(\014)2791 1464 y(\014)2791 1523 y(\014)2791 1583 y(\014)2857 1548 y Fo(d\034)34 b Fs(\001)22 b(j)p Fo(u)3118 1563 y Fp(p)3179 1548 y Fs(\000)g Fo(v)3325 1563 y Fp(p)3365 1548 y Fs(j)1296 1872 y(\024)28 b Fo(C)1511 1777 y Fn(X)1518 1989 y Fp(p)p Fm(2)p Fl(\003)1672 1872 y Fs(j)p Fo(u)1756 1887 y Fp(p)1817 1872 y Fs(\000)22 b Fo(v)1963 1887 y Fp(p)2003 1872 y Fs(j)1296 2153 y Fw(=)28 b Fo(C)1575 2059 y Fn(X)1510 2270 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1755 2278 y Fk(\017)1801 2153 y Fs(j)p Fo(u)1885 2168 y Fp(p)1946 2153 y Fs(\000)23 b Fo(u)2102 2112 y Fl(\(0\))2102 2178 y Fp(p)2195 2153 y Fs(j)1296 2399 y(\024)28 b Fo(C)7 b(\017)17 b(:)168 1606 y Fw(\(57\))168 2657 y(F)-8 b(urthermore,)32 b(since)h Fo(u)1055 2621 y Fl(\(0\))1181 2657 y Fw(is)f(a)h(ground)f(state,)h(w)m(e)h(deduce)g (from)d(\(55\))h(that)1381 2826 y Fn(X)1387 3026 y Fk(B)r Fj(2S)1328 3082 y Fk(B)r Fj(\\)p Fk(X)1468 3090 y(\017)1499 3082 y Fj(6)p Fi(=)p Fj(;)1587 2920 y Fw(\()p Fo(H)1706 2935 y Fp(B)1766 2920 y Fw(\()p Fo(v)t Fw(\))22 b Fs(\000)h Fo(H)2096 2935 y Fp(B)2156 2920 y Fw(\()p Fo(u)2250 2879 y Fl(\(0\))2344 2920 y Fw(\)\))44 b Fs(\025)h Fw(0)17 b Fo(:)168 3310 y Fw(This,)33 b Fz(H8)p Fw(,)f(\(56\))g(and)g(\(57\))g (yield)g(that)422 3572 y(2)p Fo(C)7 b(\017)22 b Fw(+)g Fo(S)767 3587 y Fp(\033)814 3572 y Fw(\()p Fo(u)p Fw(\))27 b Fs(\025)h Fo(C)7 b(\017)22 b Fw(+)g Fo(S)1374 3587 y Fp(\033)1421 3572 y Fw(\()p Fo(v)t Fw(\))28 b Fs(\025)g Fo(C)7 b(\017)22 b Fw(+)1980 3478 y Fn(X)1986 3678 y Fk(B)r Fj(2S)1927 3734 y Fk(B)r Fj(\\)p Fk(X)2067 3742 y(\017)2097 3734 y Fi(=)p Fj(;)2186 3572 y Fw(\()p Fo(H)2305 3587 y Fp(B)2365 3572 y Fw(\()p Fo(v)t Fw(\))g Fs(\000)h Fo(H)2695 3587 y Fp(B)2755 3572 y Fw(\()p Fo(u)2849 3531 y Fl(\(0\))2943 3572 y Fw(\)\))973 3942 y Fs(\025)28 b Fo(C)7 b(\017)22 b Fs(\000)1378 3847 y Fn(X)1385 4047 y Fk(B)r Fj(2S)1326 4104 y Fk(B)r Fj(\\)p Fk(X)1466 4112 y(\017)1496 4104 y Fi(=)p Fj(;)1601 3847 y Fn(X)1604 4059 y Fp(p)p Fm(2)p Fp(B)1762 3806 y Fn(Z)1861 3833 y Fl(1)1817 4032 y(0)1917 3797 y Fn(\014)1917 3857 y(\014)1917 3917 y(\014)1917 3977 y(\014)1961 3874 y Fo(@)5 b(H)2098 3889 y Fp(B)p 1961 3919 V 1984 4010 a Fo(@)g(u)2096 4025 y Fp(p)2169 3942 y Fw(\()p Fo(\034)11 b(v)26 b Fw(+)c(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2787 3901 y Fl(\(0\))2881 3942 y Fw(\))2919 3797 y Fn(\014)2919 3857 y(\014)2919 3917 y(\014)2919 3977 y(\014)2985 3942 y Fo(d\034)34 b Fs(\001)21 b(j)p Fo(v)3236 3957 y Fp(p)3298 3942 y Fs(\000)i Fo(u)3454 3901 y Fl(\(0\))3454 3966 y Fp(p)3548 3942 y Fs(j)973 4264 y(\025)28 b Fo(C)7 b(\017)22 b Fs(\000)h Fo(C)1475 4169 y Fn(X)1410 4381 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1655 4389 y Fk(\017)1701 4264 y Fs(j)p Fo(v)1776 4279 y Fp(p)1838 4264 y Fs(\000)f Fo(u)1993 4223 y Fl(\(0\))1993 4289 y Fp(p)2087 4264 y Fs(j)973 4510 y(\025)28 b Fw(0)17 b Fo(:)168 4757 y Fw(Since)33 b Fo(\017)h Fw(ma)m(y)g(b)s(e)g(tak)m(en)g(as)g(small)d(as)j(w)m(e)h (wish,)f(w)m(e)h(ha)m(v)m(e)g(just)f(pro)m(v)m(ed)h(that)e Fo(S)3158 4772 y Fp(\033)3205 4757 y Fw(\()p Fo(u)p Fw(\))c Fs(\025)h Fw(0,)k(whic)m(h)168 4873 y(completes)e(the)h(pro)s(of)f(of) 39 b(\(54\))o(.)168 4989 y(Let)29 b(no)m(w)i Fo(u)c Fs(2)h(C)36 b Fw(suc)m(h)31 b(that)e Fo(S)1290 5004 y Fp(\033)1337 4989 y Fw(\()p Fo(u)p Fw(\))e(=)h(0.)42 b(Let)30 b Fo(X)37 b Fw(b)s(e)30 b(a)f(\014nite)g(set)i(and)e(let)g Fo(w)j Fw(b)s(e)e(so)g(that)f Fo(w)3563 5004 y Fp(p)3630 4989 y Fw(=)f Fo(u)3790 5004 y Fp(p)168 5105 y Fw(for)k(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003)22 b Fs(\000)g Fo(X)8 b Fw(.)43 b(Let)33 b Fo(X)1276 5120 y Fp(\017)1337 5105 y Fs(\033)28 b Fo(X)40 b Fw(as)33 b(in)f(\(55\))g(and)g(\(56\))g(and)h(set)1299 5408 y Fo(z)t Fw(\()p Fo(p)p Fw(\))c(:=)1632 5268 y Fn(\032)1746 5346 y Fo(w)s Fw(\()p Fo(p)p Fw(\))248 b(if)31 b Fo(p)d Fs(2)g Fo(X)2533 5361 y Fp(\017)2566 5346 y Fw(,)1707 5467 y Fo(u)1763 5431 y Fl(\(0\))1857 5467 y Fw(\()p Fo(p)p Fw(\))115 b(if)32 b Fo(p)27 b Fs(2)h Fw(\003)22 b Fs(\000)h Fo(X)2628 5482 y Fp(\017)2660 5467 y Fw(.)p eop %%Page: 41 41 41 40 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(41)0 450 y Fw(Arguing)32 b(as)g(in)g(the)h(pro)s(of)f(of)39 b(\(57\))o(,)33 b(w)m(e)h(get)e(that)600 548 y Fn(\014)600 608 y(\014)600 668 y(\014)712 568 y(X)718 768 y Fk(B)r Fj(2S)660 825 y Fk(B)r Fj(\\)p Fk(X)800 833 y(\017)830 825 y Fj(6)p Fi(=)p Fj(;)918 663 y Fw(\()p Fo(H)1037 678 y Fp(B)1098 663 y Fw(\()p Fo(w)s Fw(\))21 b Fs(\000)i Fo(H)1449 678 y Fp(B)1509 663 y Fw(\()p Fo(u)1603 622 y Fl(\(0\))1697 663 y Fw(\)\))39 b Fs(\000)1990 568 y Fn(X)1996 768 y Fk(B)r Fj(2S)1938 825 y Fk(B)r Fj(\\)p Fk(X)2078 833 y(\017)2108 825 y Fj(6)p Fi(=)p Fj(;)2196 663 y Fw(\()p Fo(H)2315 678 y Fp(B)2376 663 y Fw(\()p Fo(z)t Fw(\))22 b Fs(\000)h Fo(H)2704 678 y Fp(B)2765 663 y Fw(\()p Fo(u)2859 622 y Fl(\(0\))2952 663 y Fw(\)\))3028 548 y Fn(\014)3028 608 y(\014)3028 668 y(\014)946 986 y Fs(\024)28 b Fo(C)1145 892 y Fn(X)1151 1103 y Fp(p)p Fm(2)p Fl(\003)1305 986 y Fs(j)p Fo(w)1403 1001 y Fp(p)1464 986 y Fs(\000)23 b Fo(z)1609 1001 y Fp(p)1649 986 y Fs(j)946 1268 y Fw(=)k Fo(C)1208 1173 y Fn(X)1143 1385 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1388 1393 y Fk(\017)1434 1268 y Fs(j)p Fo(u)1518 1227 y Fl(\(0\))1518 1293 y Fp(p)1634 1268 y Fs(\000)c Fo(z)1779 1283 y Fp(p)1819 1268 y Fs(j)946 1514 y(\024)28 b Fo(C)7 b(\017)0 1696 y Fw(and)33 b(that)595 1789 y Fn(\014)595 1849 y(\014)628 1874 y Fo(S)688 1889 y Fp(\033)735 1874 y Fw(\()p Fo(u)p Fw(\))21 b Fs(\000)1050 1779 y Fn(X)1056 1979 y Fk(B)r Fj(2S)998 2036 y Fk(B)r Fj(\\)p Fk(X)1138 2044 y(\017)1168 2036 y Fj(6)p Fi(=)p Fj(;)1256 1874 y Fw(\()p Fo(H)1375 1889 y Fp(B)1436 1874 y Fw(\()p Fo(u)p Fw(\))g Fs(\000)i Fo(H)1770 1889 y Fp(B)1830 1874 y Fw(\()p Fo(u)1924 1832 y Fl(\(0\))2018 1874 y Fw(\)\))2094 1789 y Fn(\014)2094 1849 y(\014)2155 1874 y Fs(\024)28 b Fo(C)2419 1779 y Fn(X)2354 1990 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)2599 1998 y Fk(\017)2645 1874 y Fs(j)p Fo(u)2729 1889 y Fp(p)2790 1874 y Fs(\000)22 b Fo(u)2945 1832 y Fl(\(0\))2945 1898 y Fp(p)3039 1874 y Fs(j)1284 2161 y(\024)28 b Fo(C)7 b(\017)17 b(:)0 2360 y Fw(F)-8 b(urthermore,)32 b(since)h Fo(u)887 2324 y Fl(\(0\))1013 2360 y Fw(is)g(a)f(ground)g(state,)1214 2463 y Fn(X)1220 2663 y Fk(B)r Fj(2S)1161 2719 y Fk(B)r Fj(\\)p Fk(X)1301 2727 y(\017)1331 2719 y Fj(6)p Fi(=)p Fj(;)1420 2557 y Fw(\()p Fo(H)1539 2572 y Fp(B)1599 2557 y Fw(\()p Fo(z)t Fw(\))23 b Fs(\000)g Fo(H)1928 2572 y Fp(B)1988 2557 y Fw(\()p Fo(u)2082 2516 y Fl(\(0\))2176 2557 y Fw(\)\))44 b Fs(\025)h Fw(0)17 b Fo(:)0 2898 y Fw(By)36 b(collecting)e(the)i(ab)s(o)m(v)m(e)g(estimates,)g(and)g (exploiting)d(the)j(fact)g(that)f Fo(S)2792 2913 y Fp(\033)2839 2898 y Fw(\()p Fo(u)p Fw(\))d(=)h(0)f(=)h Fo(S)3362 2913 y Fp(\033)3409 2898 y Fw(\()p Fo(u)3503 2862 y Fl(\(0\))3596 2898 y Fw(\),)0 3014 y(w)m(e)h(conclude)f(that)0 3196 y(\(58\))513 3101 y Fn(X)520 3301 y Fk(B)r Fj(2S)461 3357 y Fk(B)r Fj(\\)p Fk(X)601 3365 y(\017)631 3357 y Fj(6)p Fi(=)p Fj(;)720 3196 y Fw(\()p Fo(H)839 3211 y Fp(B)899 3196 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)g Fo(H)1233 3211 y Fp(B)1294 3196 y Fw(\()p Fo(u)1388 3154 y Fl(\(0\))1481 3196 y Fw(\)\))39 b Fs(\000)1775 3101 y Fn(X)1781 3301 y Fk(B)r Fj(2S)1722 3357 y Fk(B)r Fj(\\)p Fk(X)1862 3365 y(\017)1892 3357 y Fj(6)p Fi(=)p Fj(;)1981 3196 y Fw(\()p Fo(H)2100 3211 y Fp(B)2160 3196 y Fw(\()p Fo(w)s Fw(\))22 b Fs(\000)g Fo(H)2511 3211 y Fp(B)2572 3196 y Fw(\()p Fo(u)2666 3154 y Fl(\(0\))2760 3196 y Fw(\)\))44 b Fs(\024)g Fw(3)p Fo(C)7 b(\017)17 b(:)0 3520 y Fw(Moreo)m(v)m(er,)34 b(from)e Fz(H1)p Fw(,)1252 3582 y Fn(X)1258 3782 y Fk(B)r Fj(2S)1213 3838 y Fk(B)r Fj(\\)p Fk(X)t Fi(=)p Fj(;)1462 3676 y Fs(j)p Fo(H)1571 3691 y Fp(B)1631 3676 y Fw(\()p Fo(u)p Fw(\))21 b Fs(\000)i Fo(H)1965 3691 y Fp(B)2026 3676 y Fw(\()p Fo(w)s Fw(\))p Fs(j)43 b Fw(=)h(0)17 b Fo(:)0 3979 y Fw(This)33 b(and)f(\(58\))g(imply)f(that)1487 4087 y Fn(X)1493 4287 y Fk(B)r Fj(2S)1448 4344 y Fk(B)r Fj(\\)p Fk(X)t Fj(6)p Fi(=)p Fj(;)1680 4182 y Fw(\()p Fo(H)1799 4197 y Fp(B)1860 4182 y Fw(\()p Fo(u)p Fw(\))21 b Fs(\000)i Fo(H)2194 4197 y Fp(B)2254 4182 y Fw(\()p Fo(w)s Fw(\)\))1278 4505 y(=)1500 4411 y Fn(X)1507 4611 y Fk(B)r Fj(2S)1448 4667 y Fk(B)r Fj(\\)p Fk(X)1588 4675 y(\017)1618 4667 y Fj(6)p Fi(=)p Fj(;)1707 4505 y Fw(\()p Fo(H)1826 4520 y Fp(B)1886 4505 y Fw(\()p Fo(u)p Fw(\))f Fs(\000)g Fo(H)2220 4520 y Fp(B)2281 4505 y Fw(\()p Fo(w)s Fw(\)\))1278 4792 y Fs(\024)83 b Fw(3)p Fo(C)7 b(\017)17 b(:)0 4973 y Fw(Since)33 b Fo(\017)g Fw(ma)m(y)f(b)s(e)h(tak)m(en)g (arbitrarily)d(small,)g(this)j(giv)m(es)g(that)f Fo(u)g Fw(is)g(a)g(ground)h(state.)0 5090 y(Let)41 b(us)h(no)m(w)f(supp)s(ose) h(that)f Fo(u)g Fs(2)i(C)k Fw(is)40 b(a)h(ground)g(state.)68 b(Fix)41 b Fo(\017)h(>)f Fw(0)g(and)g(let)f Fo(X)3164 5105 y Fp(\017)3238 5090 y Fw(as)h(in)f(\(55\))0 5206 y(and)33 b(\(56\))o(.)44 b(W)-8 b(e)33 b(set)1135 5408 y(~)-55 b Fo(u)o Fw(\()p Fo(p)p Fw(\))28 b(:=)1467 5268 y Fn(\032)1542 5351 y Fo(u)1598 5314 y Fl(\(0\))1692 5351 y Fw(\()p Fo(p)p Fw(\))210 b(if)32 b Fo(p)27 b Fs(2)h Fo(X)2368 5366 y Fp(\017)2401 5351 y Fw(,)1589 5467 y Fo(u)p Fw(\()p Fo(p)p Fw(\))162 b(if)32 b Fo(p)27 b Fs(2)h Fw(\003)22 b Fs(\000)h Fo(X)2463 5482 y Fp(\017)2496 5467 y Fw(.)p eop %%Page: 42 42 42 41 bop 168 251 a Ft(42)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Then,)34 b(since)f Fo(u)f Fw(is)g(a)g(ground)h(state,)1398 539 y Fn(X)1404 739 y Fk(B)r Fj(2S)1346 795 y Fk(B)r Fj(\\)p Fk(X)1486 803 y(\017)1516 795 y Fj(6)p Fi(=)p Fj(;)1621 634 y Fo(H)1702 649 y Fp(B)1762 634 y Fw(\()p Fo(u)p Fw(\))44 b Fs(\024)2122 539 y Fn(X)2128 739 y Fk(B)r Fj(2S)2070 795 y Fk(B)r Fj(\\)p Fk(X)2210 803 y(\017)2240 795 y Fj(6)p Fi(=)p Fj(;)2345 634 y Fo(H)2426 649 y Fp(B)2486 634 y Fw(\()6 b(~)-55 b Fo(u)p Fw(\))17 b Fo(:)168 949 y Fw(Moreo)m(v)m(er,)34 b(b)m(y)f Fz(H8)p Fw(,)g(the)g(fact)f(that)g Fo(u)c Fs(2)g(C)39 b Fw(and)32 b(\(56\),)984 1044 y Fn(X)981 1255 y Fp(B)s Fm(2S)1148 1138 y Fs(j)p Fo(H)1257 1153 y Fp(B)1318 1138 y Fw(\()6 b(~)-55 b Fo(u)o Fw(\))22 b Fs(\000)h Fo(H)1652 1153 y Fp(B)1712 1138 y Fw(\()p Fo(u)1806 1097 y Fl(\(0\))1900 1138 y Fw(\))p Fs(j)83 b(\024)g Fo(C)2303 1044 y Fn(X)2309 1255 y Fp(p)p Fm(2)p Fl(\003)2463 1138 y Fs(j)6 b Fw(~)-55 b Fo(u)2547 1153 y Fp(p)2608 1138 y Fs(\000)23 b Fo(u)2764 1097 y Fl(\(0\))2764 1163 y Fp(p)2858 1138 y Fs(j)2049 1420 y Fw(=)84 b Fo(C)2368 1325 y Fn(X)2303 1537 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)2548 1545 y Fk(\017)2594 1420 y Fs(j)6 b Fw(~)-55 b Fo(u)2678 1435 y Fp(p)2739 1420 y Fs(\000)23 b Fo(u)2895 1379 y Fl(\(0\))2895 1445 y Fp(p)2988 1420 y Fs(j)2049 1665 y(\024)83 b Fo(C)7 b(\017)17 b(;)168 1832 y Fw(for)32 b(some)g Fo(C)j(>)27 b Fw(0,)33 b(and)f(so)1029 2016 y Fo(S)1089 2031 y Fp(\033)1136 2016 y Fw(\()p Fo(u)p Fw(\))82 b Fs(\024)i Fo(C)7 b(\017)22 b Fw(+)1751 1921 y Fn(X)1747 2133 y Fp(B)s Fm(2S)1898 2016 y Fw(\()p Fo(H)2017 2031 y Fp(B)2078 2016 y Fw(\()p Fo(u)p Fw(\))f Fs(\000)i Fo(H)2412 2031 y Fp(B)2472 2016 y Fw(\()6 b(~)-55 b Fo(u)p Fw(\)\))1350 2289 y Fs(\024)84 b Fo(C)7 b(\017)22 b Fw(+)g(0)g(+)1978 2194 y Fn(X)1985 2394 y Fk(B)r Fj(2S)1926 2451 y Fk(B)r Fj(\\)p Fk(X)2066 2459 y(\017)2096 2451 y Fi(=)p Fj(;)2201 2289 y Fs(j)p Fo(H)2310 2304 y Fp(B)2370 2289 y Fw(\()p Fo(u)p Fw(\))g Fs(\000)h Fo(H)2705 2304 y Fp(B)2765 2289 y Fw(\()6 b(~)-55 b Fo(u)p Fw(\))p Fs(j)17 b Fo(:)168 2603 y Fw(Since,)30 b(if)f Fo(B)22 b Fs(\\)17 b Fo(X)794 2618 y Fp(\017)855 2603 y Fw(=)27 b Fs(;)p Fw(,)k(w)m(e)g(ha)m(v)m(e)g (that)f Fo(u)1694 2618 y Fp(p)1760 2603 y Fw(=)k(~)-55 b Fo(u)1920 2618 y Fp(p)1989 2603 y Fw(for)29 b(an)m(y)i Fo(p)d Fs(2)g Fo(B)5 b Fw(,)30 b(it)f(th)m(us)i(follo)m(ws)e(from)g Fz(H1)g Fw(that)1547 2771 y Fo(S)1607 2786 y Fp(\033)1654 2771 y Fw(\()p Fo(u)p Fw(\))44 b Fs(\024)h Fo(C)7 b(\017)22 b Fw(+)g(0)g(+)g(0)17 b Fo(:)168 2938 y Fw(Since)25 b Fo(\017)g Fw(ma)m(y)f(b)s(e)h(tak)m(en)h(arbitrary)e(small,)g(w)m(e)h (obtain)f(that)h Fo(S)2443 2953 y Fp(\033)2489 2938 y Fw(\()p Fo(u)p Fw(\))i Fs(\024)i Fw(0,)d(and,)g(in)e(fact,)h Fo(S)3444 2953 y Fp(\033)3491 2938 y Fw(\()p Fo(u)p Fw(\))i(=)g(0,)168 3054 y(thanks)33 b(to)g(\(54\))o(,)g(whic)m(h)g(has)g(b)s(een)g (already)f(pro)m(v)m(ed.)45 b(This)33 b(ends)h(the)f(pro)s(of)e(of)h (Lemma)g(40.)46 b Fg(\003)168 3222 y Fw(The)40 b(renormalized)e(energy) i(\\b)s(eha)m(v)m(es)h(nicely")e(with)g(resp)s(ect)h(to)f(p)s(oin)m(t)m (wise)g(con)m(v)m(ergence,)44 b(as)168 3338 y(w)m(e)33 b(no)m(w)h(sho)m(w:)168 3480 y Fz(Lemma)j(41.)42 b Fr(L)-5 b(et)35 b Fo(u)974 3495 y Fp(n)1048 3480 y Fs(2)28 b(C)41 b Fr(b)-5 b(e)35 b(such)f(that)1614 3647 y Fw(lim)1563 3707 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1818 3647 y Fo(u)1874 3662 y Fp(n)1921 3647 y Fw(\()p Fo(p)p Fw(\))44 b(=)g Fo(u)p Fw(\()p Fo(p)p Fw(\))17 b Fo(;)168 3852 y Fr(for)34 b(any)h Fo(p)28 b Fs(2)g Fw(\003)p Fr(.)44 b(Then,)168 4036 y Fw(\(59\))1079 b(lim)1370 4096 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1625 3941 y Fn(X)1631 4153 y Fp(p)p Fm(2)p Fl(\003)1786 4036 y Fs(j)p Fo(u)1870 4051 y Fp(n)1916 4036 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)g Fo(u)p Fw(\()p Fo(p)p Fw(\))p Fs(j)44 b Fw(=)g(0)17 b Fo(;)168 4361 y Fw(\(60\))1236 b(lim)1527 4421 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1782 4361 y Fo(S)1842 4376 y Fp(\033)1889 4361 y Fw(\()p Fo(u)1983 4376 y Fp(n)2029 4361 y Fw(\))45 b(=)f Fo(S)2292 4376 y Fp(\033)2339 4361 y Fw(\()p Fo(u)p Fw(\))168 4566 y Fr(and)168 4762 y Fw(\(61\))616 b(lim)906 4822 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1162 4668 y Fn(X)1168 4879 y Fp(p)p Fm(2)p Fl(\003)1322 4652 y Fn(\020)1402 4668 y(X)1408 4868 y Fk(B)r Fj(2S)1413 4924 y Fk(B)r Fj(3)p Fk(p)1577 4695 y Fo(@)5 b(H)1714 4710 