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Fs(x)25 b Fk(2)g Fm(Z)2010 4462 y Fr(d)352 4625 y Ft(\(H2\))47 b Fs(J)632 4639 y Fr(A)715 4625 y Ft(=)25 b(0)30 b(if)g(the)g(diameter)g(of)h Fs(A)f Ft(is)g(greater)h(than)f Fs(r)352 4754 y Ft(\(H3\))47 b Fk(k)p Fs(J)9 b Fk(k)26 b Ft(:=)878 4686 y Fj(P)974 4781 y Fr(A)p Fq(2)p Fo(F)p Fp(:)16 b Fr(A)p Fq(3)p Fp(0)1301 4754 y Fk(k)p Fs(J)1396 4768 y Fr(A)1454 4754 y Fk(k)1499 4768 y Fr(u)1569 4754 y Fs(<)25 b Fk(1)382 4904 y Ft(Conditions)c (\(H1\),)26 b(\(H2\),)h(\(H3\))d(will)d(alw)m(a)m(ys)j(b)s(e)e(assumed) h(without)f(explicit)g(men)m(tion.)38 b(The)382 5017 y(Hamiltonian)29 b(\()p Fs(H)1016 5031 y Fp(\003)1069 5017 y Ft(\))1104 5031 y Fp(\003)p Fq(2)p Fo(F)1275 5017 y Ft(asso)s(ciated)i(with)e Fs(J)39 b Ft(is)30 b(de\014ned)f(as)1274 5219 y Fs(H)1350 5233 y Fp(\003)1428 5219 y Ft(:)c(\012)g Fk(3)g Fs(\033)j Fk(!)1991 5132 y Fj(X)1851 5334 y Fr(A)p Fq(2)p Fo(F)p Fp(:)16 b Fr(A)p Fq(\\)p Fp(\003)p Fq(6)p Fp(=)p Fq(;)2277 5219 y Fs(J)2327 5233 y Fr(A)2385 5219 y Ft(\()p Fs(\033)s Ft(\))26 b Fk(2)f Fm(R)18 b Fs(:)1978 5652 y Ft(4)p eop %%Page: 5 6 5 5 bop 382 548 a Ft(Clearly)31 b Fk(k)p Fs(H)821 562 y Fp(\003)874 548 y Fk(k)919 562 y Fr(u)994 548 y Fk(\024)e(j)p Ft(\003)p Fk(jk)p Fs(J)9 b Fk(k)p Ft(.)50 b(F)-8 b(or)34 b Fs(\033)n(;)15 b(\034)40 b Fk(2)28 b Ft(\012)33 b(w)m(e)g(also)g(let) g Fs(H)2493 515 y Fr(\034)2486 575 y Fp(\003)2538 548 y Ft(\()p Fs(\033)s Ft(\))e(:=)e Fs(H)2895 562 y Fp(\003)2948 548 y Ft(\()p Fs(\033)3035 562 y Fr(V)3096 548 y Fs(\034)3136 562 y Fr(V)3192 543 y Fn(c)3229 548 y Ft(\))k(and)f Fs(\034)43 b Ft(is)382 661 y(called)34 b(the)h Fl(b)-5 b(oundary)38 b(c)-5 b(ondition)p Ft(.)55 b(F)-8 b(or)36 b(eac)m(h)g(\003)c Fk(2)g Fm(F)5 b Ft(,)42 b Fs(\034)g Fk(2)32 b Ft(\012)j(the)g (\(\014nite)f(v)m(olume\))g(Gibbs)382 774 y(measure)c(on)g(\(\012)p Fs(;)15 b Fk(F)9 b Ft(\),)32 b(are)f(giv)m(en)f(b)m(y)1149 955 y Fs(\026)1204 918 y Fr(\034)1204 978 y Fp(\003)1257 955 y Ft(\()p Fs(\033)s Ft(\))c(:=)1530 882 y Fj(\000)1571 955 y Fs(Z)1640 918 y Fr(\034)1633 978 y Fp(\003)1686 882 y Fj(\001)1728 904 y Fq(\000)p Fp(1)1838 955 y Ft(exp[)p Fk(\000)p Fs(H)2156 918 y Fr(\034)2149 978 y Fp(\003)2201 