y Fp(B)p 1577 4739 199 4 v 1600 4831 a Fo(@)g(u)1712 4846 y Fp(p)1785 4762 y Fw(\()p Fo(u)1879 4777 y Fp(n)1925 4762 y Fw(\))1963 4652 y Fn(\021)2023 4674 y Fl(2)2090 4762 y Fw(=)2193 4668 y Fn(X)2200 4879 y Fp(p)p Fm(2)p Fl(\003)2354 4652 y Fn(\020)2434 4668 y(X)2440 4868 y Fk(B)r Fj(2S)2445 4924 y Fk(B)r Fj(3)p Fk(p)2608 4695 y Fo(@)g(H)2745 4710 y Fp(B)p 2608 4739 V 2631 4831 a Fo(@)g(u)2743 4846 y Fp(p)2816 4762 y Fw(\()p Fo(u)p Fw(\))2948 4652 y Fn(\021)3007 4674 y Fl(2)3063 4762 y Fo(:)168 5104 y Fz(Pro)s(of.)59 b Fw(By)40 b(\(51\))o(,)f(\014xed)g Fo(\017)f(>)e Fw(0,)j(w)m(e)g(can)f (\014nd)h(a)e(\014nite)h(set)h Fo(X)2588 5119 y Fp(\017)2657 5104 y Fw(=)e Fs(f)p Fo(p)2869 5053 y Fl(\(1\))2869 5114 y Fp(\017)2963 5104 y Fo(;)17 b(:)g(:)g(:)e(;)i(p)3230 5053 y Fl(\()p Fp(`)3286 5061 y Fk(\017)3317 5053 y Fl(\))3230 5114 y Fp(\017)3348 5104 y Fs(g)38 b Fw(in)f(suc)m(h)i(a)168 5220 y(w)m(a)m(y)34 b(that)1555 5268 y Fn(X)1490 5480 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1735 5488 y Fk(\017)1781 5363 y Fs(j)p Fo(u)1865 5322 y Fl(\(1\))1981 5363 y Fs(\000)22 b Fo(u)2136 5322 y Fl(\(0\))2230 5363 y Fs(j)44 b(\024)h Fo(\017)17 b(:)p eop %%Page: 43 43 43 42 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(43)0 450 y Fw(Then,)1310 934 y(lim)1258 994 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1514 839 y Fn(X)1520 1051 y Fp(p)p Fm(2)p Fl(\003)1674 934 y Fs(j)p Fo(u)1758 949 y Fp(n)1804 934 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)p Fw(\()p Fo(p)p Fw(\))p Fs(j)1098 1215 y(\024)135 b Fw(lim)1258 1275 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1527 1121 y Fn(X)1514 1332 y Fp(p)p Fm(2)p Fp(X)1655 1340 y Fk(\017)1701 1215 y Fs(j)p Fo(u)1785 1230 y Fp(n)1831 1215 y Fw(\()p Fo(p)p Fw(\))22 b Fs(\000)h Fo(u)p Fw(\()p Fo(p)p Fw(\))p Fs(j)38 b Fw(+)g Fo(\017)1098 1573 y Fw(=)136 b(lim)1258 1633 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1556 1449 y Fp(`)1585 1457 y Fk(\017)1514 1478 y Fn(X)1521 1691 y Fp(k)r Fl(=1)1674 1573 y Fs(j)p Fo(u)1758 1588 y Fp(n)1804 1573 y Fw(\()p Fo(p)1891 1532 y Fl(\()p Fp(k)r Fl(\))1891 1598 y Fp(\017)1989 1573 y Fw(\))22 b Fs(\000)g Fo(u)p Fw(\()p Fo(p)2291 1532 y Fl(\()p Fp(k)r Fl(\))2291 1598 y Fp(\017)2388 1573 y Fw(\))p Fs(j)39 b Fw(+)f Fo(\017)1098 1802 y Fw(=)84 b(0)22 b(+)g Fo(\017)17 b(;)0 2270 y Fw(due)33 b(to)f(\(38\).)0 2386 y(By)h(taking)f Fo(\017)h Fw(as)g(small)d(as)j(w)m(e)g(wish,)g(w)m (e)h(th)m(us)f(obtain)f(the)h(pro)s(of)e(of)39 b(\(59\).)0 2502 y(F)-8 b(urthermore,)32 b(since,)h(b)m(y)h(Lemma)d(40,)h(the)h (series)g(in)f(\(53\))g(is)g(absolutely)g(con)m(v)m(ergen)m(t,)920 3002 y Fo(S)980 3017 y Fp(\033)1027 3002 y Fw(\()p Fo(u)p Fw(\))21 b Fs(\000)i Fo(S)1340 3017 y Fp(\033)1387 3002 y Fw(\()p Fo(u)1481 3017 y Fp(n)1527 3002 y Fw(\))28 b(=)1700 2907 y Fn(X)1696 3118 y Fp(B)s Fm(2S)1864 2891 y Fn(\020)1924 3002 y Fo(H)2005 3017 y Fp(B)2065 3002 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)g Fo(H)2399 3017 y Fp(B)2460 3002 y Fw(\()p Fo(u)2554 3017 y Fp(n)2600 3002 y Fw(\))2638 2891 y Fn(\021)2714 3002 y Fo(:)0 3561 y Fw(Hence,)34 b(using)e Fz(H8)p Fw(,)845 4028 y Fs(j)p Fo(S)933 4043 y Fp(\033)979 4028 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)h Fo(S)1293 4043 y Fp(\033)1339 4028 y Fw(\()p Fo(u)1433 4043 y Fp(n)1480 4028 y Fw(\))p Fs(j)684 4243 y(\024)849 4148 y Fn(X)845 4359 y Fp(B)s Fm(2S)1013 4148 y Fn(X)1015 4359 y Fp(p)p Fm(2)p Fp(B)1173 4107 y Fn(Z)1273 4133 y Fl(1)1229 4333 y(0)1329 4098 y Fn(\014)1329 4158 y(\014)1329 4218 y(\014)1329 4278 y(\014)1372 4175 y Fo(@)5 b(H)1509 4190 y Fp(B)p 1372 4220 199 4 v 1395 4311 a Fo(@)g(u)1507 4326 y Fp(p)1580 4243 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)h Fo(\034)11 b(u)2165 4258 y Fp(n)2212 4243 y Fw(\)\))2288 4098 y Fn(\014)2288 4158 y(\014)2288 4218 y(\014)2288 4278 y(\014)2354 4243 y Fo(d\034)28 b Fs(j)p Fo(u)p Fw(\()p Fo(p)p Fw(\))21 b Fs(\000)i Fo(u)2861 4258 y Fp(n)2907 4243 y Fw(\()p Fo(p)p Fw(\))p Fs(j)685 4572 y Fw(=)845 4477 y Fn(X)851 4689 y Fp(p)p Fm(2)p Fl(\003)1009 4477 y Fn(X)1015 4677 y Fk(B)r Fj(2S)1020 4734 y Fk(B)r Fj(3)p Fk(p)1173 4436 y Fn(Z)1273 4463 y Fl(1)1229 4662 y(0)1329 4427 y Fn(\014)1329 4487 y(\014)1329 4547 y(\014)1329 4607 y(\014)1372 4504 y Fo(@)5 b(H)1509 4519 y Fp(B)p 1372 4549 V 1395 4640 a Fo(@)g(u)1507 4655 y Fp(p)1581 4572 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)g Fo(\034)11 b(u)2165 4587 y Fp(n)2212 4572 y Fw(\)\))2288 4427 y Fn(\014)2288 4487 y(\014)2288 4547 y(\014)2288 4607 y(\014)2354 4572 y Fo(d\034)28 b Fs(j)p Fo(u)p Fw(\()p Fo(p)p Fw(\))21 b Fs(\000)i Fo(u)2861 4587 y Fp(n)2908 4572 y Fw(\()p Fo(p)p Fw(\))p Fs(j)684 4895 y(\024)84 b Fo(C)955 4801 y Fn(X)961 5012 y Fp(p)p Fm(2)p Fl(\003)1116 4895 y Fs(j)p Fo(u)p Fw(\()p Fo(p)p Fw(\))21 b Fs(\000)h Fo(u)1501 4910 y Fp(n)1548 4895 y Fw(\()p Fo(p)p Fw(\))p Fs(j)17 b Fo(;)0 5469 y Fw(for)32 b(a)g(suitable)g Fo(C)7 b Fw(,)33 b(whic)m(h)g(yields) f(\(60\))o(,)h(due)g(to)f(\(59\).)p eop %%Page: 44 44 44 43 bop 168 251 a Ft(44)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Moreo)m(v)m(er,)34 b(b)m(y)f(using)g Fz(H8)f Fw(t)m(wice,)h(w)m(e)h(ha)m(v)m(e:)620 573 y Fn(\014)620 632 y(\014)620 692 y(\014)620 752 y(\014)620 812 y(\014)620 872 y(\014)620 931 y(\014)653 712 y(X)659 924 y Fp(p)p Fm(2)p Fl(\003)814 577 y Fn(0)814 752 y(B)814 816 y(@)905 712 y(X)911 912 y Fk(B)r Fj(2S)915 969 y Fk(B)r Fj(3)p Fk(p)1079 739 y Fo(@)5 b(H)1216 754 y Fp(B)p 1079 784 199 4 v 1102 875 a Fo(@)g(u)1214 890 y Fp(p)1287 807 y Fw(\()p Fo(u)1381 822 y Fp(n)1428 807 y Fw(\))1466 577 y Fn(1)1466 752 y(C)1466 816 y(A)1553 599 y Fl(2)1614 807 y Fs(\000)1714 712 y Fn(X)1720 924 y Fp(p)p Fm(2)p Fl(\003)1874 577 y Fn(0)1874 752 y(B)1874 816 y(@)1965 712 y(X)1972 912 y Fk(B)r Fj(2S)1976 969 y Fk(B)r Fj(3)p Fk(p)2140 739 y Fo(@)g(H)2277 754 y Fp(B)p 2140 784 V 2163 875 a Fo(@)g(u)2275 890 y Fp(p)2348 807 y Fw(\()p Fo(u)p Fw(\))2480 577 y Fn(1)2480 752 y(C)2480 816 y(A)2566 599 y Fl(2)2606 573 y Fn(\014)2606 632 y(\014)2606 692 y(\014)2606 752 y(\014)2606 812 y(\014)2606 872 y(\014)2606 931 y(\014)459 1263 y Fs(\024)620 1168 y Fn(X)626 1379 y Fp(p)p Fm(2)p Fl(\003)780 1032 y Fn(2)780 1208 y(6)780 1272 y(4)847 1032 y(0)847 1208 y(B)847 1272 y(@)938 1168 y(X)944 1368 y Fk(B)r Fj(2S)949 1424 y Fk(B)r Fj(3)p Fk(p)1102 1118 y Fn(\014)1102 1178 y(\014)1102 1238 y(\014)1102 1297 y(\014)1145 1195 y Fo(@)g(H)1282 1210 y Fp(B)p 1145 1240 V 1168 1331 a Fo(@)g(u)1280 1346 y Fp(p)1354 1263 y Fw(\()p Fo(u)1448 1278 y Fp(n)1494 1263 y Fw(\))1532 1118 y Fn(\014)1532 1178 y(\014)1532 1238 y(\014)1532 1297 y(\014)1587 1263 y Fw(+)1685 1118 y Fn(\014)1685 1178 y(\014)1685 1238 y(\014)1685 1297 y(\014)1728 1195 y Fo(@)g(H)1865 1210 y Fp(B)p 1728 1240 V 1752 1331 a Fo(@)g(u)1864 1346 y Fp(p)1937 1263 y Fw(\()p Fo(u)p Fw(\))2069 1118 y Fn(\014)2069 1178 y(\014)2069 1238 y(\014)2069 1297 y(\014)2101 1032 y(1)2101 1208 y(C)2101 1272 y(A)2211 1263 y Fs(\001)2261 1032 y Fn(0)2261 1208 y(B)2261 1272 y(@)2351 1168 y(X)2358 1368 y Fk(B)r Fj(2S)2362 1424 y Fk(B)r Fj(3)p Fk(p)2516 1118 y Fn(\014)2516 1178 y(\014)2516 1238 y(\014)2516 1297 y(\014)2559 1195 y Fo(@)g(H)2696 1210 y Fp(B)p 2559 1240 V 2582 1331 a Fo(@)g(u)2694 1346 y Fp(p)2767 1263 y Fw(\()p Fo(u)2861 1278 y Fp(n)2908 1263 y Fw(\))22 b Fs(\000)3077 1195 y Fo(@)5 b(H)3214 1210 y Fp(B)p 3077 1240 V 3100 1331 a Fo(@)g(u)3212 1346 y Fp(p)3286 1263 y Fw(\()p Fo(u)p Fw(\))3418 1118 y Fn(\014)3418 1178 y(\014)3418 1238 y(\014)3418 1297 y(\014)3450 1032 y(1)3450 1208 y(C)3450 1272 y(A)3537 1032 y(3)3537 1208 y(7)3537 1272 y(5)459 1636 y Fs(\024)84 b Fw(2)p Fo(C)779 1542 y Fn(X)785 1753 y Fp(p)p Fm(2)p Fl(\003)943 1542 y Fn(X)949 1742 y Fk(B)r Fj(2S)954 1798 y Fk(B)r Fj(3)p Fk(p)1107 1542 y Fn(X)1111 1753 y Fp(q)r Fm(2)p Fp(B)1268 1496 y Fn(\022)1341 1501 y(Z)1441 1527 y Fl(1)1397 1726 y(0)1497 1492 y Fn(\014)1497 1552 y(\014)1497 1612 y(\014)1497 1671 y(\014)1572 1569 y Fo(@)1628 1533 y Fl(2)1669 1569 y Fo(H)1750 1584 y Fp(B)p 1540 1614 303 4 v 1540 1705 a Fo(@)5 b(u)1652 1720 y Fp(p)1692 1705 y Fo(@)g(u)1804 1720 y Fp(q)1853 1636 y Fw(\()p Fo(\034)11 b(u)2000 1651 y Fp(n)2069 1636 y Fw(+)22 b(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(u)p Fw(\))2560 1492 y Fn(\014)2560 1552 y(\014)2560 1612 y(\014)2560 1671 y(\014)2626 1636 y Fo(d\034)34 b Fs(\001)22 b(j)p Fo(u)2887 1651 y Fp(n)2933 1636 y Fw(\()p Fo(q)t Fw(\))g Fs(\000)g Fo(u)p Fw(\()p Fo(q)t Fw(\))p Fs(j)3384 1496 y Fn(\023)460 2007 y Fw(=)84 b(2)p Fo(C)779 1913 y Fn(X)786 2124 y Fp(q)r Fm(2)p Fl(\003)943 1913 y Fn(X)949 2113 y Fk(B)r Fj(2S)955 2169 y Fk(B)r Fj(3)p Fk(q)1107 1913 y Fn(X)1110 2124 y Fp(p)p Fm(2)p Fp(B)1284 1867 y Fn(\022)1358 1872 y(Z)1457 1898 y Fl(1)1413 2097 y(0)1513 1863 y Fn(\014)1513 1923 y(\014)1513 1982 y(\014)1513 2042 y(\014)1589 1940 y Fo(@)1645 1904 y Fl(2)1685 1940 y Fo(H)1766 1955 y Fp(B)p 1557 1984 V 1557 2076 a Fo(@)5 b(u)1669 2091 y Fp(p)1709 2076 y Fo(@)g(u)1821 2091 y Fp(q)1869 2007 y Fw(\()p Fo(\034)11 b(u)2016 2022 y Fp(n)2085 2007 y Fw(+)22 b(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)p Fw(\))2577 1863 y Fn(\014)2577 1923 y(\014)2577 1982 y(\014)2577 2042 y(\014)2643 2007 y Fo(d\034)33 b Fs(\001)22 b(j)p Fo(u)2903 2022 y Fp(n)2949 2007 y Fw(\()p Fo(q)t Fw(\))g Fs(\000)h Fo(u)p Fw(\()p Fo(q)t Fw(\))p Fs(j)3401 1867 y Fn(\023)459 2331 y Fs(\024)84 b Fw(2)p Fo(C)746 2290 y Fl(2)802 2236 y Fn(X)809 2448 y Fp(q)r Fm(2)p Fl(\003)962 2331 y Fs(j)p Fo(u)1046 2346 y Fp(n)1092 2331 y Fw(\()p Fo(q)t Fw(\))22 b Fs(\000)h Fo(u)p Fw(\()p Fo(q)t Fw(\))p Fs(j)17 b Fo(;)168 2632 y Fw(for)32 b(some)g Fo(C)j(>)27 b Fw(0.)44 b(This)32 b(completes)h(the)g(pro)s(of)e(of)39 b(\(61\),)32 b(b)m(y)i(exploiting) e(\(59\))g(once)h(more.)127 b Fg(\003)168 2960 y Fw(4.2.)48 b Fz(The)41 b(heat)f(\015o)m(w.)49 b Fw(W)-8 b(e)35 b(no)m(w)h (consider)f(the)h(solution)d(of)h(a)h(\\heat)g(\015o)m(w")g(problem,)f (whic)m(h,)168 3076 y(indeed,)f(is)f(the)h(\\gradien)m(t)f(\015o)m(w")h (of)f(our)g(renormalized)f(energy)j(functional.)168 3193 y(Giv)m(en)c Fo(u)d Fs(2)h(C)37 b Fw(and)31 b Fo(t)d(>)f Fw(0,)k(w)m(e)g(denote)h(b)m(y)f(\010)1832 3157 y Fp(t)1862 3193 y Fw(\()p Fo(u)p Fw(\))f(the)h(con\014guration)e Fo(p)f Fs(7!)f Fo(U)10 b Fw(\()p Fo(p;)17 b(t)p Fw(\))31 b(that)f(satis\014es)168 3605 y(\(62\))539 3311 y Fn(8)539 3400 y(>)539 3430 y(>)539 3460 y(>)539 3490 y(<)539 3669 y(>)539 3699 y(>)539 3729 y(>)539 3759 y(:)711 3441 y Fo(@)762 3456 y Fp(t)792 3441 y Fo(U)10 b Fw(\()p Fo(p;)17 b(t)p Fw(\))28 b(=)g Fs(\000)1301 3346 y Fn(X)1308 3546 y Fk(B)r Fj(2S)1312 3603 y Fk(B)r Fj(3)p Fk(p)1476 3374 y Fo(@)5 b(H)1613 3389 y Fp(B)p 1476 3418 199 4 v 1499 3509 a Fo(@)g(u)1611 3524 y Fp(p)1684 3441 y Fw(\()p Fo(U)10 b Fw(\()p Fs(\001)p Fo(;)17 b(t)p Fw(\)\))376 b(for)32 b(an)m(y)h Fo(p)28 b Fs(2)g Fw(\003)k(and)h Fo(t)28 b(>)f Fw(0)17 b Fo(;)1062 3820 y(U)10 b Fw(\()p Fo(p;)17 b Fw(0\))28 b(=)f Fo(u)p Fw(\()p Fo(p)p Fw(\))945 b(for)33 b(an)m(y)g Fo(p)27 b Fs(2)h Fw(\003)17 b Fo(:)168 4011 y Fw(Then,)34 b(the)f(follo)m(wing)c(prop)s(erties)k(hold:)168 4166 y Fz(Lemma)k(42.)42 b Fr(If)34 b Fo(u)27 b Fs(2)h(C)6 b Fr(,)35 b(then)g Fw(\010)1439 4130 y Fp(t)1504 4166 y Fr(is)f(wel)5 b(l-de\014ne)-5 b(d)33 b(for)i(any)g Fo(t)28 b(>)f Fw(0)p Fr(.)45 b(A)n(lso,)34 b Fw(\010)3088 4130 y Fp(t)3118 4166 y Fw(\()p Fs(C)6 b Fw(\))28 b Fs(\032)g(C)41 b Fr(and)168 4414 y Fw(\(63\))610 b Fs(\000)1057 4347 y Fo(d)p 1039 4392 86 4 v 1039 4483 a(dt)1135 4414 y(S)1195 4429 y Fp(\033)1242 4304 y Fn(\020)1302 4414 y Fw(\010)1372 4373 y Fp(t)1402 4414 y Fw(\()p Fo(u)p Fw(\))1534 4304 y Fn(\021)1637 4414 y Fw(=)1757 4320 y Fn(X)1763 4531 y Fp(p)p Fm(2)p Fl(\003)1918 4304 y Fn(\020)1997 4320 y(X)2004 4520 y Fk(B)r Fj(2S)2008 4576 y Fk(B)r Fj(3)p Fk(p)2172 4347 y Fo(@)5 b(H)2309 4362 y Fp(B)p 2172 4392 199 4 v 2195 4483 a Fo(@)g(u)2307 4498 y Fp(p)2380 4414 y Fw(\(\010)2488 4373 y Fp(t)2518 4414 y Fw(\()p Fo(u)p Fw(\)\))2688 4304 y Fn(\021)2747 4326 y Fl(2)2831 4414 y Fs(\025)44 b Fw(0)17 b Fo(:)168 4752 y Fr(F)-7 b(urthermor)i(e,)34 b(if)g Fo(u)27 b Fs(2)i(C)6 b Fr(,)35 b(the)f(map)1771 4906 y Fo(t)44 b Fs(7!)g Fw(\010)2064 4865 y Fp(t)2094 4906 y Fw(\()p Fo(u)p Fw(\))168 5080 y Fr(b)-5 b(elongs)33 b(to)j Fo(C)7 b Fw(\([0)p Fo(;)17 b Fw(1])p Fo(;)g Fs(C)6 b Fw(\))p Fr(.)43 b(Mor)-5 b(e)g(over,)35 b(\014xe)-5 b(d)34 b Fo(T)42 b(>)27 b Fw(0)p Fr(,)35 b(the)f(map)1671 5275 y Fs(C)g(3)28 b Fo(u)44 b Fs(7!)g Fw(\010)2165 5234 y Fp(t)2195 5275 y Fw(\()p Fo(u)p Fw(\))168 5469 y Fr(is)34 b(c)-5 b(ontinuous,)35 b(with)g(uniform)f(mo)-5 b(dulus)35 b(of)f(c)-5 b(ontinuity)35 b(for)g Fo(t)28 b Fs(2)g Fw([0)p Fo(;)17 b(T)d Fw(])p Fr(.)p eop %%Page: 45 45 45 44 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(45)0 450 y Fz(Pro)s(of.)43 b Fw(Fixed)33 b(an)m(y)g Fo(p)27 b Fs(2)i Fw(\003,)154 714 y Fn(X)161 914 y Fk(B)r Fj(2S)165 970 y Fk(B)r Fj(3)p Fk(p)319 664 y Fn(\014)319 724 y(\014)319 783 y(\014)319 843 y(\014)362 741 y Fo(@)5 b(H)499 756 y Fp(B)p 362 785 199 4 v 385 877 a Fo(@)g(u)497 892 y Fp(p)570 808 y Fw(\()p Fo(u)p Fw(\))22 b Fs(\000)833 741 y Fo(@)5 b(H)970 756 y Fp(B)p 833 785 V 856 877 a Fo(@)g(u)968 892 y Fp(p)1042 808 y Fw(\()p Fo(v)t Fw(\))1169 664 y Fn(\014)1169 724 y(\014)1169 783 y(\014)1169 843 y(\014)1284 808 y Fs(\024)1449 714 y Fn(X)1455 914 y Fk(B)r Fj(2S)1460 970 y Fk(B)r Fj(3)p Fk(p)1613 714 y Fn(X)1616 925 y Fp(q)r Fm(2)p Fp(B)1774 673 y Fn(Z)1873 699 y Fl(1)1829 898 y(0)1929 664 y Fn(\014)1929 724 y(\014)1929 783 y(\014)1929 843 y(\014)2005 741 y Fo(@)2061 705 y Fl(2)2101 741 y Fo(H)2182 756 y Fp(B)p 1972 785 303 4 v 1972 877 a Fo(@)g(u)2084 892 y Fp(p)2125 877 y Fo(@)g(u)2237 892 y Fp(q)2285 808 y Fw(\()p Fo(\034)11 b(u)22 b Fw(+)g(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(v)t Fw(\))2941 664 y Fn(\014)2941 724 