955 y Ft(\()p Fs(\033)s Ft(\)])15 b(1)-25 b(I)2420 974 y Fq(f)p Fr(\034)2486 987 y Fi(\003)2527 973 y Fn(c)2565 974 y Fq(g)2604 955 y Ft(\()p Fs(\033)2691 969 y Fp(\003)2740 951 y Fn(c)2776 955 y Ft(\))15 b Fs(;)582 b Ft(\(2.3\))382 1125 y(where)27 b Fs(Z)711 1092 y Fr(\034)704 1152 y Fp(\003)784 1125 y Ft(is)f(the)h(prop)s(er)f(normalization)g(factor)i (called)e(partition)g(function,)h(and)f(1)-25 b(I)27 b(is)f(the)382 1238 y(indicator)f(function.)39 b(In)26 b(the)h(future)f(w)m(e)h(are)g(going)g(to)g(consider)f(an)h(in)m (teraction)f Fs(J)36 b Ft(with)26 b(an)382 1351 y(explicit)j (additional)g Fl(chemic)-5 b(al)35 b(p)-5 b(otential)42 b Fs(\025)p Ft(.)h(In)30 b(particular)g(w)m(e)h(will)d(consider)i(a)i (c)m(hemical)382 1464 y(p)s(oten)m(tial)c(on)g(a)h(p)s(olycub)s(e)d (\(\003)p Fs(;)15 b(`;)g Fk(A)p Ft(\))29 b(suc)m(h)f(that)h 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Fp(0)1140 5235 y Fs(;)g(m;)g(`)1338 5249 y Fp(0)1378 5235 y Ft(.)63 b(Analogously)37 b(\\for)h Fs(x)f Ft(large)h(enough")g(means)f Fl(for)j Fs(x)f Fl(lar)-5 b(ger)p 382 5318 1296 4 v 486 5371 a Fi(3)520 5403 y Fv(again)27 b(with)f(some)f(abuse)h(of)h(notation)1978 5652 y Ft(7)p eop %%Page: 8 9 8 8 bop 382 548 a Fl(than)31 b(some)f(c)-5 b(onstant)p Ft(.)41 b(F)-8 b(or)28 b(simplicit)m(y)c(w)m(e)k(write)e(things)g(lik)m (e)g(\\Assume)h(\(USM\).)h(Then)e(for)382 661 y(all)34 b Fs(")g(>)f Ft(0)j(there)f(is)f Fs(C)40 b(>)33 b Ft(0)j(suc)m(h)f (that)h(.)15 b(.)h(.)f(")55 b(without)35 b(reiterating)g(that)g Fs(C)42 b Ft(dep)s(ends)33 b(not)382 774 y(only)c(on)i Fs(")p Ft(,)g(but)e(in)g(principle,)f(on)i(all)f(the)i(paramen)m(ters)f (men)m(tioned)g(ab)s(o)m(v)m(e.)382 937 y Ff(Theorem)38 b(2.2.)44 b Fl(Assume)36 b(\(USM\).)f(Then)i(for)f(al)5 b(l)37 b Fs(")32 b(>)f Ft(0)36 b Fl(and)h(for)g(al)5 b(l)36 b(lo)-5 b(c)g(al)38 b(functions)e Fs(f)382 1050 y Fl(on)d Ft(\012)f Fl(ther)-5 b(e)34 b(is)e 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Fr(K)875 3350 y Fn(t)907 3328 y Fs(f)952 3342 y Fr(s)989 3328 y Fk(k)1034 3290 y Fp(2)1099 3328 y Ft(=)g Fk(k)p Fs(P)1298 3342 y Fr(s)1335 3328 y Fs(R)1404 3342 y Fr(K)1464 3350 y Fn(t)1496 3328 y Fs(f)10 b Fk(k)1596 3290 y Fp(2)1660 3328 