y(\014)2941 783 y(\014)2941 843 y(\014)3007 808 y Fo(d\034)33 b Fs(\001)22 b(k)p Fo(u)g Fs(\000)g Fo(v)t Fs(k)1284 1102 y(\024)84 b Fo(C)24 b Fs(k)p Fo(u)d Fs(\000)i Fo(v)t Fs(k)17 b Fo(;)0 1396 y Fw(for)32 b(an)m(y)h Fo(u;)g(v)f Fs(2)c(C)6 b Fw(,)33 b(due)g(to)f Fz(H8)p Fw(.)43 b(Accordingly)-8 b(,)32 b(the)h(map)1360 1739 y Fs(C)h(3)28 b Fo(u)44 b Fs(7!)1787 1644 y Fn(X)1794 1844 y Fk(B)r Fj(2S)1798 1901 y Fk(B)r Fj(3)p Fk(p)1962 1672 y Fo(@)5 b(H)2099 1687 y Fp(B)p 1962 1716 199 4 v 1985 1807 a Fo(@)g(u)2097 1822 y Fp(p)2170 1739 y Fw(\()p Fo(u)p Fw(\))0 2176 y(is)35 b(Lipsc)m(hitz)g(con)m(tin)m(uous)h(with)f (resp)s(ect)i(to)d(the)i(sup)q(-norm.)50 b(Then,)38 b(the)d(short)h (time)e(existence)0 2293 y(of)27 b(the)h(heat)g(\015o)m(w)g(with)f(con) m(tin)m(uous)h(dep)s(endence)i(on)e(the)f(initial)d(data)j(follo)m(ws)g (from)f(the)i(theory)0 2409 y(of)k(ODE's)i(in)e(Banac)m(h)h(spaces)i (\(see,)f(e.g.,)f([Hal80)o(]\).)44 b(The)34 b(short)f(time)e(\015o)m(w) j(ma)m(y)e(b)s(e)i(con)m(tin)m(ued)0 2525 y(pro)m(vided)i(that)f(it)f (do)s(es)i(not)f(exit)g(from)f Fs(C)41 b Fw(\(see,)c(e.g.,)g(the)e (Prolongation)e(Lemma)h(in)h([Hal80)n(]\).)0 2641 y(The)j(comparison)e (principle)g(\(see,)j(e.g.,)g(Lemma)c(4.3)i(in)f([KdlLR97)o(]\))h(sho)m (ws)i(that)e(\010)3291 2605 y Fp(t)3321 2641 y Fw(\()p Fs(C)6 b Fw(\))35 b Fs(\032)h(C)0 2758 y Fw(and)d(th)m(us)g(that)g(it)e (is)h(p)s(ossible)g(to)g(con)m(tin)m(ue)i(\010)1758 2721 y Fp(t)1820 2758 y Fw(in)e Fs(C)6 b Fw(.)0 2874 y(It)35 b(th)m(us)g(remains)f(to)g(pro)m(v)m(e)i(\(63\))o(.)50 b(T)-8 b(o)34 b(this)g(exten)m(t,)j(recalling)32 b Fz(H8)p Fw(,)j(w)m(e)h(\014x)f Fo(C)2945 2889 y Fl(1)2984 2874 y Fo(;)17 b(C)3098 2889 y Fl(2)3168 2874 y Fo(>)31 b Fw(0)j(in)g(suc)m(h)0 2990 y(a)e(w)m(a)m(y)1169 3341 y(sup)1152 3422 y Fp(v)r Fm(2O)1291 3430 y Fk(\033)1349 3341 y Fw(sup)1357 3422 y Fp(p)p Fm(2)p Fl(\003)1517 3246 y Fn(X)1523 3446 y Fk(B)r Fj(2S)1528 3502 y Fk(B)r Fj(3)p Fk(p)1681 3196 y Fn(\014)1681 3256 y(\014)1681 3316 y(\014)1681 3375 y(\014)1724 3273 y Fo(@)5 b(H)1861 3288 y Fp(B)p 1724 3318 V 1747 3409 a Fo(@)g(u)1859 3424 y Fp(p)1932 3341 y Fw(\()p Fo(v)t Fw(\))2059 3196 y Fn(\014)2059 3256 y(\014)2059 3316 y(\014)2059 3375 y(\014)2137 3341 y Fs(\024)44 b Fo(C)2328 3356 y Fl(1)2384 3341 y Fo(;)1169 3706 y Fw(sup)1152 3787 y Fp(v)r Fm(2O)1291 3795 y Fk(\033)1349 3706 y Fw(sup)1357 3787 y Fp(p)p Fm(2)p Fl(\003)1517 3611 y Fn(X)1523 3811 y Fk(B)r Fj(2S)1528 3868 y Fk(B)r Fj(3)p Fk(p)1681 3611 y Fn(X)1684 3823 y Fp(q)r Fm(2)p Fp(B)1841 3561 y Fn(\014)1841 3621 y(\014)1841 3681 y(\014)1841 3741 y(\014)1917 3638 y Fo(@)1973 3602 y Fl(2)2013 3638 y Fo(H)2094 3653 y Fp(B)p 1885 3683 303 4 v 1885 3774 a Fo(@)5 b(u)1997 3789 y Fp(p)2037 3774 y Fo(@)g(u)2149 3789 y Fp(q)2197 3706 y Fw(\()p Fo(v)t Fw(\))2324 3561 y Fn(\014)2324 3621 y(\014)2324 3681 y(\014)2324 3741 y(\014)2401 3706 y Fs(\024)45 b Fo(C)2593 3721 y Fl(2)2649 3706 y Fo(:)0 4149 y Fw(F)-8 b(rom)40 b(\(51\))o(,)k(w)m(e)e(also)e(ha) m(v)m(e)j(that)e(there)h(exits)f(a)g(collection)f(of)g(\014nite)h(sets) i Fo(X)3007 4164 y Fp(n)3096 4149 y Fs(\032)g Fw(\003,)g(so)f(that)0 4265 y Fo(X)81 4280 y Fp(n)156 4265 y Fs(\032)28 b Fo(X)342 4280 y Fp(n)p Fl(+1)511 4265 y Fw(for)k(an)m(y)i Fo(n)28 b Fs(2)g Fq(N)9 b Fw(,)38 b Fo(X)1237 4280 y Fl(1)1299 4265 y Fs([)22 b Fo(X)1468 4280 y Fl(2)1530 4265 y Fs([)g(\001)17 b(\001)g(\001)26 b Fw(=)i(\003,)k(and)1368 4510 y Fn(X)1297 4722 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1542 4730 y Fk(n)1600 4605 y Fs(j)p Fo(u)1684 4564 y Fl(\(1\))1684 4630 y Fp(p)1800 4605 y Fs(\000)22 b Fo(u)1955 4564 y Fl(\(0\))1955 4630 y Fp(p)2049 4605 y Fs(j)44 b(\024)2258 4538 y Fw(1)p 2253 4582 59 4 v 2253 4673 a Fo(n)2338 4605 y(:)0 5001 y Fw(F)-8 b(or)32 b(an)m(y)h Fo(t)28 b Fs(\025)g Fw(0,)k(w)m(e)i(de\014ne)0 5321 y(\(64\))781 b Fo(f)1003 5336 y Fp(n)1050 5321 y Fw(\()p Fo(t)p Fw(\))44 b(:=)1421 5226 y Fn(X)1427 5427 y Fk(B)r Fj(2S)1362 5483 y Fk(B)r Fj(\\)p Fk(X)1502 5491 y(n)1545 5483 y Fj(6)p Fi(=)p Fj(;)1650 5211 y Fn(\020)1709 5321 y Fo(H)1790 5336 y Fp(B)1851 5321 y Fw(\(\010)1959 5280 y Fp(t)1989 5321 y Fw(\()p Fo(u)p Fw(\)\))21 b Fs(\000)i Fo(H)2361 5336 y Fp(B)2421 5321 y Fw(\()p Fo(u)2515 5280 y Fl(\(0\))2609 5321 y Fw(\))2647 5211 y Fn(\021)p eop %%Page: 46 46 46 45 bop 168 251 a Ft(46)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 459 y Fw(and)32 b(w)m(e)i(no)m(w)f(study)h(the)f(con)m(v)m(ergence)i(prop)s(erties)e (of)f Fo(f)2298 474 y Fp(n)2377 459 y Fw(and)2599 433 y(_)2567 459 y Fo(f)2615 474 y Fp(n)2662 459 y Fw(.)44 b(First)31 b(of)h(all,)784 728 y Fs(j)p Fo(f)860 743 y Fp(n)907 728 y Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)h Fo(S)1200 743 y Fp(\033)1247 728 y Fw(\(\010)1355 686 y Fp(t)1385 728 y Fw(\()p Fo(u)p Fw(\)\))p Fs(j)625 1024 y Fw(=)784 790 y Fn(\014)784 850 y(\014)784 909 y(\014)784 969 y(\014)784 1029 y(\014)784 1089 y(\014)784 1149 y(\014)886 929 y(X)892 1129 y Fk(B)r Fj(2S)828 1186 y Fk(B)r Fj(\\)p Fk(X)968 1194 y(n)1010 1186 y Fi(=)p Fj(;)1115 913 y Fn(\020)1175 1024 y Fo(H)1256 1039 y Fp(B)1316 1024 y Fw(\(\010)1424 983 y Fp(t)1454 1024 y Fw(\()p Fo(u)p Fw(\)\))e Fs(\000)i Fo(H)1826 1039 y Fp(B)1887 1024 y Fw(\()p Fo(u)1981 983 y Fl(\(0\))2074 1024 y Fw(\))2112 913 y Fn(\021)2172 790 y(\014)2172 850 y(\014)2172 909 y(\014)2172 969 y(\014)2172 1029 y(\014)2172 1089 y(\014)2172 1149 y(\014)624 1398 y Fs(\024)853 1303 y Fn(X)859 1503 y Fk(B)r Fj(2S)794 1560 y Fk(B)r Fj(\\)p Fk(X)934 1568 y(n)977 1560 y Fi(=)p Fj(;)1082 1303 y Fn(X)1084 1515 y Fp(p)p Fm(2)p Fp(B)1242 1262 y Fn(Z)1342 1288 y Fl(1)1298 1488 y(0)1398 1283 y Fn(\014)1398 1343 y(\014)1398 1403 y(\014)1441 1330 y Fo(@)5 b(H)1578 1345 y Fp(B)p 1441 1375 199 4 v 1464 1466 a Fo(@)g(u)1576 1481 y Fp(p)1650 1398 y Fw(\()p Fo(\034)11 b Fw(\010)1811 1357 y Fp(t)1841 1398 y Fw(\()p Fo(u)p Fw(\)\))21 b(+)h(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2486 1357 y Fl(\(0\))2580 1398 y Fw(\))2618 1283 y Fn(\014)2618 1343 y(\014)2618 1403 y(\014)2668 1398 y Fo(d\034)33 b Fs(\001)22 b(j)p Fw(\010)2942 1357 y Fp(t)2972 1398 y Fw(\()p Fo(u)p Fw(\))3104 1413 y Fp(p)3165 1398 y Fs(\000)h Fo(u)3321 1357 y Fl(\(0\))3321 1422 y Fp(p)3415 1398 y Fs(j)624 1767 y(\024)856 1673 y Fn(X)784 1884 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1029 1892 y Fk(n)1092 1673 y Fn(X)1098 1873 y Fk(B)r Fj(2S)1103 1929 y Fk(B)r Fj(3)p Fk(p)1256 1632 y Fn(Z)1356 1658 y Fl(1)1311 1857 y(0)1412 1653 y Fn(\014)1412 1713 y(\014)1412 1772 y(\014)1455 1700 y Fo(@)5 b(H)1592 1715 y Fp(B)p 1455 1744 V 1478 1836 a Fo(@)g(u)1590 1851 y Fp(p)1663 1767 y Fw(\()p Fo(\034)11 b Fw(\010)1824 1726 y Fp(t)1855 1767 y Fw(\()p Fo(u)p Fw(\)\))21 b(+)h(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2500 1726 y Fl(\(0\))2594 1767 y Fw(\))2632 1653 y Fn(\014)2632 1713 y(\014)2632 1772 y(\014)2682 1767 y Fo(d\034)33 b Fs(\001)22 b(j)p Fw(\010)2956 1726 y Fp(t)2986 1767 y Fw(\()p Fo(u)p Fw(\))3118 1782 y Fp(p)3179 1767 y Fs(\000)g Fo(u)3334 1726 y Fl(\(0\))3334 1792 y Fp(p)3428 1767 y Fs(j)624 2091 y(\024)83 b Fo(C)854 2106 y Fl(1)998 1996 y Fn(X)927 2208 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1172 2216 y Fk(n)1230 2091 y Fs(j)p Fw(\010)1328 2050 y Fp(t)1358 2091 y Fw(\()p Fo(u)p Fw(\))1490 2106 y Fp(p)1551 2091 y Fs(\000)23 b Fo(u)1707 2050 y Fl(\(0\))1707 2115 y Fp(p)1801 2091 y Fs(j)624 2403 y(\024)794 2336 y Fo(C)864 2351 y Fl(1)p 794 2380 110 4 v 820 2472 a Fo(n)930 2403 y(;)168 2701 y Fw(and)32 b(so)168 2969 y(\(65\))509 b Fo(f)899 2984 y Fp(n)946 2969 y Fw(\()p Fo(t)p Fw(\))33 b(con)m(v)m(erges)h(to)f Fo(S)1706 2984 y Fp(\033)1752 2969 y Fw(\(\010)1860 2933 y Fp(t)1890 2969 y Fw(\()p Fo(u)p Fw(\)\))f(uniformly)f(in)h Fo(t)27 b Fs(2)i Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\).)168 3237 y(Let)32 b(no)m(w)168 3534 y(\(66\))896 b Fo(g)t Fw(\()p Fo(t)p Fw(\))44 b(:=)g Fs(\000)1685 3439 y Fn(X)1692 3651 y Fp(p)p Fm(2)p Fl(\003)1846 3423 y Fn(\020)1926 3439 y(X)1932 3639 y Fk(B)r Fj(2S)1937 3696 y Fk(B)r Fj(3)p Fk(p)2100 3467 y Fo(@)5 b(H)2237 3482 y Fp(B)p 2100 3511 199 4 v 2123 3602 a Fo(@)g(u)2235 3617 y Fp(p)2308 3534 y Fw(\(\010)2416 3493 y Fp(t)2446 3534 y Fw(\()p Fo(u)p Fw(\)\))2616 3423 y Fn(\021)2675 3446 y Fl(2)2731 3534 y Fo(:)168 3962 y Fw(Since)41 b Fo(u)487 3926 y Fl(\(0\))622 3962 y Fw(is)f(a)h(ground)g(state,)i(and)e(so)g(it)f(solv) m(es)i(the)f(v)-5 b(ariational)38 b(equation,)43 b(estimate)d(\(51\)) 168 4078 y(yields)32 b(that,)g(\014xed)i(an)m(y)f Fo(p)28 b Fs(2)g Fw(\003,)982 4257 y Fn(\014)982 4317 y(\014)982 4377 y(\014)982 4437 y(\014)982 4497 y(\014)982 4556 y(\014)982 4616 y(\014)1019 4397 y(X)1025 4597 y Fk(B)r Fj(2S)1029 4653 y Fk(B)r Fj(3)p Fk(p)1193 4424 y Fo(@)5 b(H)1330 4439 y Fp(B)p 1193 4469 V 1216 4560 a Fo(@)g(u)1328 4575 y Fp(p)1401 4492 y Fw(\()p Fo(w)s Fw(\))1550 4257 y Fn(\014)1550 4317 y(\014)1550 4377 y(\014)1550 4437 y(\014)1550 4497 y(\014)1550 4556 y(\014)1550 4616 y(\014)822 4947 y Fw(=)982 4713 y Fn(\014)982 4773 y(\014)982 4833 y(\014)982 4893 y(\014)982 4952 y(\014)982 5012 y(\014)982 5072 y(\014)1019 4853 y(X)1025 5053 y Fk(B)r Fj(2S)1029 5109 y Fk(B)r Fj(3)p Fk(p)1193 4880 y Fo(@)g(H)1330 4895 y Fp(B)p 1193 4924 V 1216 5016 a Fo(@)g(u)1328 5031 y Fp(p)1401 4947 y Fw(\()p Fo(w)s Fw(\))22 b Fs(\000)1675 4853 y Fn(X)1681 5053 y Fk(B)r Fj(2S)1686 5109 y Fk(B)r Fj(3)p Fk(p)1849 4880 y Fo(@)5 b(H)1986 4895 y Fp(B)p 1849 4924 V 1872 5016 a Fo(@)g(u)1984 5031 y Fp(p)2058 4947 y Fw(\()p Fo(u)2152 4906 y Fl(\(0\))2245 4947 y Fw(\))2283 4713 y Fn(\014)2283 4773 y(\014)2283 4833 y(\014)2283 4893 y(\014)2283 4952 y(\014)2283 5012 y(\014)2283 5072 y(\014)821 5321 y Fs(\024)985 5226 y Fn(X)992 5427 y Fk(B)r Fj(2S)996 5483 y Fk(B)r Fj(3)p Fk(p)1150 5226 y Fn(X)1153 5438 y Fp(q)r Fm(2)p Fp(B)1310 5186 y Fn(Z)1410 5212 y Fl(1)1365 5411 y(0)1466 5177 y Fn(\014)1466 5236 y(\014)1466 5296 y(\014)1466 5356 y(\014)1541 5254 y Fo(@)1597 5218 y Fl(2)1638 5254 y Fo(H)1719 5269 y Fp(B)p 1509 5298 303 4 v 1509 5389 a Fo(@)g(u)1621 5404 y Fp(p)1661 5389 y Fo(@)g(u)1773 5404 y Fp(q)1822 5321 y Fw(\()p Fo(\034)11 b(w)24 b Fw(+)f(\(1)e Fs(\000)i Fo(\034)11 b Fw(\))p Fo(u)2461 5280 y Fl(\(0\))2555 5321 y Fw(\))2593 5177 y Fn(\014)2593 5236 y(\014)2593 5296 y(\014)2593 5356 y(\014)2660 5321 y Fo(d\034)27 b Fs(j)p Fo(w)2878 5336 y Fp(q)2938 5321 y Fs(\000)c Fo(u)3094 5280 y Fl(\(0\))3094 5346 y Fp(q)3187 5321 y Fs(j)17 b Fo(;)p eop %%Page: 47 47 47 46 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(47)0 450 y Fw(for)32 b(an)m(y)h Fo(w)d Fs(2)e(C)6 b Fw(.)44 b(And)33 b(so)755 726 y Fn(X)762 938 y Fp(p)p Fm(2)p Fl(\003)916 587 y Fn(\014)916 647 y(\014)916 706 y(\014)916 766 y(\014)916 826 y(\014)916 886 y(\014)916 945 y(\014)953 726 y(X)959 926 y Fk(B)r Fj(2S)964 983 y Fk(B)r Fj(3)p Fk(p)1127 754 y Fo(@)5 b(H)1264 769 y Fp(B)p 1127 798 199 4 v 1150 889 a Fo(@)g(u)1262 904 y Fp(p)1336 821 y Fw(\()p Fo(w)s Fw(\))1485 587 y Fn(\014)1485 647 y(\014)1485 706 y(\014)1485 766 y(\014)1485 826 y(\014)1485 886 y(\014)1485 945 y(\014)595 1195 y Fs(\024)755 1100 y Fn(X)762 1312 y Fp(p)p Fm(2)p Fl(\003)920 1100 y Fn(X)926 1300 y Fk(B)r Fj(2S)931 1357 y Fk(B)r Fj(3)p Fk(p)1084 1100 y Fn(X)1087 1312 y Fp(q)r Fm(2)p Fp(B)1245 1059 y Fn(Z)1344 1085 y Fl(1)1300 1285 y(0)1400 1050 y Fn(\014)1400 1110 y(\014)1400 1170 y(\014)1400 1230 y(\014)1476 1127 y Fo(@)1532 1091 y Fl(2)1572 1127 y Fo(H)1653 1142 y Fp(B)p 1443 1172 303 4 v 1443 1263 a Fo(@)g(u)1555 1278 y Fp(p)1595 1263 y Fo(@)g(u)1707 1278 y Fp(q)1756 1195 y Fw(\()p Fo(\034)11 b(w)25 b Fw(+)d(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(u)2395 1154 y Fl(\(0\))2490 1195 y Fw(\))2528 1050 y Fn(\014)2528 1110 y(\014)2528 1170 y(\014)2528 1230 y(\014)2594 1195 y Fo(d\034)28 b Fs(j)p Fo(w)2813 1210 y Fp(q)2872 1195 y Fs(\000)23 b Fo(u)3028 1154 y Fl(\(0\))3028 1219 y Fp(q)3122 1195 y Fs(j)596 1566 y Fw(=)755 1471 y Fn(X)762 1683 y Fp(q)r Fm(2)p Fl(\003)920 1471 y Fn(X)926 1671 y Fk(B)r Fj(2S)932 1728 y Fk(B)r Fj(3)p Fk(q)1084 1471 y Fn(X)1086 1683 y Fp(p)p Fm(2)p Fp(B)1245 1430 y Fn(Z)1344 1456 y Fl(1)1300 1656 y(0)1400 1421 y Fn(\014)1400 1481 y(\014)1400 1541 y(\014)1400 1601 y(\014)1476 1498 y Fo(@)1532 1462 y Fl(2)1572 1498 y Fo(H)1653 1513 y Fp(B)p 1443 1543 V 1443 1634 a Fo(@)5 b(u)1555 1649 y Fp(p)1595 1634 y Fo(@)g(u)1707 1649 y Fp(q)1756 1566 y Fw(\()p Fo(\034)11 b(w)25 b Fw(+)d(\(1)g Fs(\000)g Fo(\034)11 b Fw(\))p Fo(u)2395 1525 y Fl(\(0\))2490 1566 y Fw(\))2528 1421 y Fn(\014)2528 1481 y(\014)2528 1541 y(\014)2528 1601 y(\014)2594 1566 y Fo(d\034)28 b Fs(j)p Fo(w)2813 1581 y Fp(q)2872 1566 y Fs(\000)23 b Fo(u)3028 1525 y Fl(\(0\))3028 1590 y Fp(q)3122 1566 y Fs(j)-3150 b Fw(\(67\))595 1889 y Fs(\024)83 b Fo(C)825 1904 y Fl(2)898 1795 y Fn(X)905 2006 y Fp(q)r Fm(2)p Fl(\003)1058 1889 y Fs(j)p Fo(w)1156 1904 y Fp(q)1216 1889 y Fs(\000)23 b Fo(u)1372 1848 y Fl(\(0\))1372 1914 y Fp(q)1465 1889 y Fs(j)595 2135 y(\024)83 b Fo(C)825 2150 y Fl(2)881 2135 y Fo(]F)31 b(;)0 2361 y Fw(for)h(an)m(y)h Fo(w)d Fs(2)e(C)6 b Fw(,)33 b(where)h(\(51\))e(has)h(b)s(een)g(used)h (once)f(more.)0 2477 y(As)g(a)f(consequence,)0 2750 y(\(68\))1063 2656 y Fn(X)1069 2867 y Fp(p)p Fm(2)p Fl(\003)1223 2636 y Fn(\014)1223 2695 y(\014)1223 2755 y(\014)1277 2656 y(X)1283 2856 y Fk(B)r Fj(2S)1288 2912 y Fk(B)r Fj(3)p Fk(p)1451 2683 y Fo(@)5 b(H)1588 2698 y Fp(B)p 1451 2727 199 4 v 1474 2819 a Fo(@)g(u)1586 2834 y Fp(p)1659 2750 y Fw(\(\010)1767 2709 y Fp(t)1797 2750 y Fw(\()p Fo(u)p Fw(\)\))1967 2636 y Fn(\014)1967 2695 y(\014)1967 2755 y(\014)2000 2662 y Fl(2)2084 2750 y Fs(\024)44 b Fo(C)2275 2765 y Fl(1)2315 2750 y Fo(C)2385 2765 y Fl(2)2440 2750 y Fo(]F)31 b(;)0 3124 y Fw(whic)m(h)36 b(is)g(a)f(b)s(ound)i(for)e(the) h(righ)m(t)f(hand)h(side)g(of)43 b(\(66\))o(.)54 b(F)-8 b(rom)34 b(this)h(and)h(the)h(de\014nition)e(of)g