y Fk(\024)25 b(k)p Fs(R)1870 3342 y Fr(K)1930 3350 y Fn(t)1962 3328 y Fs(f)10 b Fk(k)2062 3290 y Fp(2)2127 3328 y Ft(=)2335 3266 y(1)p 2233 3307 251 4 v 2233 3390 a Fk(j)p Fs(B)2327 3404 y Fr(K)2387 3412 y Fn(t)2419 3390 y Fk(j)2444 3364 y Fp(2)2583 3241 y Fj(X)2508 3438 y Fr(x;y)r Fq(2)p Fr(B)2705 3449 y Fn(K)2756 3462 y(t)2804 3328 y Fs(\026)p Ft(\()p Fs(#)2948 3342 y Fr(x)2991 3328 y Fs(f)25 b(#)3115 3342 y Fr(y)3156 3328 y Fs(f)10 b Ft(\))1099 3625 y(=)1307 3564 y(1)p 1205 3604 V 1205 3688 a Fk(j)p Fs(B)1299 3702 y Fr(K)1359 3710 y Fn(t)1391 3688 y Fk(j)1416 3661 y Fp(2)1555 3539 y Fj(X)1480 3736 y Fr(x;y)r Fq(2)p Fr(B)1677 3747 y Fn(K)1728 3760 y(t)1775 3625 y Fs(\026)p Ft(\(\()p Fs(#)1954 3639 y Fr(x)p Fq(\000)p Fr(y)2091 3625 y Fs(f)g Ft(\))p Fs(f)g Ft(\))24 b Fk(\024)2484 3564 y Ft(1)p 2401 3604 212 4 v 2401 3688 a Fk(j)p Fs(B)2495 3702 y Fr(K)2555 3710 y Fn(t)2587 3688 y Fk(j)2696 3539 y Fj(X)2637 3736 y Fr(z)s Fq(2)p Fr(B)2773 3747 y Fi(2)p Fn(K)2854 3760 y(t)2902 3625 y Fs(\026)p Ft(\(\()p Fs(#)3081 3639 y Fr(z)3121 3625 y Fs(f)10 b Ft(\))p Fs(f)g Ft(\))382 3892 y(so,)35 b(using)e(our)g(mixing)g(assumption)f(\(2.10\),)37 b(w)m(e)e(obtain)e(that)i(there)f(is)f Fs(A)3035 3906 y Fp(1)3106 3892 y Ft(=)e Fs(A)3276 3906 y Fp(1)3316 3892 y Ft(\()p Fs(f)10 b Ft(\))31 b Fs(>)g Ft(0)382 4004 y(suc)m(h)f(that)1316 4109 y Fk(k)p Fs(R)1430 4123 y Fr(K)1490 4131 y Fn(t)1522 4109 y Fs(f)1567 4123 y Fr(s)1603 4109 y Fk(k)1648 4072 y Fp(2)1713 4109 y Fk(\024)25 b Fs(A)1877 4123 y Fp(1)1932 4109 y Fs(t)1965 4072 y Fq(\000)p Fr(d=)p Fp(2)2313 4109 y Fk(8)p Fs(s;)15 b(t)24 b(>)h Ft(0)15 b Fs(:)748 b Ft(\(3.2\))382 4253 y(Th)m(us,)29 b(letting)h Fs(')p Ft(\()p Fs(t)p Ft(\))d(:=)e Fk(k)p Fs(f)1325 4267 y Fr(t)1375 4253 y Fk(\000)20 b Fs(R)1535 4267 y Fr(K)1595 4275 y Fn(t)1626 4253 y Fs(f)1671 4267 y Fr(t)1700 4253 y Fk(k)1745 4220 y Fp(2)1785 4253 y Ft(,)31 b(what)f(w)m(e)h(need)f(to)h(do)f(is)g(to)h(sho)m(w)f(that)1634 4412 y Fs(')p Ft(\()p Fs(t)p Ft(\))c Fk(\024)f Fs(A)1986 4426 y Fp(2)2041 4412 y Fs(t)2074 4374 y Fq(\000)p Fr(d=)p Fp(2+)p Fr(")2342 4412 y Fs(:)1066 b Ft(\(3.3\))382 4570 y(Inequalit)m(y)29 b(\(3.3\))j(is)e(implied,)d(b)m(y)k(iteration,)f(b)m (y)g(the)h(follo)m(wing)d(statemen)m(t)814 4729 y Fk(9)p Fs(\016)h(<)c Ft(4)1075 4691 y Fq(\000)p Fr(d=)p Fp(2)1272 4729 y Ft(suc)m(h)30 b(that)h Fk(8)p Fs(t)24 b Fk(\025)h Ft(0)122 b Fs(')p Ft(\()p Fs(t)p Ft(\))26 b Fk(\024)f Fs(\016)s(')p Ft(\()p Fs(t=)p Ft(4\))e(+)d Fs(A)2806 4743 y Fp(3)2861 4729 y Fs(t)2894 4691 y Fq(\000)p Fr(d=)p Fp(2+)p Fr(")3162 4729 y Fs(:)246 b Ft(\(3.4\))382 4887 y(In)30 b(order)f(to)j(pro)m(v)m(e)f(\(3.4\))h(w)m(e)f(write,)f(using)e (\(3.2\))977 5046 y Fs(')p Ft(\()p Fs(t)p Ft(\))e Fk(\024)f Ft(2)p Fk(k)p Fs(f)1396 5060 y Fr(t)1446 5046 y Fk(\000)20 b Fs(R)1606 5060 y Fr(K)1666 5074 y Fn(t=)p Fi(4)1760 5046 y Fs(f)1805 5060 y Fr(t)1834 5046 y Fk(k)1879 5008 y Fp(2)1939 5046 y Ft(+)g(2)p Fk(k)p Fs(R)2189 5060 y Fr(K)2249 5068 y Fn(t)2282 5046 y Fs(f)2327 5060 y Fr(t)2376 5046 y Fk(\000)g Fs(R)2536 5060 y Fr(K)2596 5074 y Fn(t=)p Fi(4)2690 5046 y Fs(f)2735 5060 y Fr(t)2764 5046 y Fk(k)2809 5008 y Fp(2)1165 5201 y Fk(\024)25 b Ft(2)p Fk(k)p Fs(f)1396 5215 y Fr(t)1446 5201 y Fk(\000)20 b Fs(R)1606 5215 y Fr(K)1666 5229 y Fn(t=)p Fi(4)1760 5201 y Fs(f)1805 5215 y Fr(t)1834 5201 y Fk(k)1879 5163 y Fp(2)1939 5201 y Ft(+)g(4)p Fk(k)p Fs(R)2189 5215 y Fr(K)2249 5229 y Fn(t=)p Fi(4)2344 5201 y Fs(f)2389 5215 y Fr(t)2418 5201 y Fk(k)2463 5163 y Fp(2)2523 5201 y Ft(+)g(4)p Fk(k)p Fs(R)2773 5215 y Fr(K)2833 5223 y Fn(t)2866 5201 y Fs(f)2911 5215 y Fr(t)2940 5201 y Fk(k)2985 5163 y Fp(2)3433 5201 y Ft(\(3.5\))1165 5362 y Fk(\024)25 b Ft(2)p Fk(k)p Fs(f)1396 5376 y Fr(t)1446 5362 y Fk(\000)20 b Fs(R)1606 5376 y Fr(K)1666 5390 y Fn(t=)p Fi(4)1760 5362 y Fs(f)1805 5376 y Fr(t)1834 5362 y Fk(k)1879 5325 y Fp(2)1939 5362 y Ft(+)g Fs(A)2098 5325 y Fq(0)2098 5385 y Fp(1)2153 5362 y Fs(t)2186 5325 y Fq(\000)p Fr(d=)p Fp(2)2367 5362 y Fs(:)1978 5652 y Ft(8)p eop %%Page: 9 10 9 9 bop 382 548 a Ft(Let)31 b(then)1116 653 y Fs( )s Ft(\()p Fs(t;)15 b(K)7 b Ft(\))26 b(:=)f Fk(k)p Fs(f)1642 667 y Fr(t)1692 653 y Fk(\000)20 b Fs(R)1852 667 y Fr(K)1920 653 y Fs(f)1965 667 y Fr(t)1994 653 y Fk(k)2039 615 y Fp(2)2261 653 y Fs(t)25 b Fk(\025)g Ft(0)p Fs(;)46 b(K)32 b Fk(2)25 b Fm(Z)2791 667 y Fp(+)2861 653 y Fs(:)382 801 y Ft(W)-8 b(e)35 b(claim)d(that)i(in)f(order)g(to)h(pro)m(v)m(e)h (\(3.4\))g(it)e(is)g(su\016cien)m(t)g(to)h(sho)m(w)f(that)i(for)e(some) h Fs(A)3479 815 y Fp(4)3549 801 y Ft(=)382 913 y Fs(A)450 927 y Fp(4)490 913 y Ft(\()p Fs(";)15 b(f)10 b Ft(\))31 b(w)m(e)f(ha)m(v)m(e)427 1138 y Fk(9)p Fs(\016)f(<)c Ft(4)688 1101 y Fq(\000)p Fr(d=)p Fp(2)880 1138 y Fs(s:t:)g Fk(8)p Fs(K)32 b Fk(\024)25 b Fs(K)1364 1152 y Fr(t)1500 1015 y Fj(Z)1590 1041 y Fp(2)p Fr(t)1550 1221 y(t)1655 1138 y Fs( )s Ft(\()p Fs(s;)15 b(K)7 b Ft(\))15 b Fs(ds)27 b Fk(\024)2192 1077 y Fs(\016)p 2192 1117 46 4 v 2192 1201 a Ft(2)2262 1015 y Fj(Z)2353 1041 y Fp(2)p Fr(t)2313 1221 y(t)2418 1138 y Fs( )s Ft(\()p Fs(s=)p Ft(2)p Fs(;)15 b(K)7 b Ft(\))15 b Fs(ds)c Ft(+)3123 1077 y Fs(A)3191 1091 y Fp(4)p 3016 1117 322 4 v 3016 1204 a Fs(t)3049 1178 y Fr(d=)p Fp(2)p Fq(\000)p Fp(1)p Fq(\000)p Fr(")3362 1138 y Fs(:)46 b Ft(\(3.6\))382 1351 y(In)30 b(fact,)h(since)f Fs( )s Ft(\()p Fk(\001)p Fs(;)15 b(K)7 b Ft(\))32 b(is)d(nonincreasing) g(\(3.6\))j(implies)1360 1560 y Fs( )s Ft(\(2)p Fs(t;)15 b(K)7 b Ft(\))27 b Fk(\024)1828 1498 y Fs(\016)p 1827 1539 46 4 v 1827 1622 a Ft(2)1898 1560 y Fs( )s Ft(\()p Fs(t=)p Ft(2)p Fs(;)15 b(K)7 b Ft(\))22 b(+)2462 1498 y Fs(A)2530 1512 y Fp(4)p 2400 1539 232 4 v 2400 1626 a Fs(t)2433 1600 y Fr(d=)p Fp(2)p Fq(\000)p Fr(")382 1756 y Ft(and,)30 b(using)f(\(3.5\),)j(w)m(e)f(\014nd)457 1972 y Fs(')p Ft(\()p Fs(t)p Ft(\))26 b Fk(\024)f Ft(2)p Fs( )s Ft(\()p Fs(t;)15 b(K)1033 1991 y Fr(t=)p Fp(4)1135 1972 y Ft(\))21 b(+)1310 1911 y Fs(A)1378 1878 y Fq(0)1378 1935 y Fp(1)p 1292 1951 144 4 v 1292 2038 a Fs(t)1325 2012 y Fr(d=)p Fp(2)1471 1972 y Fk(\024)k Fs(\016)s( )s Ft(\()p Fs(t=)p Ft(4)p Fs(;)15 b(K)1947 1991 y Fr(t=)p Fp(4)2050 1972 y Ft(\))21 b(+)2225 1911 y Fs(A)2293 1878 y Fq(0)2293 1935 y Fp(1)p 2206 1951 V 2206 2038 a Fs(t)2239 2012 y Fr(d=)p Fp(2)2381 1972 y Ft(+)2543 1911 y Fs(A)2611 1878 y Fq(0)2611 1935 y Fp(4)p 2481 1951 232 4 v 2481 2038 a Fs(t)2514 2012 y Fr(d=)p Fp(2)p Fq(\000)p Fr(")2748 1972 y Fk(\024)k Fs(\016)s(')p Ft(\()p Fs(t=)p Ft(4\))e(+)3325 1911 y Fs(A)3393 1925 y Fp(3)p 3263 1951 V 3263 2038 a Fs(t)3296 2012 y Fr(d=)p Fp(2)p Fq(\000)p Fr(")3519 1972 y Fs(:)382 2168 y Ft(Hence)31 