Fo(C)3595 3139 y Fl(1)3634 3124 y Fw(,)0 3240 y(w)m(e)f(ha)m(v)m(e)g(that)e(all)e (the)j(series)h(de\014ning)e Fo(g)k Fw(con)m(v)m(erge)e(absolutely)-8 b(.)0 3356 y(Hence,)34 b(b)m(y)f(rearranging)f(the)h(terms,)f(it)g (follo)m(ws)f(that)0 3637 y(\(69\))573 b Fs(\000)844 3543 y Fn(X)841 3754 y Fp(B)s Fm(2S)1008 3543 y Fn(X)1011 3754 y Fp(p)p Fm(2)p Fp(B)1171 3543 y Fn(X)1179 3743 y Fk(A)p Fj(2S)1184 3799 y Fk(A)p Fj(3)p Fk(p)1334 3497 y Fn(\022)1418 3570 y Fo(@)5 b(H)1555 3585 y Fp(B)p 1418 3614 V 1441 3706 a Fo(@)g(u)1553 3721 y Fp(p)1626 3637 y Fw(\(\010)1734 3596 y Fp(t)1764 3637 y Fw(\()p Fo(u)p Fw(\)\))1960 3570 y Fo(@)g(H)2097 3585 y Fp(A)p 1960 3614 195 4 v 1981 3706 a Fo(@)g(u)2093 3721 y Fp(p)2165 3637 y Fw(\(\010)2273 3596 y Fp(t)2303 3637 y Fw(\()p Fo(u)p Fw(\)\))2473 3497 y Fn(\023)2590 3637 y Fw(=)44 b Fo(g)t Fw(\()p Fo(t)p Fw(\))17 b Fo(:)0 4005 y Fw(W)-8 b(e)33 b(no)m(w)g(sho)m(w)h(that)734 4233 y(_)702 4259 y Fo(f)750 4274 y Fp(n)797 4259 y Fw(\()p Fo(t)p Fw(\))83 b(=)g Fs(\000)1313 4165 y Fn(X)1319 4365 y Fk(B)r Fj(2S)1254 4421 y Fk(B)r Fj(\\)p Fk(X)1394 4429 y(n)1437 4421 y Fj(6)p Fi(=)p Fj(;)1542 4165 y Fn(X)1544 4376 y Fp(p)p Fm(2)p Fp(B)1705 4165 y Fn(X)1712 4365 y Fk(A)p Fj(2S)1717 4421 y Fk(A)p Fj(3)p Fk(p)1878 4192 y Fo(@)5 b(H)2015 4207 y Fp(B)p 1878 4236 199 4 v 1901 4328 a Fo(@)g(u)2013 4343 y Fp(p)2086 4259 y Fw(\(\010)2194 4218 y Fp(t)2224 4259 y Fw(\()p Fo(u)p Fw(\)\))2404 4192 y Fo(@)g(H)2541 4207 y Fp(A)p 2404 4236 195 4 v 2425 4328 a Fo(@)g(u)2537 4343 y Fp(p)2608 4259 y Fw(\(\010)2716 4218 y Fp(t)2746 4259 y Fw(\()p Fo(u)p Fw(\)\))17 b Fo(:)-2960 b Fw(\(70\))0 4633 y(F)-8 b(or)32 b(this,)g(w)m(e)i(observ)m(e)g(that,)f(\014xed)g (an)m(y)h Fo(t)27 b Fs(2)i Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\),)1337 4862 y(lim)1335 4925 y Fp(\016)r Fm(!)p Fl(0)1491 4862 y Fw(\010)1561 4821 y Fp(t)p Fl(+)p Fp(\016)1680 4862 y Fw(\()p Fo(u)p Fw(\))1812 4877 y Fp(q)1893 4862 y Fw(=)44 b(\010)2083 4821 y Fp(t)2114 4862 y Fw(\()p Fo(u)p Fw(\))2246 4877 y Fp(q)2300 4862 y Fo(;)0 5122 y Fw(for)32 b(an)m(y)h Fo(q)f Fs(2)c Fw(\003,)k(thanks)i(to)e(Lemma)f (42.)43 b(Therefore,)34 b(b)m(y)f(Lemma)f(41,)0 5363 y(\(71\))970 b(lim)1142 5426 y Fp(\016)r Fm(!)p Fl(0)1298 5268 y Fn(X)1305 5480 y Fp(q)r Fm(2)p Fl(\003)1458 5363 y Fs(j)p Fw(\010)1556 5322 y Fp(t)p Fl(+)p Fp(\016)1675 5363 y Fw(\()p Fo(u)p Fw(\))1807 5378 y Fp(q)1866 5363 y Fs(\000)23 b Fw(\010)2036 5322 y Fp(t)2066 5363 y Fw(\()p Fo(u)p Fw(\))2198 5378 y Fp(q)2235 5363 y Fs(j)44 b Fw(=)g(0)17 b Fo(:)p eop %%Page: 48 48 48 47 bop 168 251 a Ft(48)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(Moreo)m(v)m(er,)34 b(for)e(an)m(y)h Fo(t)28 b(>)g Fw(0)k(and)h(an)m(y)g Fo(\016)n(;)17 b(\016)1710 414 y Fm(0)1761 450 y Fs(2)28 b Fq(R)44 b Fw(suc)m(h)34 b(that)e Fo(t)22 b Fw(+)g Fo(\016)n(;)34 b(t)23 b Fw(+)f Fo(\016)2851 414 y Fm(0)2901 450 y Fo(>)28 b Fw(0,)k(w)m(e)i(ha)m(v)m(e)g(that)638 588 y Fn(X)644 799 y Fp(p)p Fm(2)p Fl(\003)802 588 y Fn(X)808 788 y Fk(B)r Fj(2S)813 844 y Fk(B)r Fj(3)p Fk(p)969 588 y Fn(X)976 788 y Fk(A)p Fj(2S)981 844 y Fk(A)p Fj(3)p Fk(p)1132 547 y Fn(Z)1232 573 y Fl(1)1187 772 y(0)1288 538 y Fn(\014)1288 598 y(\014)1288 657 y(\014)1288 717 y(\014)1331 615 y Fo(@)5 b(H)1468 630 y Fp(B)p 1331 659 199 4 v 1354 751 a Fo(@)g(u)1466 766 y Fp(p)1539 682 y Fw(\(\010)1647 641 y Fp(t)p Fl(+)p Fp(\016)r(\034)1805 682 y Fw(\()p Fo(u)p Fw(\)\))21 b Fs(\000)2106 615 y Fo(@)5 b(H)2243 630 y Fp(B)p 2106 659 V 2129 751 a Fo(@)g(u)2241 766 y Fp(p)2314 682 y Fw(\(\010)2422 641 y Fp(t)2452 682 y Fw(\()p Fo(u)p Fw(\)\))2622 538 y Fn(\014)2622 598 y(\014)2622 657 y(\014)2622 717 y(\014)2677 682 y Fs(\001)2727 538 y Fn(\014)2727 598 y(\014)2727 657 y(\014)2727 717 y(\014)2770 615 y Fo(@)g(H)2907 630 y Fp(A)p 2770 659 195 4 v 2791 751 a Fo(@)g(u)2903 766 y Fp(p)2975 682 y Fw(\(\010)3083 641 y Fp(t)p Fl(+)p Fp(\016)3196 618 y Fj(0)3219 641 y Fp(\034)3263 682 y Fw(\()p Fo(u)p Fw(\)\))3433 538 y Fn(\014)3433 598 y(\014)3433 657 y(\014)3433 717 y(\014)3498 682 y Fo(d\034)477 1053 y Fs(\024)638 959 y Fn(X)644 1170 y Fp(p)p Fm(2)p Fl(\003)802 959 y Fn(X)808 1159 y Fk(B)r Fj(2S)813 1215 y Fk(B)r Fj(3)p Fk(p)969 959 y Fn(X)976 1159 y Fk(A)p Fj(2S)981 1215 y Fk(A)p Fj(3)p Fk(p)1132 959 y Fn(X)1135 1170 y Fp(q)r Fm(2)p Fp(B)1292 918 y Fn(Z)1392 944 y Fl(1)1348 1143 y(0)1448 918 y Fn(Z)1548 944 y Fl(1)1504 1143 y(0)1604 909 y Fn(\014)1604 969 y(\014)1604 1028 y(\014)1604 1088 y(\014)1679 986 y Fo(@)1735 950 y Fl(2)1776 986 y Fo(H)1857 1001 y Fp(B)p 1647 1030 303 4 v 1647 1122 a Fo(@)g(u)1759 1137 y Fp(p)1799 1122 y Fo(@)g(u)1911 1137 y Fp(q)1960 1053 y Fw(\()p Fo(\030)g Fw(\010)2116 1012 y Fp(t)p Fl(+)p Fp(\016)r(\034)2273 1053 y Fw(\()p Fo(u)p Fw(\))21 b(+)h(\(1)g Fs(\000)h Fo(\030)5 b Fw(\)\010)2889 1012 y Fp(t)2918 1053 y Fw(\()p Fo(u)p Fw(\)\))3088 909 y Fn(\014)3088 969 y(\014)3088 1028 y(\014)3088 1088 y(\014)638 1417 y Fs(\001)683 1272 y Fn(\014)683 1332 y(\014)683 1392 y(\014)683 1452 y(\014)725 1349 y Fo(@)g(H)862 1364 y Fp(A)p 725 1394 195 4 v 747 1485 a Fo(@)g(u)859 1500 y Fp(p)930 1417 y Fw(\(\010)1038 1376 y Fp(t)p Fl(+)p Fp(\016)1151 1352 y Fj(0)1175 1376 y Fp(\034)1218 1417 y Fw(\()p Fo(u)p Fw(\)\))1388 1272 y Fn(\014)1388 1332 y(\014)1388 1392 y(\014)1388 1452 y(\014)1443 1417 y Fs(\001)22 b(j)p Fw(\010)1591 1376 y Fp(t)p Fl(+)p Fp(\016)r(\034)1748 1417 y Fw(\()p Fo(u)p Fw(\))1880 1432 y Fp(q)1939 1417 y Fs(\000)h Fw(\010)2109 1376 y Fp(t)2139 1417 y Fw(\()p Fo(u)p Fw(\))2271 1432 y Fp(q)2309 1417 y Fs(j)17 b Fo(d\030)j(d\034)478 1702 y Fw(=)638 1608 y Fn(X)645 1819 y Fp(q)r Fm(2)p Fl(\003)798 1608 y Fn(X)805 1819 y Fp(p)p Fm(2)p Fl(\003)983 1608 y Fn(X)989 1808 y Fk(B)r Fj(2S)969 1864 y Fk(B)r Fj(3)p Fk(p;q)1169 1608 y Fn(X)1177 1808 y Fk(A)p Fj(2S)1181 1864 y Fk(A)p Fj(3)p Fk(p)1332 1567 y Fn(Z)1432 1593 y Fl(1)1387 1792 y(0)1488 1567 y Fn(Z)1587 1593 y Fl(1)1543 1792 y(0)1643 1558 y Fn(\014)1643 1618 y(\014)1643 1677 y(\014)1643 1737 y(\014)1719 1635 y Fo(@)1775 1599 y Fl(2)1815 1635 y Fo(H)1896 1650 y Fp(B)p 1687 1679 303 4 v 1687 1771 a Fo(@)5 b(u)1799 1786 y Fp(p)1839 1771 y Fo(@)g(u)1951 1786 y Fp(q)1999 1702 y Fw(\()p Fo(\030)g Fw(\010)2155 1661 y Fp(t)p Fl(+)p Fp(\016)r(\034)2312 1702 y Fw(\()p Fo(u)p Fw(\))22 b(+)g(\(1)g Fs(\000)g Fo(\030)5 b Fw(\)\010)2928 1661 y Fp(t)2958 1702 y Fw(\()p Fo(u)p Fw(\)\))3128 1558 y Fn(\014)3128 1618 y(\014)3128 1677 y(\014)3128 1737 y(\014)638 2066 y Fs(\001)683 1921 y Fn(\014)683 1981 y(\014)683 2041 y(\014)683 2101 y(\014)725 1998 y Fo(@)g(H)862 2013 y Fp(A)p 725 2043 195 4 v 747 2134 a Fo(@)g(u)859 2149 y Fp(p)930 2066 y Fw(\(\010)1038 2025 y Fp(t)p Fl(+)p Fp(\016)1151 2001 y Fj(0)1175 2025 y Fp(\034)1218 2066 y Fw(\()p Fo(u)p Fw(\)\))1388 1921 y Fn(\014)1388 1981 y(\014)1388 2041 y(\014)1388 2101 y(\014)1443 2066 y Fs(\001)22 b(j)p Fw(\010)1591 2025 y Fp(t)p Fl(+)p Fp(\016)r(\034)1748 2066 y Fw(\()p Fo(u)p Fw(\))1880 2081 y Fp(q)1939 2066 y Fs(\000)h Fw(\010)2109 2025 y Fp(t)2139 2066 y Fw(\()p Fo(u)p Fw(\))2271 2081 y Fp(q)2309 2066 y Fs(j)17 b Fo(d\030)j(d\034)477 2351 y Fs(\024)84 b Fo(C)708 2366 y Fl(1)764 2257 y Fn(X)771 2468 y Fp(q)r Fm(2)p Fl(\003)924 2257 y Fn(X)930 2468 y Fp(p)p Fm(2)p Fl(\003)1108 2257 y Fn(X)1115 2457 y Fk(B)r Fj(2S)1095 2513 y Fk(B)r Fj(3)p Fk(p;q)1292 2216 y Fn(Z)1392 2242 y Fl(1)1348 2441 y(0)1448 2216 y Fn(Z)1548 2242 y Fl(1)1503 2441 y(0)1604 2207 y Fn(\014)1604 2267 y(\014)1604 2326 y(\014)1604 2386 y(\014)1679 2284 y Fo(@)1735 2248 y Fl(2)1776 2284 y Fo(H)1857 2299 y Fp(B)p 1647 2328 303 4 v 1647 2420 a Fo(@)5 b(u)1759 2435 y Fp(p)1799 2420 y Fo(@)g(u)1911 2435 y Fp(q)1959 2351 y Fw(\()p Fo(\030)g Fw(\010)2115 2310 y Fp(t)p Fl(+)p Fp(\016)r(\034)2273 2351 y Fw(\()p Fo(u)p Fw(\))21 b(+)h(\(1)g Fs(\000)h Fo(\030)5 b Fw(\)\010)2889 2310 y Fp(t)2918 2351 y Fw(\()p Fo(u)p Fw(\)\))3088 2207 y Fn(\014)3088 2267 y(\014)3088 2326 y(\014)3088 2386 y(\014)638 2657 y Fs(\001j)p Fw(\010)764 2616 y Fp(t)p Fl(+)p Fp(\016)r(\034)921 2657 y Fw(\()p Fo(u)p Fw(\))1053 2672 y Fp(q)1112 2657 y Fs(\000)23 b Fw(\010)1282 2616 y Fp(t)1312 2657 y Fw(\()p Fo(u)p Fw(\))1444 2672 y Fp(q)1482 2657 y Fs(j)17 b Fo(d\030)j(d\034) 477 2827 y Fs(\024)84 b Fo(C)708 2842 y Fl(1)747 2827 y Fo(C)817 2842 y Fl(2)873 2733 y Fn(X)880 2944 y Fp(q)r Fm(2)p Fl(\003)1033 2827 y Fs(j)p Fw(\010)1131 2786 y Fp(t)p Fl(+)p Fp(\016)r(\034)1289 2827 y Fw(\()p Fo(u)p Fw(\))1421 2842 y Fp(q)1480 2827 y Fs(\000)23 b Fw(\010)1650 2786 y Fp(t)1680 2827 y Fw(\()p Fo(u)p Fw(\))1812 2842 y Fp(q)1850 2827 y Fs(j)17 b Fo(:)168 3102 y Fw(This)32 b(estimate)g(and)h(\(71\))f(giv)m(e)g(that)922 3340 y(lim)920 3403 y Fp(\016)r Fm(!)p Fl(0)1076 3245 y Fn(X)1083 3457 y Fp(p)p Fm(2)p Fl(\003)1241 3245 y Fn(X)1247 3446 y Fk(B)r Fj(2S)1252 3502 y Fk(B)r Fj(3)p Fk(p)1407 3245 y Fn(X)1415 3446 y Fk(A)p Fj(2S)1420 3502 y Fk(A)p Fj(3)p Fk(p)1570 3205 y Fn(Z)1670 3231 y Fl(1)1626 3430 y(0)1726 3196 y Fn(\014)1726 3255 y(\014)1726 3315 y(\014)1726 3375 y(\014)1769 3273 y Fo(@)5 b(H)1906 3288 y Fp(B)p 1769 3317 199 4 v 1792 3408 a Fo(@)g(u)1904 3423 y Fp(p)1978 3340 y Fw(\(\010)2086 3299 y Fp(t)p Fl(+)p Fp(\016)r(\034)2243 3340 y Fw(\()p Fo(u)p Fw(\)\))22 b Fs(\000)2544 3273 y Fo(@)5 b(H)2681 3288 y Fp(B)p 2544 3317 V 2567 3408 a Fo(@)g(u)2679 3423 y Fp(p)2753 3340 y Fw(\(\010)2861 3299 y Fp(t)2891 3340 y Fw(\()p Fo(u)p Fw(\)\))3061 3196 y Fn(\014)3061 3255 y(\014)3061 3315 y(\014)3061 3375 y(\014)926 3703 y Fs(\001)976 3559 y Fn(\014)976 3619 y(\014)976 3679 y(\014)976 3738 y(\014)1019 3636 y Fo(@)g(H)1156 3651 y Fp(A)p 1019 3681 195 4 v 1040 3772 a Fo(@)g(u)1152 3787 y Fp(p)1223 3703 y Fw(\(\010)1331 3662 y Fp(t)p Fl(+)p Fp(\016)r(\034)1489 3703 y Fw(\()p Fo(u)p Fw(\)\))1659 3559 y Fn(\014)1659 3619 y(\014)1659 3679 y(\014)1659 3738 y(\014)1725 3703 y Fo(d\034)55 b Fw(=)44 b(0)17 b Fo(:)168 3518 y Fw(\(72\))168 3930 y(Also,)959 4084 y Fo(f)1007 4099 y Fp(n)1054 4084 y Fw(\()p Fo(t)22 b Fw(+)g Fo(\016)t Fw(\))g Fs(\000)h Fo(f)1502 4099 y Fp(n)1549 4084 y Fw(\()p Fo(t)p Fw(\))p 959 4129 702 4 v 1286 4220 a Fo(\016)790 4407 y Fw(=)1018 4313 y Fn(X)1024 4513 y Fk(B)r Fj(2S)959 4569 y Fk(B)r Fj(\\)p Fk(X)1099 4577 y(n)1142 4569 y Fj(6)p Fi(=)p Fj(;)1256 4340 y Fo(H)1337 4355 y Fp(B)1398 4340 y Fw(\(\010)1506 4304 y Fp(t)p Fl(+)p Fp(\016)1625 4340 y Fw(\()p Fo(u)p Fw(\)\))e Fs(\000)i Fo(H)1997 4355 y Fp(B)2057 4340 y Fw(\(\010)2165 4304 y Fp(t)2195 4340 y Fw(\()p Fo(u)p Fw(\)\))p 1256 4384 1109 4 v 1787 4476 a Fo(\016)790 4781 y Fw(=)1018 4687 y Fn(X)1024 4887 y Fk(B)r Fj(2S)959 4943 y Fk(B)r Fj(\\)p Fk(X)1099 4951 y(n)1142 4943 y Fj(6)p Fi(=)p Fj(;)1246 4687 y Fn(X)1249 4898 y Fp(p)p Fm(2)p Fp(B)1417 4714 y Fw(1)p 1417 4759 49 4 v 1418 4850 a Fo(\016)1492 4646 y Fn(Z)1592 4672 y Fp(\016)1548 4871 y Fl(0)1656 4714 y Fo(@)5 b(H)1793 4729 y Fp(B)p 1656 4759 199 4 v 1679 4850 a Fo(@)g(u)1791 4865 y Fp(p)1865 4781 y Fw(\(\010)1973 4740 y Fp(t)p Fl(+)p Fp(\034)2097 4781 y Fw(\()p Fo(u)p Fw(\)\))2267 4671 y Fn(\020)2353 4714 y Fo(@)p 2335 4759 93 4 v 2335 4850 a(@)g(t)2438 4781 y Fw(\010)2508 4740 y Fp(t)p Fl(+)p Fp(\034)2631 4781 y Fw(\()p Fo(u)p Fw(\))2763 4671 y Fn(\021)2822 4850 y Fp(p)2879 4781 y Fo(d\034)790 5152 y Fw(=)83 b Fs(\000)1112 5058 y Fn(X)1118 5258 y Fk(B)r Fj(2S)1053 5314 y Fk(B)r Fj(\\)p Fk(X)1193 5322 y(n)1236 5314 y Fj(6)p Fi(=)p Fj(;)1341 5058 y Fn(X)1343 5269 y Fp(p)p Fm(2)p Fp(B)1504 5058 y Fn(X)1511 5258 y Fk(A)p Fj(2S)1516 5314 y Fk(A)p Fj(3)p Fk(p)1667 5017 y Fn(Z)1766 5043 y Fl(1)1722 5242 y(0)1832 5085 y Fo(@)5 b(H)1969 5100 y Fp(B)p 1832 5130 199 4 v 1855 5221 a Fo(@)g(u)1967 5236 y Fp(p)2041 5152 y Fw(\(\010)2149 5111 y Fp(t)p Fl(+)p Fp(\016)r(\034)2306 5152 y Fw(\()p Fo(u)p Fw(\)\))2486 5085 y Fo(@)g(H)2623 5100 y Fp(A)p 2486 5130 195 4 v 2507 5221 a Fo(@)g(u)2619 5236 y Fp(p)2690 5152 y Fw(\(\010)2798 5111 y Fp(t)p Fl(+)p Fp(\016)r(\034)2956 5152 y Fw(\()p Fo(u)p Fw(\)\))17 b Fo(d\034)27 b(;)168 5469 y Fw(thanks)33 b(to)g(\(62\))o(.)p eop %%Page: 49 49 49 48 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(49)0 450 y Fw(Therefore,)34 b(b)m(y)h(\(72\))o(,)405 866 y(lim)403 929 y Fp(\016)r Fm(!)p Fl(0)559 632 y Fn(\014)559 691 y(\014)559 751 y(\014)559 811 y(\014)559 871 y(\014)559 931 y(\014)559 990 y(\014)603 798 y Fo(f)651 813 y Fp(n)698 798 y Fw(\()p Fo(t)22 b Fw(+)g Fo(\016)t Fw(\))g Fs(\000)h Fo(f)1146 813 y Fp(n)1193 798 y Fw(\()p Fo(t)p Fw(\))p 603 843 702 4 v 930 934 a Fo(\016)1336 866 y Fw(+)1502 771 y Fn(X)1509 971 y Fk(B)r Fj(2S)1444 1028 y Fk(B)r Fj(\\)p Fk(X)1584 1036 y(n)1626 1028 y Fj(6)p Fi(=)p Fj(;)1731 771 y Fn(X)1734 983 y Fp(p)p Fm(2)p Fp(B)1894 771 y Fn(X)1902 971 y Fk(A)p Fj(2S)1906 1028 y Fk(A)p Fj(3)p Fk(p)2067 798 y Fo(@)5 b(H)2204 813 y Fp(B)p 2067 843 199 4 v 2090 934 a Fo(@)g(u)2202 949 y Fp(p)2276 866 y Fw(\(\010)2384 825 y Fp(t)2414 866 y Fw(\()p Fo(u)p Fw(\)\))2594 798 y Fo(@)g(H)2731 813 y Fp(A)p 2594 843 195 4 v 2614 934 a Fo(@)g(u)2726 949 y Fp(p)2798 866 y Fw(\(\010)2906 825 y Fp(t)2936 866 y Fw(\()p Fo(u)p Fw(\)\))3106 632 y Fn(\014)3106 691 y(\014)3106 751 y(\014)3106 811 y(\014)3106 871 y(\014)3106 931 y(\014)3106 990 y(\014)243 1240 y Fs(\024)85 b Fw(lim)403 1302 y Fp(\016)r Fm(!)p Fl(0)559 1145 y Fn(X)566 1356 y Fp(p)p Fm(2)p Fl(\003)724 1145 y Fn(X)730 1345 y Fk(B)r Fj(2S)735 1402 y Fk(B)r Fj(3)p Fk(p)891 1145 y Fn(X)898 1345 y Fk(A)p Fj(2S)903 1402 y Fk(A)p Fj(3)p Fk(p)1054 1104 y Fn(Z)1153 1130 y Fl(1)1109 1330 y(0)1209 1095 y Fn(\014)1209 1155 y(\014)1209 1215 y(\014)1209 1275 y(\014)1252 1172 y Fo(@)5 b(H)1389 1187 y Fp(B)p 1252 1217 199 4 v 1276 1308 a Fo(@)g(u)1388 1323 y Fp(p)1461 1240 y Fw(\(\010)1569 1198 y Fp(t)p Fl(+)p Fp(\016)r(\034)1726 1240 y Fw(\()p Fo(u)p Fw(\)\))22 b Fs(\000)2027 1172 y Fo(@)5 b(H)2164 1187 y Fp(B)p 2027 1217 V 2051 1308 a Fo(@)g(u)2163 1323 y Fp(p)2236 1240 y Fw(\(\010)2344 1198 y Fp(t)2374 1240 y Fw(\()p Fo(u)p Fw(\)\))2544 1095 y Fn(\014)2544 1155 y(\014)2544 1215 y(\014)2544 1275 y(\014)2598 1240 y Fs(\001)2648 1095 y Fn(\014)2648 1155 y(\014)2648 1215 y(\014)2648 1275 y(\014)2691 1172 y Fo(@)g(H)2828 1187 y Fp(A)p 2691 1217 195 4 v 2713 1308 a Fo(@)g(u)2825 1323 y Fp(p)2896 1240 y Fw(\(\010)3004 1198 y Fp(t)p Fl(+)p Fp(\016)r(\034)3162 1240 y Fw(\()p Fo(u)p Fw(\)\))3332 1095 y Fn(\014)3332 1155 y(\014)3332 1215 y(\014)3332 1275 y(\014)3398 1240 y Fo(d\034)403 1611 y Fw(+)496 1516 y Fn(X)502 1727 y Fp(p)p Fm(2)p Fl(\003)660 1516 y Fn(X)666 1716 y Fk(B)r Fj(2S)671 1772 y Fk(B)r Fj(3)p Fk(p)827 1516 y Fn(X)834 1716 y Fk(A)p Fj(2S)839 1772 y Fk(A)p Fj(3)p Fk(p)990 1475 y Fn(Z)1089 1501 y Fl(1)1045 1701 y(0)1145 1466 y Fn(\014)1145 1526 y(\014)1145 1586 y(\014)1145 1645 y(\014)1189 1543 y Fo(@)g(H)1326 1558 y Fp(B)p 1189 1588 199 4 v 1212 1679 a Fo(@)g(u)1324 1694 y Fp(p)1397 1611 y Fw(\(\010)1505 1569 y Fp(t)p Fl(+)p Fp(\016)r(\034)1663 1611 y Fw(\()p Fo(u)p Fw(\)\))21 b Fs(\000)1964 1543 y Fo(@)5 b(H)2101 1558 y Fp(B)p 1964 1588 V 1987 1679 a Fo(@)g(u)2099 1694 y Fp(p)2172 1611 y Fw(\(\010)2280 1569 y Fp(t)2310 1611 y Fw(\()p Fo(u)p Fw(\)\))2480 1466 y Fn(\014)2480 1526 y(\014)2480 1586 y(\014)2480 1645 y(\014)2535 1611 y Fs(\001)2585 1466 y Fn(\014)2585 1526 y(\014)2585 1586 y(\014)2585 1645 y(\014)2628 1543 y Fo(@)g(H)2765 1558 y Fp(A)p 2628 1588 195 4 v 2649 1679 a Fo(@)g(u)2761 1694 y Fp(p)2832 1611 y Fw(\(\010)2940 1569 y Fp(t)2970 1611 y Fw(\()p Fo(u)p Fw(\)\))3140 1466 y Fn(\014)3140 1526 y(\014)3140 1586 y(\014)3140 1645 y(\014)3206 1611 y Fo(d\034)244 1894 y Fw(=)83 b(0)17 b Fo(;)0 2158 y Fw(th)m(us)34 b(yielding)c(the)j(pro)s(of)f(of)39 b(\(70\).)0 2275 y(Th)m(us,)34 b(from)e(\(69\))f(and)i(\(70\),)1043 2549 y Fs(j)1103 2522 y Fw(_)1071 2549 y Fo(f)1119 2564 y Fp(n)1166 2549 y Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)g Fo(g)t Fw(\()p Fo(t)p Fw(\))p Fs(j)884 2845 y Fw(=)1043 2611 y Fn(\014)1043 2671 y(\014)1043 2730 y(\014)1043 2790 y(\014)1043 2850 y(\014)1043 2910 y(\014)1043 2970 y(\014)1144 2750 y(X)1151 2950 y Fk(B)r Fj(2S)1086 3007 y Fk(B)r Fj(\\)p Fk(X)1226 3015 y(n)1268 3007 y Fi(=)p Fj(;)1373 2750 y Fn(X)1376 2962 y Fp(p)p Fm(2)p Fp(B)1536 2750 y Fn(X)1544 2950 y Fk(A)p Fj(2S)1549 3007 y Fk(A)p Fj(3)p Fk(p)1709 2778 y Fo(@)5 b(H)1846 2793 y Fp(B)p 1709 2822 199 4 v 1733 2913 a Fo(@)g(u)1845 2928 y Fp(p)1918 2845 y Fw(\(\010)2026 2804 y Fp(t)2056 2845 y Fw(\()p Fo(u)p Fw(\)\))2236 2778 y Fo(@)g(H)2373 2793 y Fp(A)p 2236 2822 195 4 v 2256 2913 a Fo(@)g(u)2368 2928 y Fp(p)2440 2845 y Fw(\(\010)2548 2804 y Fp(t)2578 2845 y Fw(\()p Fo(u)p Fw(\)\))2748 2611 y Fn(\014)2748 2671 y(\014)2748 2730 y(\014)2748 2790 y(\014)2748 2850 y(\014)2748 2910 y(\014)2748 2970 y(\014)0 2845 y Fw(\(73\))884 3331 y(=)1043 3067 y Fn(\014)1043 3126 y(\014)1043 3186 y(\014)1043 3246 y(\014)1043 3306 y(\014)1043 3366 y(\014)1043 3425 y(\014)1043 3485 y(\014)1076 3236 y(X)1082 3447 y Fp(p)p Fm(2)p Fl(\003)1315 3236 y Fn(X)1321 3436 y Fk(B)r Fj(2S)1326 3484 y Fk(B)r Fj(3)p Fk(p)1256 3538 y(B)r Fj(\\)p Fk(X)1396 3546 y(n)1439 3538 y Fi(=)p Fj(;)1556 3236 y Fn(X)1564 3436 y Fk(A)p Fj(2S)1568 3493 y Fk(A)p Fj(3)p Fk(p)1729 3263 y Fo(@)g(H)1866 3278 y Fp(B)p 1729 3308 199 4 v 1752 3399 a Fo(@)g(u)1864 3414 y Fp(p)1938 3331 y Fw(\(\010)2046 3290 y Fp(t)2076 3331 y Fw(\()p Fo(u)p Fw(\)\))2256 3263 y Fo(@)g(H)2393 3278 y Fp(A)p 2256 3308 195 4 v 2276 3399 a Fo(@)g(u)2388 3414 y Fp(p)2460 3331 y Fw(\(\010)2568 3290 y Fp(t)2598 3331 y Fw(\()p Fo(u)p Fw(\)\))2768 3067 y Fn(\014)2768 3126 y(\014)2768 3186 y(\014)2768 3246 y(\014)2768 3306 y(\014)2768 3366 y(\014)2768 3425 y(\014)2768 3485 y(\014)2833 3331 y Fo(:)0 3782 y Fw(Thence,)35 b(if)c(w)m(e)j(set) 1263 4054 y Fo(\013)1325 4069 y Fp(p)1365 4054 y Fw(\()p Fo(t)p Fw(\))44 b(:=)1670 3959 y Fn(X)1677 4159 y Fk(A)p Fj(2S)1682 4216 y Fk(A)p Fj(3)p Fk(p)1843 3987 y Fo(@)5 b(H)1980 4002 y Fp(A)p 1843 4031 V 1864 4122 a Fo(@)g(u)1976 4137 y Fp(p)2047 4054 y Fw(\(\010)2155 4013 y Fp(t)2185 4054 y Fw(\()p Fo(u)p Fw(\)\))17 b Fo(;)0 4441 y Fw(w)m(e)34 b(ha)m(v)m(e)g(that)0 4706 y(\(74\))1391 4611 y Fn(X)1397 4822 y Fp(p)p Fm(2)p Fl(\003)1552 4706 y Fs(j)p Fo(\013)1642 4721 y Fp(p)1681 4706 y Fw(\()p Fo(t)p Fw(\))p Fs(j)44 b(\024)h Fo(C)2056 4721 y Fl(2)2112 4706 y Fo(]F)30 b(;)0 5077 y Fw(for)i(an)m(y)h Fo(t)28 b Fs(2)g Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\),)32 b(due)h(to)f(\(67\),)g(and)h(so)0 5363 y(\(75\))1166 b(lim)1289 5423 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1616 5268 y Fn(X)1544 5480 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1789 5488 y Fk(n)1848 5363 y Fs(j)p Fo(\013)1938 5378 y Fp(p)1977 5363 y Fw(\()p Fo(t)p Fw(\))p Fs(j)44 b Fw(=)g(0)17 b Fo(;)p eop %%Page: 50 50 50 49 bop 168 251 a Ft(50)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fw(for)32 b(an)m(y)h Fo(t)28 b Fs(2)g Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\).)42 b(Also,)32 b(from)g(\(70\))o(,)807 835 y Fs(j)867 809 y Fw(_)835 835 y Fo(f)883 850 y Fp(n)930 835 y Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)h Fo(g)t Fw(\()p Fo(t)p Fw(\))p Fs(j)82 b(\024)1595 571 y Fn(\014)1595 631 y(\014)1595 691 y(\014)1595 751 y(\014)1595 810 y(\014)1595 870 y(\014)1595 930 y(\014)1595 990 y(\014)1628 741 y(X)1635 952 y Fp(p)p Fm(2)p Fl(\003)1867 741 y Fn(X)1873 941 y Fk(B)r Fj(2S)1878 988 y Fk(B)r Fj(3)p Fk(p)1809 1042 y(B)r Fj(\\)p Fk(X)1949 1050 y(n)1991 1042 y Fi(=)p Fj(;)2106 725 y Fn(\020)2166 835 y Fo(\013)2228 850 y Fp(p)2267 835 y Fw(\()p Fo(t)p Fw(\))2388 768 y Fo(@)5 b(H)2525 783 y Fp(B)p 2388 812 199 4 v 2412 904 a Fo(@)g(u)2524 919 y Fp(p)2597 835 y Fw(\(\010)2705 794 y Fp(t)2735 835 y Fw(\()p Fo(u)p Fw(\)\))2905 725 y Fn(\021)2964 571 y(\014)2964 631 y(\014)2964 691 y(\014)2964 751 y(\014)2964 810 y(\014)2964 870 y(\014)2964 930 y(\014)2964 990 y(\014)1435 1362 y Fw(=)1595 1098 y Fn(\014)1595 1158 y(\014)1595 1218 y(\014)1595 1277 y(\014)1595 1337 y(\014)1595 1397 y(\014)1595 1457 y(\014)1595 1516 y(\014)1700 1267 y(X)1628 1479 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1873 1487 y Fk(n)2010 1267 y Fn(X)2016 1467 y Fk(B)r Fj(2S)2021 1515 y Fk(B)r Fj(3)p Fk(p)1952 1569 y(B)r Fj(\\)p Fk(X)2092 1577 y(n)2134 1569 y Fi(=)p Fj(;)2249 1251 y Fn(\020)2309 1362 y Fo(\013)2371 1377 y Fp(p)2410 1362 y Fw(\()p Fo(t)p Fw(\))2531 1295 y Fo(@)g(H)2668 1310 y Fp(B)p 2531 1339 V 2555 1430 a Fo(@)g(u)2667 1445 y Fp(p)2740 1362 y Fw(\(\010)2848 1321 y Fp(t)2878 1362 y Fw(\()p Fo(u)p Fw(\)\))3048 1251 y Fn(\021)3107 1098 y(\014)3107 1158 y(\014)3107 1218 y(\014)3107 1277 y(\014)3107 1337 y(\014)3107 1397 y(\014)3107 1457 y(\014)3107 1516 y(\014)1435 1859 y Fs(\024)1667 1764 y Fn(X)1595 1976 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1840 1984 y Fk(n)1899 1629 y Fn(0)1899 1804 y(B)1899 1868 y(@)1986 1859 y Fs(j)p Fo(\013)2076 1874 y Fp(p)2115 1859 y Fw(\()p Fo(t)p Fw(\))p Fs(j)22 b(\001)2330 1764 y Fn(X)2336 1964 y Fk(B)r Fj(2S)2341 2021 y Fk(B)r Fj(3)p Fk(p)2494 1714 y Fn(\014)2494 1774 y(\014)2494 1834 y(\014)2494 1894 y(\014)2537 1791 y Fo(@)5 b(H)2674 1806 y Fp(B)p 2537 1836 V 2560 1927 a Fo(@)g(u)2672 1942 y Fp(p)2746 1859 y Fw(\(\010)2854 1818 y Fp(t)2884 1859 y Fw(\()p Fo(u)p Fw(\)\))3054 1714 y Fn(\014)3054 1774 y(\014)3054 1834 y(\014)3054 1894 y(\014)3086 1629 y(1)3086 1804 y(C)3086 1868 y(A)1435 2185 y Fs(\024)83 b Fo(C)1665 2200 y Fl(1)1792 2090 y Fn(X)1721 2302 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1966 2310 y Fk(n)2025 2185 y Fs(j)p Fo(\013)2115 2200 y Fp(p)2154 2185 y Fw(\()p Fo(t)p Fw(\))p Fs(j)17 b Fo(:)168 2513 y Fw(F)-8 b(rom)31 b(this)h(and)h (\(75\))o(,)g(w)m(e)g(conclude)g(that)1825 2487 y(_)1793 2513 y Fo(f)1841 2528 y Fp(n)1920 2513 y Fw(con)m(v)m(erges)i(to)d Fo(g)k Fw(p)s(oin)m(t)m(wise)d(in)e(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\).)168 2630 y(Moreo)m(v)m(er,)34 b(b)m(y)f(the)g(latter)f (estimate,)g(\(68\))g(and)g(\(74\),)g(w)m(e)i(ha)m(v)m(e)g(that)1076 2855 y Fs(j)1136 2829 y Fw(_)1104 2855 y Fo(f)1152 2870 y Fp(n)1199 2855 y Fw(\()p Fo(t)p Fw(\))p Fs(j)28 b(\024)g(j)p Fo(g)t Fw(\()p Fo(t)p Fw(\))p Fs(j)21 b Fw(+)h Fo(C)1878 2870 y Fl(1)1933 2761 y Fn(X)1940 2972 y Fp(p)p Fm(2)p Fl(\003)2094 2855 y Fs(j)p Fo(\013)2184 2870 y Fp(p)2223 2855 y Fw(\()p Fo(t)p Fw(\))p Fs(j)28 b(\024)g Fw(2)p Fo(C)2614 2870 y Fl(1)2653 2855 y Fo(C)2723 2870 y Fl(2)2762 2855 y Fo(]F)i(:)168 3160 y Fw(Thence,)k(b)m(y)g(the)f(Dominated)d(Con) m(v)m(ergence)36 b(Theorem,)1111 3425 y(lim)1060 3485 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1315 3290 y Fn(Z)1415 3316 y Fl(+)p Fm(1)1370 3515 y Fl(0)1593 3399 y Fw(_)1561 3425 y Fo(f)1609 3440 y Fp(n)1656 3425 y Fw(\()p Fo(t)p Fw(\))p Fo(')p Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)44 b Fw(=)2209 3290 y Fn(Z)2309 3316 y Fl(+)p Fm(1)2264 3515 y Fl(0)2455 3425 y Fo(g)t Fw(\()p Fo(t)p Fw(\))p Fo(')p Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)g(;)168 3686 y Fw(for)32 b(an)m(y)h Fo(')28 b Fs(2)g Fo(C)764 3650 y Fm(1)757 3711 y Fl(0)838 3686 y Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\).)43 b(Then,)33 b(b)m(y)i(\(65\),)1013 3821 y Fn(Z)1113 3847 y Fl(+)p Fm(1)1069 4047 y Fl(0)1259 3957 y Fo(S)1319 3972 y Fp(\033)1366 3957 y Fw(\(\010)1474 3916 y Fp(t)1504 3957 y Fw(\()p Fo(u)p Fw(\)\))26 b(_)-53 b Fo(')o Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)28 b Fw(=)79 b(lim)2082 4017 y Fp(n)p Fm(!)p Fl(+)p Fm(1)2338 3821 y Fn(Z)2437 3847 y Fl(+)p Fm(1)2393 4047 y Fl(0)2583 3957 y Fo(f)2631 3972 y Fp(n)2678 3957 y Fw(\()p Fo(t)p Fw(\))27 b(_)-54 b Fo(')p Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)1013 4234 y Fw(=)28 b Fs(\000)69 b Fw(lim)1211 4293 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1466 4098 y Fn(Z)1566 4124 y Fl(+)p Fm(1)1522 4323 y Fl(0)1744 4207 y Fw(_)1712 4234 y Fo(f)1760 4249 y Fp(n)1807 4234 y Fw(\()p Fo(t)p Fw(\))p Fo(')p Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)28 b Fw(=)f Fs(\000)2421 4098 y Fn(Z)2521 4124 y Fl(+)p Fm(1)2477 4323 y Fl(0)2667 4234 y Fo(g)t Fw(\()p Fo(t)p Fw(\))p Fo(')p Fw(\()p Fo(t)p Fw(\))17 b Fo(dt)g(;)168 4494 y Fw(for)32 b(an)m(y)h Fo(')28 b Fs(2)g Fo(C)764 4458 y Fm(1)757 4519 y Fl(0)838 4494 y Fw(\(0)p Fo(;)17 b Fw(+)p Fs(1)p Fw(\).)168 4613 y(This)30 b(implies)e(that)948 4574 y Fp(d)p 935 4590 62 4 v 935 4647 a(dt)1007 4613 y Fo(S)1067 4628 y Fp(\033)1114 4613 y Fw(\(\010)1222 4577 y Fp(t)1252 4613 y Fw(\()p Fo(u)p Fw(\)\))f(=)h Fo(g)33 b Fw(\(in)c(the)i(sense)g(of)f(distributions)f(and)h(thence)h (p)s(oin)m(t)m(wise\),)168 4730 y(whic)m(h,)i(together)g(with)f(\(66\)) g(giv)m(es)h(\(63\))o(.)1993 b Fg(\003)168 4937 y Fw(Giv)m(en)29 b(an)m(y)i Fo(s)d Fs(2)g Fw([0)p Fo(;)17 b Fw(1],)29 b(w)m(e)i(no)m(w)g(consider)f(the)g(linear)e(in)m(terp)s(olation)g(b)s (et)m(w)m(een)j Fo(u)3208 4901 y Fl(\(0\))3332 4937 y Fw(and)f Fo(u)3575 4901 y Fl(\(1\))3669 4937 y Fw(,)g(b)m(y)168 5054 y(de\014ning)1395 5219 y Fo(u)1451 5178 y Fl(\()p Fp(s)p Fl(\))1586 5219 y Fw(:=)45 b(\(1)22 b Fs(\000)g Fo(s)p Fw(\))17 b Fo(u)2099 5178 y Fl(\(0\))2231 5219 y Fw(+)39 b Fo(s)17 b(u)2465 5178 y Fl(\(1\))2575 5219 y Fo(:)168 5406 y Fw(It)32 b(is)g(easily)g(seen)i(that)f Fo(u)1120 5370 y Fl(\()p Fp(s)p Fl(\))1238 5406 y Fs(2)28 b(C)6 b Fw(.)44 b(F)-8 b(urthermore,)32 b(the)h(follo)m(wing)d (compactness)k(result)e(holds:)p eop %%Page: 51 51 51 50 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(51)0 468 y Fz(Corollary)28 b(43.)36 b Fr(Fixe)-5 b(d)26 b Fo(s)i Fs(2)g Fw([0)p Fo(;)17 b Fw(1])p Fr(,)28 b(ther)-5 b(e)28 b(exists)f(a)g(se)-5 b(quenc)g(e)27 b Fo(t)2342 483 y Fp(n;s)2468 468 y Fr(and)g(a)h(c)-5 b(on\014gur)g(ation)26 b Fo(u)3360 417 y Fl(\()p Fp(s)p Fl(\))3360 480 y Fp(?)3479 468 y Fs(2)i(C)6 b Fr(,)0 584 y(which)34 b(is)h(a)f(solution)h(of)f (our)i(variational)d(e)-5 b(quation,)35 b(so)f(that)1424 750 y Fw(lim)1372 810 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1628 750 y Fo(t)1663 765 y Fp(n;s)1790 750 y Fw(=)27 b(+)p Fs(1)1017 979 y Fr(and)252 b Fw(lim)1372 1039 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1628 864 y Fn(\015)1628 924 y(\015)1628 984 y(\015)1683 979 y Fw(\010)1753 938 y Fp(t)1778 946 y Fk(n;s)1874 979 y Fw(\()p Fo(u)1968 938 y Fl(\()p Fp(s)p Fl(\))2059 979 y Fw(\))22 b Fs(\000)g Fo(u)2274 938 y Fl(\()p Fp(s)p Fl(\))2274 1003 y Fp(?)2366 864 y Fn(\015)2366 924 y(\015)2366 984 y(\015)2449 979 y Fw(=)27 b(0)17 b Fo(:)0 1191 y Fz(Pro)s(of.)47 b Fw(Let)34 b Fo(f)11 b Fw(\()p Fo(t)p Fw(\))29 b(:=)h Fo(S)918 1206 y Fp(\033)965 1191 y Fw(\(\010)1073 1155 y Fp(t)1103 1191 y Fw(\()p Fo(u)p Fw(\)\).)46 b(By)34 b(Lemmata)e(40)i(and)f(42,)h(w)m(e)h(kno)m (w)g(that)e Fo(f)45 b Fw(is)33 b(decreasing)0 1307 y(and)g(b)s(ounded)g (b)m(y)g(b)s(elo)m(w.)44 b(Th)m(us,)1458 1473 y(lim)1415 1533 y Fp(t)p Fm(!)p Fl(+)p Fm(1)1653 1473 y Fo(f)11 b Fw(\()p Fo(t)p Fw(\))28 b(=)55 b(inf)1955 1540 y Fl([0)p Fp(;)p Fm(1)p Fl(\))2144 1473 y Fo(f)27 b(:)0 1694 y Fw(Then,)34 b(there)f(m)m(ust)g(exists)g(a)f(div)m(ergen)m(t)i (sequence)h(of)d(times)g Fo(t)2360 1709 y Fp(n;s)2492 1694 y Fw(so)h(that)0 1860 y(\(76\))1323 b(lim)1445 1920 y Fp(n)p Fm(!)p Fl(+)p Fm(1)1700 1860 y Fo(f)1759 1819 y Fm(0)1782 1860 y Fw(\()p Fo(t)1855 1875 y Fp(n;s)1955 1860 y Fw(\))27 b(=)h(0)17 b Fo(:)0 2069 y Fw(If)28 b(not,)h(one)f(w)m (ould)f(ha)m(v)m(e)j(that)d Fo(f)1229 2033 y Fm(0)1252 2069 y Fw(\()p Fo(t)p Fw(\))h Fs(\024)g(\000)p Fo(c)g Fw(for)g(some)f Fo(c)h(>)g Fw(0)f(and)h(an)m(y)h Fo(t)f Fw(large)e(enough,)k(but)e(then)1158 2236 y(0)g(=)70 b(lim)1338 2295 y Fp(t)p Fm(!)p Fl(+)p Fm(1)1577 2236 y Fo(f)11 b Fw(\()p Fo(t)22 b Fw(+)g(1\))f Fs(\000)i Fo(f)11 b Fw(\()p Fo(t)p Fw(\))28 b Fs(\024)g(\000)p Fo(c)17 b(;)0 2440 y Fw(whic)m(h)33 b(is)f(a)g(con)m(tradiction.)0 2556 y(F)-8 b(urther,)24 b(b)m(y)f(Lemma)e(42)h(and)g(Corollary)e(39,)k (w)m(e)f(ma)m(y)f(supp)s(ose,)j(p)s(ossibly)d(taking)f(subsequences,)0 2684 y(that)36 b(\010)285 2648 y Fp(t)310 2656 y Fk(n;s)406 2684 y Fw(\()p Fo(u)500 2648 y Fl(\()p Fp(s)p Fl(\))591 2684 y Fw(\))g(con)m(v)m(erges)i(in)e(the)g(sup)q(-norm,)g(sa)m(y)h(to) f(a)g(suitable)f Fo(u)2663 2633 y Fl(\()p Fp(s)p Fl(\))2663 2697 y Fp(?)2788 2684 y Fs(2)g(C)6 b Fw(.)54 b(By)37 b(Lemma)e(41,)0 2818 y(\(76\))o(,)e(\(63\))f(and)h(\(11\))o(,)g(w)m(e)g (ha)m(v)m(e)h(that)f Fo(u)1498 2767 y Fl(\()p Fp(s)p Fl(\))1498 2831 y Fp(?)1621 2818 y Fw(is)f(a)h(solution)e(of)h(our)g(v) -5 b(ariational)29 b(equation.)219 b Fg(\003)0 3061 y Fw(4.3.)49 b Fz(The)k(basin)h(of)g(attraction)f(of)h(the)f(heat)h (\015o)m(w.)49 b Fw(W)-8 b(e)47 b(no)m(w)h(follo)m(w)d(the)i(v)-5 b(ariational)0 3177 y(pattern)37 b(p)s(erformed)f(in)f([dlL)-11 b(V05)o(].)56 b(F)-8 b(or)35 b(this,)i(w)m(e)h(consider)e(the)h (\\basin)f(of)g(attraction")f(of)h(the)0 3296 y(con\014gurations)h Fo(u)690 3260 y Fl(\(0\))822 3296 y Fw(and)h Fo(u)1073 3260 y Fl(\(1\))1204 3296 y Fw(with)f(resp)s(ect)i(to)f(the)g(heat)g (\015o)m(w)g(\010)2568 3260 y Fp(t)2598 3296 y Fw(.)59 b(Namely)-8 b(,)38 b(for)f Fo(i)g Fs(2)f(f)p Fw(0)p Fo(;)17 b Fw(1)p Fs(g)p Fw(,)0 3413 y(w)m(e)34 b(de\014ne)748 3545 y Fo(\014)803 3560 y Fp(i)875 3545 y Fw(:=)1022 3434 y Fn(n)1089 3545 y Fo(s)27 b Fs(2)i Fw([0)p Fo(;)17 b Fw(1])97 b Fs(j)156 b Fw(lim)1691 3605 y Fp(t)p Fm(!)p Fl(+)p Fm(1)1929 3430 y Fn(\015)1929 3490 y(\015)1929 3550 y(\015)1985 3545 y Fw(\010)2055 3504 y Fp(t)2085 3545 y Fw(\()p Fo(u)2179 3504 y Fl(\()p Fp(s)p Fl(\))2270 3545 y Fw(\))22 b Fs(\000)h Fo(u)2486 3504 y Fl(\()p Fp(i)p Fl(\))2568 3430 y Fn(\015)2568 3490 y(\015)2568 3550 y(\015)2651 3545 y Fw(=)28 b(0)2804 3434 y Fn(o)2887 3545 y Fo(:)0 3745 y Fw(Note)33 b(that,)h(since)f Fo(u)771 3709 y Fl(\()p Fp(i)p Fl(\))887 3745 y Fw(is)g(a)f(solution)g(of)h(our) g(v)-5 b(ariational)29 b(equation,)34 b(then)f(\010)2927 3709 y Fp(t)2958 3745 y Fw(\()p Fo(u)3052 3709 y Fl(\()p Fp(i)p Fl(\))3134 3745 y Fw(\))c(=)f Fo(u)3361 3709 y Fl(\()p Fp(i)p Fl(\))3444 3745 y Fw(,)33 b(and)0 3862 y(so)25 b Fo(i)j Fs(2)g Fo(\014)322 3877 y Fp(i)350 3862 y Fw(,)f(for)d Fo(i)k Fs(2)g(f)p Fw(0)p Fo(;)17 b Fw(1)p Fs(g)p Fw(.)40 b(W)-8 b(e)26 b(also)e(consider)h(the)g (con\014gurations)g(in)f Fs(C)31 b Fw(with)25 b(zero)g(renormalized)0 3978 y(energy)-8 b(,)34 b(that)e(is)g(the)h(set)1142 4129 y Fo(\026)1201 4144 y Fp(\033)1292 4129 y Fw(:=)1439 4019 y Fn(n)1506 4129 y Fo(u)27 b Fs(2)h(C)104 b(j)97 b Fo(S)2024 4144 y Fp(\033)2071 4129 y Fw(\()p Fo(u)p Fw(\))27 b(=)g(0)p Fo(:)2409 4019 y Fn(o)2492 4129 y Fo(:)0 4301 y Fw(W)-8 b(e)28 b(no)m(w)h(in)m(tro)s(duce)f(the)h(set)g (of)e(the)i(solutions)e(of)g(our)h(v)-5 b(ariational)25 b(problem)i(whic)m(h)h(b)s(elong)f(to)h Fs(C)0 4417 y Fw(and)33 b(whose)g Fo(S)537 4432 y Fp(\033)584 4417 y Fw(-v)-5 b(alue)32 b(is)g(less)h(than)f Fo(r)s Fw(.)44 b(More)33 b(precisely)-8 b(,)32 b(giv)m(en)h Fo(r)d(>)e Fw(0,)k(w)m(e)i(de\014ne)820 4610 y Fo(\027)868 4625 y Fp(\033)915 4610 y Fw(\()p Fo(r)s Fw(\))44 b(:=)1229 4499 y Fn(n)1296 4610 y Fo(u)27 b Fs(2)h(C)104 b(j)97 b Fo(u)32 b Fw(is)g(a)h(solution)e(of)h(our)1074 4826 y(v)-5 b(ariational)29 b(equation)k(and)f Fo(S)2207 4841 y Fp(\033)2254 4826 y Fw(\()p Fo(u)p Fw(\))27 b Fs(\024)h Fo(r)2565 4716 y Fn(o)2648 4826 y Fo(:)0 5018 y Fw(By)33 b(Lemma)e(40,)h(w)m(e)i(ha)m(v)m(e)g(that)1334 5187 y Fo(u)1390 5146 y Fl(\(0\))1484 5187 y Fo(;)45 b(u)1612 5146 y Fl(\(1\))1733 5187 y Fs(2)28 b Fo(\026)1886 5202 y Fp(\033)1961 5187 y Fs(\032)g Fo(\027)2114 5202 y Fp(\033)2161 5187 y Fw(\()p Fo(r)s Fw(\))17 b Fo(;)0 5353 y Fw(for)35 b(an)m(y)g Fo(r)g(>)d Fw(0.)50 b(W)-8 b(e)36 b(no)m(w)g(in)m(v)m (estigate)f(the)g(prop)s(erties)g(of)g(the)g(basins)g(of)g(attraction)f (when)i Fo(\026)3615 5368 y Fp(\033)0 5469 y Fw(and)d Fo(\027)238 5484 y Fp(\033)285 5469 y Fw(\()p Fo(r)s Fw(\))f(are)g(\\as)h(small)d(as)j(p)s(ossible".)p eop %%Page: 52 52 52 51 bop 168 251 a Ft(52)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 450 y Fz(Lemma)37 b(44.)42 b Fr(Assume)34 b(that)168 614 y Fw(\(77\))1201 b Fo(\026)1602 629 y Fp(\033)1670 614 y Fs(\000)23 b(f)p Fo(u)1876 573 y Fl(\(0\))1970 614 y Fo(;)33 b(u)2086 573 y Fl(\(1\))2180 614 y Fs(g)27 b Fw(=)h Fs(;)17 b Fo(:)168 782 y Fr(L)-5 b(et)26 b Fo(r)31 b(>)c Fw(0)f Fr(and)g(supp)-5 b(ose)25 b(that)h(ther)-5 b(e)27 b(exists)e Fo(\015)33 b Fs(2)28 b Fo(C)7 b Fw(\([0)p Fo(;)17 b Fw(1])p Fo(;)g Fs(C)6 b Fw(\))25 b Fr(such)h(that)h Fo(\015)5 b Fw(\(0\))27 b(=)h Fo(u)3216 746 y Fl(\(0\))3310 782 y Fr(,)f Fo(\015)5 b Fw(\(1\))28 b(=)f Fo(u)3735 746 y Fl(\(1\))168 898 y Fr(and)34 b Fo(S)417 913 y Fp(\033)464 898 y Fw(\()p Fo(\015)5 b Fw(\()p Fo(t)p Fw(\)\))28 b Fs(\024)g Fo(r)37 b Fr(for)e(any)g Fo(t)28 b Fs(2)g Fw([0)p Fo(;)17 b Fw(1])p Fr(.)44 b(Then,)34 b Fo(r)c Fs(\025)e Fo(r)2198 913 y Fl(0)2238 898 y Fr(,)34 b(for)h(some)f Fo(r)2751 913 y Fl(0)2818 898 y Fo(>)28 b Fw(0)p Fr(.)168 1038 y Fz(Pro)s(of.)43 b Fw(W)-8 b(e)33 b(argue)g(b)m(y)g(con)m (tradiction.)42 b(Supp)s(ose)34 b(that,)e(for)g(an)m(y)i Fo(n)28 b Fs(2)g Fq(N)9 b Fw(,)38 b(there)c(exists)168 1203 y(\(78\))1322 b Fo(\015)1715 1218 y Fp(n)1789 1203 y Fs(2)28 b Fo(C)7 b Fw(\([0)p Fo(;)17 b Fw(1])p Fo(;)g Fs(C)6 b Fw(\))168 1367 y(suc)m(h)34 b(that)1380 1531 y Fo(\015)1431 1546 y Fp(n)1477 1531 y Fw(\(0\))27 b(=)h Fo(u)1789 1495 y Fl(\(0\))1883 1531 y Fw(,)33 b Fo(\015)1994 1546 y Fp(n)2040 1531 y Fw(\(1\))27 b(=)h Fo(u)2352 1495 y Fl(\(1\))2478 1531 y Fw(and)-2467 b(\(79\))1347 1677 y Fo(S)1407 1692 y Fp(\033)1454 1677 y Fw(\()p Fo(\015)1543 1692 y Fp(n)1589 1677 y Fw(\()p Fo(t)p Fw(\)\))28 b Fs(\024)g Fw(1)p Fo(=n)33 b Fw(for)f(an)m(y)h Fo(t)28 b Fs(2)g Fw([0)p Fo(;)17 b Fw(1].)-2605 b(\(80\))168 1841 y(Let)32 b(us)i(de\014ne)1505 1965 y Fo(g)1552 1980 y Fp(n)1599 1965 y Fw(\()p Fo(t)p Fw(\))28 b(:=)f Fs(k)p Fo(\015)1969 1980 y Fp(n)2016 1965 y Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)g Fo(u)2304 1924 y Fl(\(0\))2398 1965 y Fs(k)17 b Fo(:)168 2112 y Fw(Then,)40 b Fo(g)503 2127 y Fp(n)587 2112 y Fs(2)d Fo(C)7 b Fw(\([0)p Fo(;)17 b Fw(1]\),)38 b(thanks)h(to)f(\(78\)) o(.)60 b(Also,)39 b Fo(g)2113 2127 y Fp(n)2159 2112 y Fw(\(0\))e(=)f(0)i(and)g Fo(g)2762 2127 y Fp(n)2809 2112 y Fw(\(1\))e(=)h Fs(k)p Fo(u)3189 2076 y Fl(\(1\))3308 2112 y Fs(\000)26 b Fo(u)3467 2076 y Fl(\(0\))3561 2112 y Fs(k)p Fw(,)39 b(due)168 2228 y(to)32 b(\(79\))o(.)44 b(Th)m(us,)34 b(there)f(exists)h Fo(t)1359 2243 y Fp(n)1434 2228 y Fs(2)28 b Fw([0)p Fo(;)17 b Fw(1])32 b(in)f(suc)m(h)j(a)f(w)m(a) m(y)1414 2463 y Fo(g)1461 2478 y Fp(n)1508 2463 y Fw(\()p Fo(t)1581 2478 y Fp(n)1628 2463 y Fw(\))28 b(=)1807 2395 y Fs(k)p Fo(u)1913 2359 y Fl(\(1\))2029 2395 y Fs(\000)23 b Fo(u)2185 2359 y Fl(\(0\))2279 2395 y Fs(k)p 1807 2440 522 4 v 2044 2531 a Fw(2)2366 2463 y(=:)28 b Fo(c)17 b(:)168 2657 y Fw(That)33 b(is,)168 2823 y(\(81\))1235 b Fs(k)p Fo(\015)1678 2838 y Fp(n)1724 2823 y Fw(\()p Fo(t)1797 2838 y Fp(n)1844 2823 y Fw(\))22 b Fs(\000)h Fo(u)2060 2782 y Fl(\(0\))2154 2823 y Fs(k)k Fw(=)h Fo(c)17 b(:)168 2991 y Fw(W)-8 b(e)33 b(recall)e(that)h Fo(c)c(>)f Fw(0,)33 b(since)g Fo(u)1384 2954 y Fl(\(0\))1505 2991 y Fs(6\021)28 b Fo(u)1666 2954 y Fl(\(1\))1760 2991 y Fw(.)168 3107 y(Moreo)m(v)m(er,)g(b)m(y)f(Corollary)c(39,)k(p)s(erhaps) f(taking)f(subsequences,)31 b(w)m(e)c(ha)m(v)m(e)f(that)g Fo(\015)3195 3122 y Fp(n)3241 3107 y Fw(\()p Fo(t)3314 3122 y Fp(n)3361 3107 y Fw(\))g(con)m(v)m(erges)168 3226 y(in)39 b(the)i(sup)q(-norm)d(to)i(some)46 b(\026)-55 b Fo(u)40 b Fs(2)h(C)6 b Fw(.)66 b(By)42 b(\(81\),)g Fs(k)6 b Fw(\026)-55 b Fo(u)27 b Fs(\000)g Fo(u)2337 3190 y Fl(\(0\))2431 3226 y Fs(k)41 b Fw(=)f Fo(c)g Fw(and)g(th)m(us)47 b(\026)-55 b Fo(u)40 b Fs(62)h(f)p Fo(u)3448 3190 y Fl(\(0\))3542 3226 y Fo(;)33 b(u)3658 3190 y Fl(\(1\))3752 3226 y Fs(g)p Fw(.)168 3342 y(Th)m(us,)e(b)m(y)g(\(77\))o(,)f Fo(S)862 3357 y Fp(\033)909 3342 y Fw(\()6 b(\026)-55 b Fo(u)o Fw(\))28 b Fo(>)g Fw(0.)42 b(On)28 b(the)i(other)f(hand,)h(b)m(y)h (\(80\))d(and)h(Lemma)e(41,)i Fo(S)3211 3357 y Fp(\033)3258 3342 y Fw(\()6 b(\026)-55 b Fo(u)p Fw(\))27 b(=)h(0.)42 b(This)168 3458 y(con)m(tradiction)31 b(yields)h(the)h(desired)g (result.)1919 b Fg(\003)168 3647 y Fz(Lemma)37 b(45.)42 b Fr(Assume)34 b(that)168 3811 y Fw(\(82\))1126 b Fo(\027)1516 3826 y Fp(\033)1563 3811 y Fw(\()p Fo(r)1645 3826 y Fl(0)1685 3811 y Fw(\))22 b Fs(\000)g(f)p Fo(u)1950 3770 y Fl(\(0\))2044 3811 y Fo(;)33 b(u)2160 3770 y Fl(\(1\))2254 3811 y Fs(g)28 b Fw(=)f Fs(;)17 b Fo(;)168 3976 y Fr(wher)-5 b(e)37 b Fo(r)490 3991 y Fl(0)568 3976 y Fr(is)h(as)f(in)h(L)-5 b(emma)37 b(44.)54 b(Then,)38 b(ther)-5 b(e)39 b(exists)e Fo(r)2306 3991 y Fl(1)2379 3976 y Fo(>)d Fw(0)k Fr(such)g(that)g(the)g (fol)5 b(lowing)37 b(holds.)168 4092 y(If)d Fo(v)d Fs(2)e(C)41 b Fr(and)34 b Fs(k)p Fo(v)26 b Fs(\000)c Fo(u)1003 4056 y Fl(\(0\))1097 4092 y Fs(k)28 b(\024)g Fo(r)1324 4107 y Fl(1)1363 4092 y Fr(,)35 b(then)1512 4264 y Fw(lim)1469 4324 y Fp(t)p Fm(!)p Fl(+)p Fm(1)1707 4264 y Fs(k)p Fw(\010)1827 4223 y Fp(t)1857 4264 y Fw(\()p Fo(v)t Fw(\))21 b Fs(\000)i Fo(u)2161 4223 y Fl(\(0\))2255 4264 y Fs(k)k Fw(=)h(0)17 b Fo(:)168 4483 y Fr(A)n(nalo)-5 b(gously,)34 b(if)g Fo(v)e Fs(2)c(C)41 b Fr(and)34 b Fs(k)p Fo(v)26 b Fs(\000)d Fo(u)1564 4447 y Fl(\(1\))1658 4483 y Fs(k)k(\024)h Fo(r)1884 4498 y Fl(1)1924 4483 y Fr(,)34 b(then)1512 4655 y Fw(lim)1469 4715 y Fp(t)p Fm(!)p Fl(+)p Fm(1)1707 4655 y Fs(k)p Fw(\010)1827 4614 y Fp(t)1857 4655 y Fw(\()p Fo(v)t Fw(\))21 b Fs(\000)i Fo(u)2161 4614 y Fl(\(1\))2255 4655 y Fs(k)k Fw(=)h(0)17 b Fo(:)168 4858 y Fz(Pro)s(of.)48 b Fw(W)-8 b(e)34 b(pro)m(v)m(e)i(the) e(\014rst)h(assertion,)f(the)h(second)h(one)e(follo)m(wing)d (analogously)-8 b(.)47 b(The)35 b(pro)s(of)168 4974 y(is)d(b)m(y)h(con) m(tradiction:)43 b(assume)33 b(that)f(there)h(exists)h Fo(v)d Fs(2)d(C)39 b Fw(suc)m(h)34 b(that)168 5141 y(\(83\))1337 b Fs(k)p Fo(v)26 b Fs(\000)d Fo(u)1958 5100 y Fl(\(0\))2051 5141 y Fs(k)28 b(\024)g Fo(r)2278 5156 y Fl(1)168 5305 y Fw(in)k(suc)m(h)i(a)e(w)m(a)m(y)168 5469 y(\(84\))759 b(\010)1171 5433 y Fp(t)1201 5469 y Fw(\()p Fo(v)t Fw(\))33 b(do)s(es)g(not)f(con)m(v)m(erge)i(to)f Fo(u)2328 5433 y Fl(\(0\))2454 5469 y Fw(uniformly)-8 b(.)p eop %%Page: 53 53 53 52 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(53)0 450 y Fw(Here,)38 b Fo(r)306 465 y Fl(1)383 450 y Fw(will)c(b)s(e)k(a)f(suitably)f(small)f(quan)m(tit)m(y)-8 b(,)39 b(de\014ned)g(as)e(follo)m(ws.)56 b(By)39 b(\(51\),)f(w)m(e)g (can)g(\014nd)g(a)0 566 y(\014nite)32 b(set)i Fo(X)483 581 y Fp(r)515 590 y Fi(0)581 566 y Fs(\032)28 b Fw(\003)k(in)g(suc)m (h)i(a)e(w)m(a)m(y)i(that)0 755 y(\(85\))1334 660 y Fn(X)1250 872 y Fp(p)p Fm(2)p Fl(\003)p Fm(\000)p Fp(X)1495 880 y Fk(r)1524 895 y Fi(0)1579 755 y Fs(j)p Fo(u)1663 714 y Fl(\(1\))1663 780 y Fp(p)1779 755 y Fs(\000)22 b Fo(u)1934 714 y Fl(\(0\))1934 780 y Fp(p)2028 755 y Fs(j)44 b(\024)2253 688 y Fo(r)2297 703 y Fl(0)p 2232 732 126 4 v 2232 823 a Fw(4)p Fo(C)2384 755 y(;)0 1034 y Fw(where)34 b Fo(C)39 b Fw(is)32 b(the)h(quan)m(tit)m(y)h(in)d(\(9\).)43 b(W)-8 b(e)33 b(then)h(set)1246 1258 y Fo(r)1290 1273 y Fl(1)1373 1258 y Fw(:=)44 b(min)1700 1117 y Fn(\032)1774 1258 y Fo(r)1818 1273 y Fl(0)1874 1258 y Fo(;)2072 1190 y(r)2116 1205 y Fl(0)p 1956 1235 316 4 v 1956 1326 a Fw(4)p Fo(C)7 b(]X)2201 1341 y Fp(r)2233 1350 y Fi(0)2281 1117 y Fn(\033)2389 1258 y Fo(:)0 1476 y Fw(It)33 b(follo)m(ws)e(that)0 1640 y(\(86\))1291 b(sup)1477 1710 y Fk(w)q Fj(2C)1344 1779 y(k)p Fk(w)q Fj(\000)p Fk(u)1506 1761 y Fi(\(0\))1589 1779 y Fj(k\024)p Fk(r)1698 1794 y Fi(1)1759 1640 y Fo(S)1819 1655 y Fp(\033)1866 1640 y Fw(\()p Fo(w)s Fw(\))43 b Fs(\024)2190 1573 y Fo(r)2234 1588 y Fl(0)p 2190 1617 84 4 v 2207 1709 a Fw(2)2300 1640 y Fo(:)0 1933 y Fw(Indeed,)34 b(if)e Fo(w)i Fw(is)f(as)f(here)i(ab)s(o)m(v)m(e,)f(w)m(e)h(get)e(from) g Fz(H8)g Fw(and)h(\(85\))e(that)468 2164 y Fo(S)528 2179 y Fp(\033)575 2164 y Fw(\()p Fo(w)s Fw(\))82 b Fs(\024)970 2069 y Fn(X)967 2281 y Fp(B)s Fm(2S)1134 2069 y Fn(X)1137 2281 y Fp(p)p Fm(2)p Fp(B)1295 2028 y Fn(Z)1395 2055 y Fl(1)1350 2254 y(0)1451 2020 y Fn(\014)1451 2079 y(\014)1451 2139 y(\014)1451 2199 y(\014)1494 2097 y Fo(@)5 b(H)1631 2112 y Fp(B)p 1494 2141 199 4 v 1517 2232 a Fo(@)g(u)1629 2247 y Fp(p)1702 2164 y Fw(\()p Fo(\034)11 b(w)25 b Fw(+)d(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2342 2123 y Fl(\(0\))2436 2164 y Fw(\))2474 2020 y Fn(\014)2474 2079 y(\014)2474 2139 y(\014)2474 2199 y(\014)2540 2164 y Fo(d\034)34 b Fs(\001)21 b(j)p Fo(w)2814 2179 y Fp(p)2876 2164 y Fs(\000)h Fo(u)3031 2123 y Fl(\(0\))3031 2189 y Fp(p)3125 2164 y Fs(j)806 2493 y(\024)967 2398 y Fn(X)973 2610 y Fp(p)p Fm(2)p Fl(\003)1131 2398 y Fn(X)1137 2599 y Fk(B)r Fj(2S)1142 2655 y Fk(B)r Fj(3)p Fk(p)1295 2358 y Fn(Z)1395 2384 y Fl(1)1351 2583 y(0)1451 2349 y Fn(\014)1451 2408 y(\014)1451 2468 y(\014)1451 2528 y(\014)1494 2426 y Fo(@)5 b(H)1631 2441 y Fp(B)p 1494 2470 V 1517 2561 a Fo(@)g(u)1629 2576 y Fp(p)1702 2493 y Fw(\()p Fo(\034)11 b(w)25 b Fw(+)d(\(1)g Fs(\000)h Fo(\034)11 b Fw(\))p Fo(u)2342 2452 y Fl(\(0\))2436 2493 y Fw(\))2474 2349 y Fn(\014)2474 2408 y(\014)2474 2468 y(\014)2474 2528 y(\014)2540 2493 y Fo(d\034)34 b Fs(\001)22 b(j)p Fo(u)2801 2452 y Fl(\(1\))2801 2518 y Fp(p)2916 2493 y Fs(\000)h Fo(u)3072 2452 y Fl(\(0\))3072 2518 y Fp(p)3166 2493 y Fs(j)806 2817 y(\024)84 b Fo(C)1060 2722 y Fn(X)1066 2933 y Fp(p)p Fm(2)p Fl(\003)1221 2817 y Fs(j)p Fo(u)1305 2775 y Fl(\(1\))1305 2841 y Fp(p)1420 2817 y Fs(\000)23 b Fo(u)1576 2775 y Fl(\(0\))1576 2841 y Fp(p)1670 2817 y Fs(j)806 3104 y(\024)84 b Fo(C)1092 3009 y Fn(X)1060 3221 y Fp(p)p Fm(2)p Fp(X)1201 3229 y Fk(r)1230 3244 y Fi(0)1285 3104 y Fs(j)p Fo(u)1369 3063 y Fl(\(1\))1369 3129 y Fp(p)1485 3104 y Fs(\000)22 b Fo(u)1640 3063 y Fl(\(0\))1640 3129 y Fp(p)1734 3104 y Fs(j)39 b Fw(+)1925 3037 y Fo(r)1969 3052 y Fl(0)p 1925 3081 84 4 v 1942 3172 a Fw(4)806 3402 y Fs(\024)84 b Fo(C)23 b Fs(k)p Fo(u)1166 3360 y Fl(\(1\))1282 3402 y Fs(\000)g Fo(u)1438 3360 y Fl(\(0\))1531 3402 y Fs(k)17 b Fo(]X)1717 3417 y Fp(r)1749 3426 y Fi(0)1826 3402 y Fw(+)1951 3334 y Fo(r)1995 3349 y Fl(0)p 1951 3379 V 1968 3470 a Fw(4)806 3618 y Fs(\024)977 3550 y Fo(r)1021 3565 y Fl(0)p 977 3595 V 994 3686 a Fw(2)1087 3618 y Fo(;)0 3815 y Fw(th)m(us)34 b(pro)m(ving)f(\(86\))0 3931 y(From)e(this)h(and)h(\(83\),)0 4121 y(\(87\))1400 b Fo(S)1634 4136 y Fp(\033)1681 4121 y Fw(\()p Fo(v)t Fw(\))27 b Fs(\024)1950 4053 y Fo(r)1994 4068 y Fl(0)p 1950 4098 V 1968 4189 a Fw(2)2060 4121 y Fo(;)0 4312 y Fw(pro)m(vided)33 b(that)f Fo(r)656 4327 y Fl(1)728 4312 y Fw(is)g(small)f(enough)i(with)f(resp)s(ect)i(to)e Fo(r)2137 4327 y Fl(0)2176 4312 y Fw(.)0 4429 y(Also,)42 b(recalling)37 b(Corollary)i(43,)i(p)s(ossibly)f(extracting)g(a)g (subsequence,)46 b(w)m(e)41 b(ha)m(v)m(e)g(that)f(\010)3462 4392 y Fp(t)3487 4400 y Fk(n)3535 4429 y Fw(\()p Fo(v)t Fw(\))0 4545 y(uniformly)30 b(con)m(v)m(erges)35 b(to)d(a)h(suitable)e (con\014guration)38 b(\026)-55 b Fo(u)27 b Fs(2)h(C)6 b Fw(,)33 b(and)g(that)0 4707 y(\(88\))758 b(\026)-55 b Fo(u)32 b Fw(is)g(a)g(solution)f(of)h(our)h(v)-5 b(ariational)29 b(equation.)0 4869 y(F)-8 b(urthermore,)32 b(it)g(follo)m(ws)f(from)g (\(87\))h(and)h(Lemma)e(42)h(that)0 5059 y(\(89\))1024 b Fo(S)1258 5074 y Fp(\033)1304 5059 y Fw(\(\010)1412 5022 y Fp(t)1443 5059 y Fw(\()p Fo(v)t Fw(\)\))27 b Fs(\024)1750 4991 y Fo(r)1794 5006 y Fl(0)p 1750 5036 V 1767 5127 a Fw(2)1876 5059 y(for)32 b(an)m(y)h Fo(t)28 b Fs(\025)g Fw(0.)0 5250 y(Consequen)m(tly)-8 b(,)35 b(b)m(y)e(Lemma)e(41,)0 5434 y(\(90\))1401 b Fo(S)1635 5449 y Fp(\033)1681 5434 y Fw(\()6 b(\026)-55 b Fo(u)p Fw(\))27 b Fs(\024)1956 5367 y Fo(r)2000 5382 y Fl(0)p 1956 5411 V 1973 5503 a Fw(2)2049 5434 y(.)p eop %%Page: 54 54 54 53 bop 168 251 a Ft(54)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)168 453 y Fw(Note)36 b(that)42 b(\026)-55 b Fo(u)35 b Fw(cannot)h(b)s(e)g(equal)g(to)g Fo(u)1612 417 y Fl(\(0\))1706 453 y Fw(,)h(b)s(ecause)g(of)43 b(\(84\))o(;)38 b(th)m(us,)f(\(88\))o(,)g(\(90\))f(and)g(\(82\))f (imply)168 572 y(that)j(\026)-55 b Fo(u)27 b Fw(=)h Fo(u)622 536 y Fl(\(1\))716 572 y Fw(.)168 688 y(Therefore,)33 b(there)h(exists)f Fo(T)42 b(>)27 b Fw(0)32 b(suc)m(h)i(that)168 862 y(\(91\))885 b Fs(k)p Fw(\010)1347 821 y Fp(T)1402 862 y Fw(\()p Fo(v)t Fw(\))22 b Fs(\000)g Fo(u)1706 821 y Fl(\(1\))1800 862 y Fs(k)28 b Fw(=)f Fs(k)p Fw(\010)2101 821 y Fp(T)2157 862 y Fw(\()p Fo(v)t Fw(\))21 b Fs(\000)29 b Fw(\026)-55 b Fo(u)o Fs(k)28 b(\024)g Fo(r)2687 877 y Fl(1)2743 862 y Fo(:)168 1034 y Fw(Let)k(no)m(w)168 1205 y(\(92\))1211 b(~)-52 b Fo(\015)5 b Fw(\()p Fo(t)p Fw(\))27 b(:=)h(\010)1945 1164 y Fl(2)p Fp(T)10 b Fl(\()p Fp(t)p Fm(\000)p Fl(1)p Fp(=)p Fl(4\))2277 1205 y Fw(\()p Fo(v)t Fw(\))17 b Fo(;)168 1377 y Fw(for)32 b(an)m(y)h Fo(t)28 b Fs(2)g Fw([1)p Fo(=)p Fw(4)p Fo(;)33 b Fw(3)p Fo(=)p Fw(4].)168 1493 y(Then,)50 b(~)-52 b Fo(\015)51 b Fs(2)46 b Fo(C)7 b Fw(\([1)p Fo(=)p Fw(4)p Fo(;)33 b Fw(3)p Fo(=)p Fw(4])p Fo(;)17 b Fs(C)6 b Fw(\),)45 b(thanks)f(to)f(Lemma)f(42.)75 b(Also,)49 b(~)-53 b Fo(\015)5 b Fw(\(1)p Fo(=)p Fw(4\))45 b(=)h Fo(v)h Fw(and)g(~)-53 b Fo(\015)5 b Fw(\(3)p Fo(=)p Fw(4\))45 b(=)168 1609 y(\010)238 1573 y Fp(T)293 1609 y Fw(\()p Fo(v)t Fw(\).)e(Let)712 1902 y Fo(\015)5 b Fw(\()p Fo(t)p Fw(\))45 b(:=)1071 1697 y Fn(8)1071 1787 y(<)1071 1966 y(:)1451 1784 y Fo(u)1507 1748 y Fl(\(0\))1623 1784 y Fw(+)22 b(4)p Fo(t)p Fw(\()p Fo(v)k Fs(\000)d Fo(u)2072 1748 y Fl(\(0\))2165 1784 y Fw(\))434 b(if)32 b Fo(t)c Fs(2)g Fw([0)p Fo(;)33 b Fw(1)p Fo(=)p Fw(4\),)1747 1900 y(~)-52 b Fo(\015)5 b Fw(\()p Fo(t)p Fw(\))683 b(if)31 b Fo(t)d Fs(2)g Fw([1)p Fo(=)p Fw(4)p Fo(;)33 b Fw(3)p Fo(=)p Fw(4],)1176 2021 y(\010)1246 1985 y Fp(T)1302 2021 y Fw(\()p Fo(v)t Fw(\))21 b(+)h(\(4)p Fo(t)g Fs(\000)h Fw(3\)\()p Fo(u)1973 1985 y Fl(\(1\))2089 2021 y Fs(\000)f Fw(\010)2258 1985 y Fp(T)2314 2021 y Fw(\()p Fo(v)t Fw(\)\))158 b(if)32 b Fo(t)c Fs(2)g Fw(\(3)p Fo(=)p Fw(4)p Fo(;)k Fw(1].)168 2189 y(Then,)168 2360 y(\(93\))734 b Fo(\015)33 b Fs(2)28 b Fo(C)7 b Fw(\([0)p Fo(;)33 b Fw(1])p Fo(;)17 b Fs(C)6 b Fw(\),)32 b Fo(\015)5 b Fw(\(0\))28 b(=)f Fo(u)2148 2324 y Fl(\(0\))2275 2360 y Fw(and)32 b Fo(\015)5 b Fw(\(1\))28 b(=)f Fo(u)2832 2324 y Fl(\(1\))2926 2360 y Fw(.)168 2532 y(Moreo)m(v)m(er,)34 b(if)d Fo(t)d Fs(2)g Fw([0)p Fo(;)34 b Fw(1)p Fo(=)p Fw(4\),)1322 2711 y Fs(k)p Fo(\015)5 b Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)h Fo(u)1717 2670 y Fl(\(0\))1810 2711 y Fs(k)28 b(\024)g(k)p Fo(v)e Fs(\000)c Fo(u)2271 2670 y Fl(\(0\))2365 2711 y Fs(k)28 b(\024)g Fo(r)2592 2726 y Fl(1)2648 2711 y Fo(;)168 2882 y Fw(due)33 b(to)f(\(83\),)g(and,)h(if)e Fo(t)d Fs(2)g Fw(\(3)p Fo(=)p Fw(4)p Fo(;)33 b Fw(1])1221 3062 y Fs(k)p Fo(\015)5 b Fw(\()p Fo(t)p Fw(\))22 b Fs(\000)h Fo(u)1616 3021 y Fl(\(1\))1710 3062 y Fs(k)k(\024)h(k)p Fw(\010)2012 3021 y Fp(T)2068 3062 y Fw(\()p Fo(v)t Fw(\))21 b Fs(\000)i Fo(u)2372 3021 y Fl(\(1\))2466 3062 y Fs(k)k(\024)i Fo(r)2693 3077 y Fl(1)2749 3062 y Fo(;)168 3233 y Fw(due)49 b(to)e(\(91\).)90 b(Therefore,)53 b(making)47 b(use)i(of)55 b(\(86\))o(,)d(w)m(e)e (gather)e(that)g Fo(S)2977 3248 y Fp(\033)3024 3233 y Fw(\()p Fo(\015)5 b Fw(\()p Fo(t)p Fw(\)\))54 b Fs(\024)h Fo(r)3497 3248 y Fl(0)3536 3233 y Fo(=)p Fw(2,)d(for)168 3350 y Fo(t)28 b Fs(2)g Fw([0)p Fo(;)33 b Fw(1)p Fo(=)p Fw(4\))10 b Fs([)g Fw(\(3)p Fo(=)p Fw(4)p Fo(;)32 b Fw(1],)27 b(pro)m(vided)g(that)f Fo(r)1750 3365 y Fl(1)1816 3350 y Fw(is)g(small)e(enough.)42 b(In)27 b(fact,)h(recalling)d(\(89\))h (and)h(\(92\))o(,)168 3521 y(\(94\))938 b Fo(S)1340 3536 y Fp(\033)1387 3521 y Fw(\()p Fo(\015)5 b Fw(\()p Fo(t)p Fw(\)\))28 b Fs(\024)g Fo(r)1807 3536 y Fl(0)1846 3521 y Fo(=)p Fw(2)k(for)g(an)m(y)i Fo(t)28 b Fs(2)g Fw([0)p Fo(;)33 b Fw(1].)168 3692 y(But)22 b(the)h(existence)h(of)e(a)g(path)h Fo(\015)k Fw(satisfying)c(\(93\))f(and)h(\(94\))e(is)h(in)g(con)m (tradiction)f(with)h(Lemma)f(44.)168 3809 y(Notice)46 b(indeed)h(that)f(the)i(assumptions)e(of)g(Lemma)g(44)g(are)g (ful\014lled)f(here,)51 b(since)c(\(82\))f(im-)168 3925 y(plies)31 b(\(77\).)168 4041 y(This)h(con)m(tradiction)g(ends)i(the)f (pro)s(of)e(of)h(Lemma)g(45.)1505 b Fg(\003)168 4240 y Fz(Lemma)36 b(46.)42 b Fr(Assume)g Fw(\(82\))o Fr(.)j(Then,)34 b Fo(\014)1707 4255 y Fl(0)1781 4240 y Fr(and)g Fo(\014)2025 4255 y Fl(1)2099 4240 y Fr(ar)-5 b(e)34 b(op)-5 b(en)34 b(in)g(the)h(r)-5 b(elative)34 b(top)-5 b(olo)g(gy)35 b(of)f Fw([0)p Fo(;)17 b Fw(1])p Fr(.)168 4384 y Fz(Pro)s(of.)59 b Fw(W)-8 b(e)38 b(pro)m(v)m(e)i(that)d Fo(\014)1243 4399 y Fl(0)1321 4384 y Fw(is)g(op)s(en,)j(the)e(argumen)m(t)g(for)f Fo(\014)2518 4399 y Fl(1)2596 4384 y Fw(b)s(eing)g(analogous.)58 b(Let)43 b(\026)-53 b Fo(s)36 b Fs(2)i Fo(\014)3790 4399 y Fl(0)168 4500 y Fw(and)32 b(tak)m(e)1468 4631 y Fo(s)27 b Fs(2)i Fw([0)p Fo(;)k Fw(1])22 b Fs(\\)g Fw([)t(\026)-53 b Fo(s)22 b Fs(\000)h Fo(\017;)38 b Fw(\026)-53 b Fo(s)22 b Fw(+)g Fo(\017)p Fw(])17 b Fo(:)168 4782 y Fw(W)-8 b(e)38 b(sho)m(w)h(that)e Fo(s)f Fs(2)h Fo(\014)1044 4797 y Fl(0)1084 4782 y Fw(,)i(pro)m(vided)f(that)f Fo(\017)g(>)f Fw(0)i(is)f(su\016cien)m(tly)h(small.)57 b(F)-8 b(or)37 b(this,)i(assume,)g(b)m(y)168 4898 y(con)m(tradiction,)31 b(that)i Fo(s)27 b Fs(62)h Fo(\014)1222 4913 y Fl(0)1262 4898 y Fw(.)43 b(Then,)34 b(there)f(exists)h(a)e(sequence)j Fo(t)2653 4913 y Fp(n)2728 4898 y Fs(!)27 b Fw(+)p Fs(1)32 b Fw(in)g(suc)m(h)i(a)f(w)m(a)m(y)168 5070 y(\(95\))669 b(\010)1081 5034 y Fp(t)1106 5042 y Fk(n)1154 5070 y Fw(\()p Fo(u)1248 5034 y Fl(\()p Fp(s)p Fl(\))1339 5070 y Fw(\))33 b(do)s(es)g(not)f(con)m(v)m(erge)i(uniformly)d(to)h Fo(u)2821 5034 y Fl(\(0\))2915 5070 y Fw(.)168 5241 y(No)m(w,)h(since)k (\026)-53 b Fo(s)28 b Fs(2)g Fo(\014)879 5256 y Fl(0)918 5241 y Fw(,)33 b(there)g(exists)38 b(\026)-54 b Fo(n)28 b Fs(2)g Fq(N)48 b Fw(suc)m(h)34 b(that)1490 5434 y Fs(k)p Fw(\010)1610 5393 y Fp(t)1639 5401 y Fi(\026)-34 b Fk(n)1682 5434 y Fw(\()p Fo(u)1776 5393 y Fl(\()s(\026)c Fp(s)p Fl(\))1868 5434 y Fw(\))22 b Fs(\000)g Fo(u)2083 5393 y Fl(\(0\))2177 5434 y Fs(k)28 b(\024)2370 5367 y Fo(r)2414 5382 y Fl(1)p 2370 5411 84 4 v 2387 5503 a Fw(2)2480 5434 y Fo(;)p eop %%Page: 55 55 55 54 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(55)0 450 y Fw(where)41 b Fo(r)333 465 y Fl(1)413 450 y Fw(is)f(the)g(quan)m(tit)m(y)h(giv)m(en)g(in)e(Lemma)g(45.)66 b(By)41 b(the)f(con)m(tin)m(uit)m(y)h(with)f(resp)s(ect)h(to)f(the)0 566 y(initial)29 b(data)j(\(recall)f(Lemma)g(42\),)h(w)m(e)i(th)m(us)g (ha)m(v)m(e)f(that)1332 753 y Fs(k)p Fw(\010)1452 712 y Fp(t)1481 720 y Fi(\026)-34 b Fk(n)1525 753 y Fw(\()p Fo(u)1619 712 y Fl(\()p Fp(s)p Fl(\))1710 753 y Fw(\))22 b Fs(\000)h Fo(u)1926 712 y Fl(\(0\))2019 753 y Fs(k)28 b(\024)g Fo(r)2246 768 y Fl(1)2302 753 y Fo(;)0 932 y Fw(pro)m(vided)33 b(that)f Fo(\017)h Fw(is)f(small)f(enough.)0 1048 y(Therefore,)d(b)m(y)e(Lemma)d(45,)j(it)e(follo)m(ws)g(that)g (\010)1758 1012 y Fp(t)1788 1048 y Fw(\(\010)1896 1012 y Fp(t)1925 1020 y Fi(\026)-34 b Fk(n)1969 1048 y Fw(\()p Fo(u)2063 1012 y Fl(\()p Fp(s)p Fl(\))2154 1048 y Fw(\)\))25 b(uniformly)e(con)m(v)m(erges)k(to)d Fo(u)3288 1012 y Fl(\(0\))3382 1048 y Fw(.)41 b(That)0 1167 y(is,)36 b(\010)199 1131 y Fp(t)224 1139 y Fk(n)271 1167 y Fw(\()p Fo(u)365 1131 y Fl(\()p Fp(s)p Fl(\))456 1167 y Fw(\))g(uniformly)e(con)m(v)m (erges)k(to)e Fo(u)1597 1131 y Fl(\(0\))1691 1167 y Fw(.)54 b(Since)36 b(the)h(latter)e(assertion)h(is)f(in)h(con)m(tradiction)0 1284 y(with)c(\(95\),)g(the)h(pro)s(of)f(of)g(the)h(desired)g(result)g (is)f(complete.)1286 b Fg(\003)0 1494 y Fz(Corollary)37 b(47.)42 b Fr(Supp)-5 b(ose)34 b(that)45 b Fw(\(82\))34 b Fr(holds.)44 b(Then,)34 b(ther)-5 b(e)35 b(exists)f Fo(s)28 b Fs(2)g Fw([0)p Fo(;)17 b Fw(1])22 b Fs(\000)g Fw(\()p Fo(\014)3132 1509 y Fl(0)3194 1494 y Fs([)h Fo(\014)3338 1509 y Fl(1)3377 1494 y Fw(\))p Fr(.)0 1641 y Fz(Pro)s(of.)43 b Fw(Assume,)34 b(b)m(y)f(con)m(tradiction,)f(that)g([0)p Fo(;)17 b Fw(1])27 b(=)h Fo(\014)2084 1656 y Fl(0)2145 1641 y Fs([)23 b Fo(\014)2289 1656 y Fl(1)2329 1641 y Fw(.)43 b(Let)0 1820 y(\(96\))1282 b(\026)-53 b Fo(s)28 b Fw(:=)g(sup)q Fs(f)p Fo(s)f Fs(2)h Fo(\014)2076 1835 y Fl(0)2116 1820 y Fs(g)17 b Fo(:)0 1999 y Fw(By)47 b(Lemma)d(46,)i Fo(\014)754 2014 y Fl(0)840 1999 y Fw(and)g Fo(\014)1098 2014 y Fl(1)1184 1999 y Fw(are)g(non-empt)m(y)-8 b(,)49 b(disjoin)m(t,)g(relativ)m(ely)c(op)s(en)h(subsets)i(of)e([0)p Fo(;)17 b Fw(1].)0 2115 y(Since)33 b(1)27 b Fs(2)h Fo(\014)480 2130 y Fl(1)552 2115 y Fw(and)33 b Fo(\014)797 2130 y Fl(1)869 2115 y Fw(is)f(op)s(en,)h(w)m(e)h(ha)m(v)m(e)f(that)0 2294 y(\(97\))1526 b(\026)-53 b Fo(s)28 b(<)f Fw(1)17 b Fo(:)0 2473 y Fw(Analogously)-8 b(,)31 b(since)i(0)28 b Fs(2)g Fo(\014)1037 2488 y Fl(0)1109 2473 y Fw(and)k Fo(\014)1353 2488 y Fl(0)1426 2473 y Fw(is)g(op)s(en,)g(w)m(e)i(ha)m(v) m(e)g(that)0 2652 y(\(98\))1526 b(\026)-53 b Fo(s)28 b(>)f Fw(0)17 b Fo(:)0 2831 y Fw(Th)m(us,)34 b(in)e(the)h(ligh)m(t)e (of)39 b(\(96\))o(,)33 b(there)g(are)g(t)m(w)m(o)g(cases)h(to)e (distinguish:)43 b(either)36 b(\026)-53 b Fo(s)28 b Fs(2)g Fo(\014)3124 2846 y Fl(0)3196 2831 y Fw(or)36 b(\026)-53 b Fo(s)28 b Fs(2)g Fo(\014)3538 2846 y Fl(1)3577 2831 y Fw(.)0 2947 y(If)k(\026)-53 b Fo(s)28 b Fs(2)g Fo(\014)316 2962 y Fl(0)355 2947 y Fw(,)i(then,)f(since)g Fo(\014)947 2962 y Fl(0)1015 2947 y Fw(is)e(relativ)m(ely)h(op)s(en,)h(it)e(follo)m (ws)g(from)g(\(97\))h(that)k(\026)-53 b Fo(s)13 b Fw(+)g Fo(\017)28 b Fs(2)g Fo(\014)3197 2962 y Fl(0)3237 2947 y Fw(,)h(pro)m(vided)0 3063 y(that)j Fo(\017)c(>)g Fw(0)k(is)g (su\016cien)m(tly)i(small.)41 b(But)33 b(this)f(con)m(tradicts)h (\(96\))o(.)0 3179 y(If,)h(on)f(the)h(other)g(hand,)j(\026)-53 b Fo(s)30 b Fs(2)f Fo(\014)1184 3194 y Fl(1)1224 3179 y Fw(,)k(the)h(fact)g(that)f Fo(\014)1914 3194 y Fl(1)1987 3179 y Fw(is)g(op)s(en)g(and)h(\(98\))f(imply)e(that)38 b(\026)-53 b Fo(s)22 b Fs(\000)i Fo(\017)29 b Fs(2)h Fo(\014)3595 3194 y Fl(1)3634 3179 y Fw(,)0 3296 y(for)i(an)m(y)h Fo(\017)28 b(>)g Fw(0)k(su\016cien)m(tly)h(small.)42 b(But)32 b(this)h(again)e(con)m(tradicts)i(\(96\))o(.)0 3412 y(These)h(con)m(tradictions)e(giv)m(e)h(the)g(pro)s(of)e(of)i (Corollary)d(47.)1347 b Fg(\003)0 3693 y Fw(4.4.)49 b Fz(End)32 b(of)h(the)g(pro)s(of)g(of)g(Theorem)f(10.)49 b Fw(W)-8 b(e)29 b(distinguish)e(t)m(w)m(o)j(alternativ)m(es:)41 b(either)28 b(\(82\))0 3809 y(is)k(violated)f(or)i(it)e(do)s(es)i (hold.)0 3925 y(If)39 b(\(82\))32 b(is)g(violated,)g(tak)m(e)1302 4071 y Fo(u)c Fs(2)g Fo(\027)1528 4086 y Fp(\033)1575 4071 y Fw(\()p Fo(r)1657 4086 y Fl(0)1696 4071 y Fw(\))22 b Fs(\000)h(f)p Fo(u)1962 4030 y Fl(\(0\))2055 4071 y Fo(;)34 b(u)2172 4030 y Fl(\(1\))2266 4071 y Fs(g)17 b Fo(:)0 4233 y Fw(Then,)50 b(b)m(y)c(construction,)f Fo(u)k Fs(2)h(B)1330 4248 y Fp(\033)1377 4233 y Fw(,)45 b Fo(u)g Fw(is)g(a)f(solution)g(of)h(our)g(v)-5 b(ariational)42 b(equation,)48 b Fo(u)3441 4196 y Fl(\(0\))3584 4233 y Fs(\024)0 4349 y Fo(u)32 b Fs(\024)i Fo(u)255 4313 y Fl(\(1\))384 4349 y Fw(and)i Fo(u)f Fw(do)s(es)h(not)g(coincide)f (with)g(either)g Fo(u)2005 4313 y Fl(\(0\))2135 4349 y Fw(or)g Fo(u)2313 4313 y Fl(\(1\))2407 4349 y Fw(.)53 b(Th)m(us,)37 b Fo(u)e Fw(ful\014lls)f(the)i(claim)d(of)0 4465 y(Theorem)g(10.)0 4581 y(If,)g(on)f(the)h(other)g(hand,)g(\(82\))f (holds,)g(then)h(there)h(exists)1379 4760 y Fo(s)27 b Fs(2)h Fw([0)p Fo(;)17 b Fw(1])22 b Fs(\000)h Fw(\()p Fo(\014)1957 4775 y Fl(0)2018 4760 y Fs([)g Fo(\014)2162 4775 y Fl(1)2201 4760 y Fw(\))17 b Fo(:)0 4939 y Fw(Therefore,)51 b(b)m(y)c(Corollary)d(39,)49 b(there)e(exists)g(a)e(sequence)k Fo(t)2346 4954 y Fp(n)2444 4939 y Fs(!)h Fw(+)p Fs(1)c Fw(in)f(suc)m(h)j(a)d(w)m(a)m(y)j(that)0 5055 y(\010)70 5019 y Fp(t)95 5027 y Fk(n)143 5055 y Fw(\()p Fo(u)237 5019 y Fl(\()p Fp(s)p Fl(\))328 5055 y Fw(\))42 b(uniformly)e(con)m(v)m (erges)k(to)e(a)g(suitable)f Fo(u)j Fs(2)h(C)6 b Fw(,)45 b(whic)m(h)d(do)s(es)h(not)f(coincide)g(with)f(ei-)0 5174 y(ther)30 b Fo(u)259 5138 y Fl(\(0\))383 5174 y Fw(or)f Fo(u)555 5138 y Fl(\(1\))649 5174 y Fw(.)42 b(Then,)32 b(the)e(fact)g(that)f Fo(u)e Fs(2)h(C)36 b Fw(giv)m(es)31 b(that)e Fo(u)e Fs(2)h(B)2513 5189 y Fp(\033)2561 5174 y Fw(,)i(and)g(that)f Fo(u)3069 5138 y Fl(\(0\))3191 5174 y Fs(\024)f Fo(u)f Fs(\024)h Fo(u)3540 5138 y Fl(\(1\))3634 5174 y Fw(.)0 5291 y(Also,)42 b(Corollary)c(43)i(implies)d(that)j Fo(u)g Fw(is)g(a)f(solution)g(of)h(our)g(v)-5 b(ariational)36 b(equation.)66 b(Hence)42 b Fo(u)0 5407 y Fw(ful\014lls)30 b(the)j(claim)e(of)h(Theorem)h(10)f(in)f(this)i(case)g(to)s(o.)1503 b Fg(\003)p eop %%Page: 56 56 56 55 bop 168 251 a Ft(56)1059 b(R.)25 b(DE)h(LA)f(LLA)-9 b(VE)25 b(AND)g(E.)h(V)-9 b(ALDINOCI)1708 450 y Fv(References)168 608 y Fy([ALD83])110 b(S.)27 b(Aubry)g(and)f(P)-7 b(.)26 b(Y.)h(Le)g(Daeron.)e(The)i(discrete)f(Frenk)n(el-Kon)n(toro)n(v)-5 b(a)23 b(mo)r(del)k(and)f(its)h(extensions.)585 707 y(I.)h(Exact)f (results)g(for)g(the)h(ground-states.)d Fu(Phys.)32 b(D)p Fy(,)27 b(8\(3\):381{422,)d(1983.)168 807 y([Ang90])137 b(Sigurd)28 b(B.)h(Angenen)n(t.)f(Monotone)g(recurrence)f(relations,)h (their)g(Birkho\013)g(orbits)g(and)h(top)r(ological)585 907 y(en)n(trop)n(y)-7 b(.)27 b Fu(Er)l(go)l(dic)k(The)l(ory)g(Dynam.)f (Systems)p Fy(,)e(10\(1\):15{41,)c(1990.)168 1006 y([Ban88])140 b(V.)27 b(Bangert.)f(Mather)h(sets)f(for)h(t)n(wist)g(maps)f(and)h(geo) r(desics)f(on)g(tori.)h(In)g Fu(Dynamics)i(r)l(ep)l(orte)l(d,)i(V)-6 b(ol.)585 1106 y(1)p Fy(,)24 b(v)n(olume)e(1)g(of)g Fu(Dynam.)j(R)l(ep) l(ort.)h(Ser.)f(Dynam.)g(Systems)f(Appl.)p Fy(,)h(pages)d(1{56.)e (Wiley)-7 b(,)24 b(Chic)n(hester,)585 1206 y(1988.)168 1305 y([Ban89])140 b(V.)41 b(Bangert.)e(On)h(minimal)h(laminations)f (of)g(the)h(torus.)f Fu(A)n(nn.)h(Inst.)g(H.)g(Poinc)l(ar)n(\023)-40 b(e)43 b(A)n(nal.)f(Non)585 1405 y(Lin)n(\023)-40 b(eair)l(e)p Fy(,)29 b(6\(2\):95{138,)c(1989.)168 1504 y([Bes05])158 b(Ugo)27 b(Bessi.)g(Man)n(y)g(plane)g(lik)n(e)h(solutions)e(of)i Fd(n)g Fy(dimensional)f(elliptic)h(problems.)f Fu(pr)l(eprint)p Fy(,)h(2005.)168 1604 y([Bet35])159 b(Hans)20 b(A.)h(Bethe.)g (Statistical)f(theory)f(of)i(sup)r(erlattices.)f Fu(Pr)l(o)l(c.)k(R.)f (So)l(c.)g(L)l(ond.)h(Ser.)g(A)p Fy(,)c(150:552{575,)585 1704 y(1935.)168 1803 y([BK04])163 b(O.)26 b(M.)g(Braun)f(and)h(Y.)h (S.)f(Kivshar.)f Fu(The)k(Fr)l(enkel-Kontor)l(ova)g(mo)l(del)p Fy(.)e(T)-7 b(exts)26 b(and)g(Monographs)e(in)585 1903 y(Ph)n(ysics.)i(Springer-V)-7 b(erlag,)26 b(Berlin,)h(2004.)f (Concepts,)h(metho)r(ds,)h(and)f(applications.)168 2003 y([BL01])176 b(Hyman)25 b(Bass)f(and)h(Alexander)f(Lub)r(otzky)-7 b(.)25 b Fu(T)-6 b(r)l(e)l(e)27 b(lattic)l(es)p Fy(,)f(v)n(olume)e(176) g(of)h Fu(Pr)l(o)l(gr)l(ess)i(in)g(Mathemat-)585 2102 y(ics)p Fy(.)36 b(Birkh\177)-42 b(auser)33 b(Boston)h(Inc.,)k(Boston,)e (MA,)f(2001.)e(With)j(app)r(endices)f(b)n(y)g(Bass,)h(L.)f(Carb)r(one,) 585 2202 y(Lub)r(otzky)-7 b(,)28 b(G.)g(Rosen)n(b)r(erg)e(and)h(J.)h (Tits.)168 2301 y([Bla89])163 b(M.)23 b(L.)g(Blank.)f(Metric)g(prop)r (erties)g(of)h(minimal)f(solutions)g(of)h(discrete)f(p)r(erio)r(dical)g (v)-5 b(ariational)21 b(prob-)585 2401 y(lems.)28 b Fu(Nonline)l(arity) p Fy(,)g(2\(1\):1{22,)e(1989.)168 2501 y([Bla90])163 b(M.)29 b(L.)f(Blank.)g(Chaos)g(and)g(order)f(in)i(the)g(m)n (ultidimensional)f(Frenk)n(el-Kon)n(toro)n(v)-5 b(a)25 b(mo)r(del.)k Fu(T)-6 b(e)l(or)l(et.)585 2600 y(Mat.)31 b(Fiz.)p Fy(,)e(85\(3\):349{367,)23 b(1990.)168 2700 y([CdlL98])106 b(A.)20 b(Candel)e(and)h(R.)g(de)g(la)g(Lla)n(v)n(e.)f (On)g(the)i(Aubry-Mather)e(theory)g(in)h(statistical)g(mec)n(hanics.)f Fu(Comm.)585 2800 y(Math.)31 b(Phys.)p Fy(,)e(192\(3\):649{669,)23 b(1998.)168 2899 y([CdlL01])106 b(Luis)28 b(A.)g(Ca\013arelli)f(and)h (Rafael)g(de)g(la)f(Lla)n(v)n(e.)g(Planelik)n(e)f(minimizers)i(in)g(p)r (erio)r(dic)g(media.)g Fu(Comm.)585 2999 y(Pur)l(e)i(Appl.)h(Math.)p Fy(,)e(54\(12\):1403{1441,)22 b(2001.)168 3099 y([CdlL05])106 b(Luis)28 b(A.)g(Ca\013arelli)e(and)i(Rafael)f(de)g(la)g(Lla)n(v)n(e.)g (In)n(terfaces)f(of)i(ground)e(states)h(in)h(Ising)f(mo)r(dels)h(with) 585 3198 y(p)r(erio)r(dic)g(co)r(e\016cien)n(ts.)f Fu(J.)j(Stat.)f (Phys.)p Fy(,)g(118\(3-4\):687{719,)22 b(2005.)168 3298 y([Chi79])158 b(Boris)32 b(V.)i(Chirik)n(o)n(v.)e(A)i(univ)n(ersal)e (instabilit)n(y)h(of)g(man)n(y-dimensional)f(oscillator)g(systems.)h Fu(Phys.)585 3397 y(R)l(ep.)p Fy(,)28 b(52\(5\):264{379,)c(1979.)168 3497 y([DD89])161 b(W)-7 b(arren)34 b(Dic)n(ks)h(and)f(M.)h(J.)g(Dun)n (w)n(o)r(o)r(dy)-7 b(.)35 b Fu(Gr)l(oups)i(acting)f(on)h(gr)l(aphs)p Fy(,)h(v)n(olume)c(17)g(of)h Fu(Cambridge)585 3597 y(Studies)30 b(in)g(A)l(dvanc)l(e)l(d)g(Mathematics)p Fy(.)f(Cam)n(bridge)d(Univ)n (ersit)n(y)h(Press,)f(Cam)n(bridge,)g(1989.)168 3696 y([dlL05])166 b(Rafael)23 b(de)g(la)f(Lla)n(v)n(e.)g(KAM)h(theory)f (for)g(equilibrium)h(states)g(in)g(1-D)g(statistical)f(mec)n(hanics)g (mo)r(dels.)585 3796 y Fu(pr)l(eprint)p Fy(,)28 b(2005.)e(MP)p 1279 3796 25 4 v 30 w(AR)n(C)h(#)h(05-310.)168 3896 y([dlL)-9 b(V05])113 b(Rafael)33 b(de)h(la)g(Lla)n(v)n(e)e(and)h(Enrico)g(V)-7 b(aldino)r(ci.)34 b(Multiplicit)n(y)g(results)f(for)g(in)n(terfaces)g (of)h(Ginzburg-)585 3995 y(Landau-Allen-Cahn)27 b(equations)f(in)i(p)r (erio)r(dic)g(media.)f Fu(pr)l(eprint)p Fy(,)h(2005.)e(MP)p 3075 3995 V 30 w(AR)n(C)h(#)h(05-225.)168 4095 y([dlL)-9 b(V06a])71 b(Rafael)33 b(de)g(la)g(Lla)n(v)n(e)f(and)h(Enrico)f(V)-7 b(aldino)r(ci.)33 b(Critical)g(p)r(oin)n(ts)g(inside)g(the)h(gaps)e(of) h(ground)f(state)585 4194 y(laminations)27 b(in)h(statistical)f(mec)n (hanics.)g Fu(pr)l(eprint)p Fy(,)h(2006.)e(MP)p 2621 4194 V 29 w(AR)n(C.)168 4294 y([dlL)-9 b(V06b])67 b(Rafael)27 b(de)f(la)h(Lla)n(v)n(e)e(and)i(Enrico)f(V)-7 b(aldino)r(ci.)27 b(Ground)f(states)h(and)f(critical)h(p)r(oin)n(ts)f(for)h(generalized) 585 4394 y(Frenk)n(el-Kon)n(toro)n(v)-5 b(a)24 b(mo)r(dels.)j Fu(pr)l(eprint)p Fy(,)i(2006.)c(MP)p 2284 4394 V 30 w(AR)n(C)j(#)g (06-262.)168 4493 y([Geo88])143 b(Hans-Otto)25 b(Georgii.)f Fu(Gibbs)29 b(me)l(asur)l(es)f(and)g(phase)i(tr)l(ansitions)p Fy(,)c(v)n(olume)f(9)g(of)h Fu(de)i(Gruyter)g(Studies)585 4593 y(in)i(Mathematics)p Fy(.)f(W)-7 b(alter)28 b(de)f(Gruyter)g(&)h (Co.,)f(Berlin,)g(1988.)168 4693 y([Gol92])157 b(Christophe)27 b(Gol)n(\023)-39 b(e.)26 b(A)h(new)h(pro)r(of)e(of)i(the)f (Aubry-Mather's)f(theorem.)h Fu(Math.)k(Z.)p Fy(,)d(210\(3\):441{448,) 585 4792 y(1992.)168 4892 y([Hal80])160 b(Jac)n(k)19 b(K.)h(Hale.)h Fu(Or)l(dinary)i(di\013er)l(ential)i(e)l(quations)p Fy(.)20 b(Rob)r(ert)h(E.)f(Krieger)e(Publishing)i(Co.)g(Inc.,)i(Hun)n (t-)585 4991 y(ington,)27 b(N.Y.,)i(second)e(edition,)h(1980.)168 5091 y([Isr79])191 b(Rob)r(ert)41 b(B.)g(Israel.)f Fu(Convexity)i(in)h (the)f(the)l(ory)h(of)f(lattic)l(e)h(gases)p Fy(.)f(Princeton)e(Univ)n (ersit)n(y)g(Press,)585 5191 y(Princeton,)f(N.J.,)h(1979.)35 b(Princeton)h(Series)h(in)h(Ph)n(ysics,)g(With)g(an)f(in)n(tro)r (duction)g(b)n(y)g(Arth)n(ur)g(S.)585 5290 y(Wigh)n(tman.)p eop %%Page: 57 57 57 56 bop 738 251 a Ft(A)n(UBR)-6 b(Y-MA)g(THER)31 b(THEOR)-6 b(Y)32 b(IN)g(ST)-6 b(A)g(TISTICAL)32 b(MECHANICS)662 b(57)0 450 y Fy([KdlLR97])40 b(Hans)26 b(Ko)r(c)n(h,)f(Rafael)g(de)h (la)f(Lla)n(v)n(e,)g(and)g(Charles)g(Radin.)g(Aubry-Mather)g(theory)g (for)g(functions)h(on)417 550 y(lattices.)i Fu(Discr)l(ete)i(Contin.)g (Dynam.)g(Systems)p Fy(,)d(3\(1\):135{151,)d(1997.)0 649 y([LS01])189 b(Roger)23 b(C.)h(Lyndon)g(and)f(P)n(aul)g(E.)h(Sc)n (h)n(upp.)g Fu(Combinatorial)29 b(gr)l(oup)d(the)l(ory)p Fy(.)f(Classics)e(in)h(Mathemat-)417 749 y(ics.)k(Springer-V)-7 b(erlag,)25 b(Berlin,)i(2001.)f(Reprin)n(t)h(of)h(the)g(1977)e (edition.)0 849 y([Mat82])137 b(John)25 b(N.)h(Mather.)e(Existence)h (of)g(quasip)r(erio)r(dic)f(orbits)g(for)h(t)n(wist)g(homeomorphisms)f (of)h(the)g(ann)n(u-)417 948 y(lus.)j Fu(T)-6 b(op)l(olo)l(gy)p Fy(,)30 b(21\(4\):457{467,)23 b(1982.)0 1048 y([Mat84])137 b(John)46 b(N.)h(Mather.)f(Nonexistence)g(of)g(in)n(v)-5 b(arian)n(t)45 b(circles.)h Fu(Er)l(go)l(dic)i(The)l(ory)g(Dynam.)g (Systems)p Fy(,)417 1147 y(4\(2\):301{309,)24 b(1984.)0 1255 y([Mat86])137 b(John)23 b(Mather.)g(A)h(criterion)e(for)h(the)h (nonexistence)f(of)g(in)n(v)-5 b(arian)n(t)22 b(circles.)h Fu(Inst.)j(Hautes)3269 1234 y(\023)3257 1255 y(Etudes)g(Sci.)417 1355 y(Publ.)31 b(Math.)p Fy(,)e(\(63\):153{204,)24 b(1986.)0 1454 y([Mat90])137 b(John)19 b(N.)g(Mather.)g(Di\013eren)n(tiabilit)n (y)g(of)g(the)h(minimal)f(a)n(v)n(erage)d(action)j(as)f(a)h(function)g (of)g(the)h(rotation)417 1554 y(n)n(um)n(b)r(er.)28 b Fu(Bol.)j(So)l(c.)f(Br)l(asil.)i(Mat.)e(\(N.S.\))p Fy(,)e (21\(1\):59{70,)d(1990.)0 1654 y([Mat91])137 b(John)38 b(N.)g(Mather.)f(V)-7 b(ariational)36 b(construction)h(of)h(orbits)f (of)g(t)n(wist)h(di\013eomorphisms.)f Fu(J.)i(A)n(mer.)417 1753 y(Math.)31 b(So)l(c.)p Fy(,)e(4\(2\):207{263,)24 b(1991.)0 1853 y([Mor24])136 b(Harold)32 b(Marston)f(Morse.)g(A)h (fundamen)n(tal)g(class)f(of)i(geo)r(desics)d(on)i(an)n(y)g(closed)f (surface)g(of)h(gen)n(us)417 1953 y(greater)26 b(than)i(one.)f Fu(T)-6 b(r)l(ans.)30 b(A)n(mer.)g(Math.)h(So)l(c.)p Fy(,)d(26\(1\):25{60,)d(1924.)0 2052 y([Mos86])136 b(J)r(\177)-44 b(urgen)19 b(Moser.)g(Minimal)g(solutions)g(of)h(v)-5 b(ariational)18 b(problems)h(on)g(a)h(torus.)f Fu(A)n(nn.)i(Inst.)h(H.) h(Poinc)l(ar)n(\023)-40 b(e)417 2152 y(A)n(nal.)30 b(Non)g(Lin)n(\023) -40 b(eair)l(e)p Fy(,)29 b(3\(3\):229{272,)24 b(1986.)0 2251 y([PV05a])126 b(Arshak)19 b(P)n(etrosy)n(an)d(and)j(Enrico)f(V)-7 b(aldino)r(ci.)19 b(Densit)n(y)g(estimates)g(for)g(a)f (degenerate/singular)e(phase-)417 2351 y(transition)27 b(mo)r(del.)h Fu(SIAM)i(J.)f(Math.)i(A)n(nal.)p Fy(,)e (36\(4\):1057{1079)22 b(\(electronic\),)28 b(2005.)0 2451 y([PV05b])122 b(Arshak)27 b(P)n(etrosy)n(an)f(and)i(Enrico)e(V)-7 b(aldino)r(ci.)29 b(Geometric)e(prop)r(erties)g(of)h(Bernoulli-t)n(yp)r (e)f(minimiz-)417 2550 y(ers.)g Fu(Interfac)l(es)j(F)-6 b(r)l(e)l(e)30 b(Bound.)p Fy(,)f(7\(1\):55{77,)c(2005.)0 2650 y([Rob82])138 b(Derek)23 b(John)e(Scott)i(Robinson.)f Fu(A)j(c)l(ourse)g(in)g(the)g(the)l(ory)h(of)g(gr)l(oups)p Fy(,)e(v)n(olume)e(80)f(of)i Fu(Gr)l(aduate)i(T)-6 b(exts)417 2750 y(in)30 b(Mathematics)p Fy(.)g(Springer-V)-7 b(erlag,)25 b(New)j(Y)-7 b(ork,)26 b(1982.)0 2849 y([RS03])180 b(P)n(aul)29 b(H.)g(Rabino)n(witz)g(and)g(Ed)g(Stredulinsky)-7 b(.)29 b(Mixed)g(states)g(for)g(an)g(Allen-Cahn)g(t)n(yp)r(e)g(equation.)417 2949 y Fu(Comm.)i(Pur)l(e)f(Appl.)h(Math.)p Fy(,)e(56\(8\):1078{1134,) 23 b(2003.)i(Dedicated)j(to)g(the)g(memory)f(of)g(J)r(\177)-44 b(urgen)27 b(K.)417 3048 y(Moser.)0 3148 y([RS04a])138 b(P)-7 b(.)35 b(H.)g(Rabino)n(witz)f(and)h(E.)g(Stredulinsky)-7 b(.)34 b(On)h(some)f(results)g(of)h(Moser)f(and)h(of)f(Bangert.)g Fu(A)n(nn.)417 3248 y(Inst.)c(H.)g(Poinc)l(ar)n(\023)-40 b(e)31 b(A)n(nal.)f(Non)f(Lin)n(\023)-40 b(eair)l(e)p Fy(,)29 b(21\(5\):673{688,)24 b(2004.)0 3347 y([RS04b])134 b(P)n(aul)29 b(H.)g(Rabino)n(witz)g(and)g(Ed)g(Stredulinsky)-7 b(.)29 b(Mixed)g(states)g(for)g(an)g(Allen-Cahn)g(t)n(yp)r(e)g (equation.)417 3447 y(I)r(I.)f Fu(Calc.)k(V)-6 b(ar.)30 b(Partial)h(Di\013er)l(ential)f(Equations)p Fy(,)f(21\(2\):157{207,)23 b(2004.)0 3547 y([RS04c])143 b(P)n(aul)33 b(H.)g(Rabino)n(witz)g(and)g (Ed)g(Stredulinsky)-7 b(.)33 b(On)g(some)g(results)f(of)i(Moser)e(and)h (of)g(Bangert.)f(I)r(I.)417 3646 y Fu(A)l(dv.)f(Nonline)l(ar)f(Stud.)p Fy(,)d(4\(4\):377{396,)d(2004.)0 3746 y([Rue99])143 b(Da)n(vid)26 b(Ruelle.)g Fu(Statistic)l(al)j(me)l(chanics)p Fy(.)e(W)-7 b(orld)26 b(Scien)n(ti\014c)g(Publishing)g(Co.)f(Inc.,)i(Riv)n(er)e (Edge,)g(NJ,)417 3845 y(1999.)h(Rigorous)g(results,)h(Reprin)n(t)h(of)f (the)h(1989)e(edition.)0 3945 y([Ser77])171 b(Jean-Pierre)23 b(Serre.)h Fu(A)n(rbr)l(es,)k(amalgames,)i Fy(SL)1924 3957 y Fc(2)1961 3945 y Fy(.)25 b(So)r(ci)n(\023)-39 b(et)n(\023)g(e)24 b(Math)n(\023)-39 b(ematique)23 b(de)j(F)-7 b(rance,)25 b(P)n(aris,)e(1977.)417 4045 y(Av)n(ec)29 b(un)h(sommaire)e(anglais,)g(R)n(\023)-39 b(edig)n(\023)g(e)27 b(a)n(v)n(ec)h(la)h(collab)r(oration)e(de)j(Hyman)f(Bass,)f(Ast)n(\023) -39 b(erisque,)29 b(No.)417 4144 y(46.)0 4244 y([Sim93])149 b(Barry)21 b(Simon.)i Fu(The)j(statistic)l(al)f(me)l(chanics)i(of)f (lattic)l(e)f(gases.)h(Vol.)g(I)p Fy(.)d(Princeton)e(Series)h(in)h(Ph)n (ysics.)417 4344 y(Princeton)k(Univ)n(ersit)n(y)g(Press,)f(Princeton,)g (NJ,)i(1993.)0 4443 y([T)-7 b(or04])159 b(Monica)34 b(T)-7 b(orres.)33 b(Plane-lik)n(e)g(minimal)i(surfaces)e(in)i(p)r(erio)r(dic) f(media)h(with)g(exclusions.)e Fu(SIAM)j(J.)417 4543 y(Math.)31 b(A)n(nal.)p Fy(,)e(36\(2\):523{551)23 b(\(electronic\),)k (2004.)0 4642 y([V)-7 b(al86])167 b(F)-7 b(ran\030)-37 b(cois)43 b(V)-7 b(allet.)44 b Fu(Thermo)l(dynamique)j(unidimensionel)t (le,)k(et)44 b(structur)l(es)f(bidimensionel)t(les)48 b(de)417 4742 y(quelques)34 b(mo)l(d)n(\022)-40 b(eles)35 b(p)l(our)f(des)h(syst)n(\022)-40 b(emes)34 b(inc)l(ommensur)l(ables)p Fy(.)e(PhD)h(thesis,)g(Univ)n(ersit)n(\023)-39 b(e)30 b(Pierre-et-)417 4842 y(Marie-Curie)c(P)n(aris)g(VI,)i(1986.)0 4941 y([V)-7 b(al01])167 b(Enrico)21 b(V)-7 b(aldino)r(ci.)22 b Fu(Plane-like)j(minimizers)g(in)g(p)l(erio)l(dic)h(me)l(dia:)37 b(jet)25 b(\015ows)f(and)h(Ginzbur)l(g-Landau)p Fy(.)417 5041 y(PhD)j(thesis,)g(Univ)n(ersit)n(y)e(of)i(T)-7 b(exas)27 b(at)g(Austin,)h(2001.)e(MP)p 2345 5041 25 4 v 30 w(AR)n(C)h(#)h (01-356.)0 5141 y([V)-7 b(al04])167 b(Enrico)20 b(V)-7 b(aldino)r(ci.)21 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