b(the)g(theorem)f(follo)m(ws)g(from) g(\(3.6\).)382 2405 y Fe(3.1)100 b(Pro)s(of)32 b(of)h(statement)f (\(3.6\))382 2577 y Ft(Let)23 b Fs(K)32 b Fk(\024)25 b Fs(K)819 2591 y Fr(t)849 2577 y Ft(,)g(let)e(\()p Fs(B)1127 2591 y Fr(L)1175 2600 y Fi(2)1214 2577 y Fs(;)15 b(`;)g Fk(A)p Ft(\))24 b(b)s(e)e(a)h(p)s(olycub)s(e,)g(and)g(c)m(ho)s(ose)h(t) m(w)m(o)g(more)f(in)m(tegers)g Fs(L;)15 b(L)3382 2591 y Fp(1)3445 2577 y Ft(suc)m(h)382 2690 y(that)34 b Fs(`)c Fk(\024)h Fs(L)f(<)g(L)1007 2704 y Fp(1)1077 2690 y Fs(<)g(L)1240 2704 y Fp(2)1280 2690 y Ft(.)50 b(W)-8 b(e)35 b(an)m(ticipate)1906 2657 y Fp(4)1979 2690 y Ft(that)f(w)m(e)g(are)g(going)g(to)g(c)m(ho)s (ose)h Fs(`)30 b Ft(=)g Fs(L)h Fk(\030)3486 2616 y(p)p 3562 2616 33 4 v 74 x Fs(t)p Ft(,)382 2803 y Fs(L)444 2817 y Fp(1)514 2803 y Fk(\030)615 2729 y(p)p 690 2729 V 690 2803 a Fs(t)15 b Ft(log)i Fs(t)33 b Ft(and)g Fs(L)1179 2817 y Fp(2)1248 2803 y Fk(\030)1349 2729 y(p)p 1425 2729 V 74 x Fs(t)15 b Ft(\(log)i Fs(t)p Ft(\))1709 2770 y Fp(2)1782 2803 y Ft(\(precise)33 b(de\014nitions)e(in)h(\(3.23\)\).) 53 b(Let)34 b(\003)3243 2817 y Fp(1)3313 2803 y Ft(:=)c Fs(B)3508 2817 y Fr(L)3556 2826 y Fi(1)3595 2803 y Ft(,)382 2916 y(\003)445 2930 y Fp(2)510 2916 y Ft(:=)25 b Fs(B)700 2930 y Fr(L)748 2939 y Fi(2)787 2916 y Ft(,)30 b(and)g(de\014ne)1297 3082 y Fs(g)1340 3096 y Fr(t)1396 3082 y Ft(:=)25 b Fs(P)1575 3096 y Fr(t)1605 3082 y Ft(\()p Fs(f)30 b Fk(\000)20 b Fs(R)1875 3096 y Fr(K)1943 3082 y Fs(f)10 b Ft(\))181 b Fs(g)2257 3096 y Fr(x;t)2372 3082 y Ft(:=)25 b Fs(#)2547 3096 y Fr(x)2591 3082 y Fs(g)2634 3096 y Fr(t)2679 3082 y Fs(:)729 b Ft(\(3.7\))382 3249 y(Thanks)29 b(to)i(translation)f(in)m (v)-5 b(ariance,)29 b(w)m(e)i(ha)m(v)m(e,)h(since)e Fs(\026)15 b(g)2439 3263 y Fr(x;t)2553 3249 y Ft(=)25 b Fs(\026f)35 b Ft(=)25 b(0,)1442 3457 y Fs( )s Ft(\()p Fs(t;)15 b(K)7 b Ft(\))26 b(=)1926 3396 y(1)p 1863 3436 172 4 v 1863 3519 a Fk(j)p Fs(B)1957 3533 y Fr(L)2010 3519 y Fk(j)2088 3371 y Fj(X)2060 3568 y Fr(x)p Fq(2)p Fr(B)2200 3579 y Fn(L)2261 3457 y Fs(\026)p Ft(\()p Fs(g)2397 3420 y Fp(2)2394 3480 y Fr(x;t)2484 3457 y Ft(\))15 b Fs(:)874 b Ft(\(3.8\))382 3700 y(F)-8 b(or)31 b(simplicit)m(y)d(w)m(e)j (de\014ne)e(the)i(follo)m(wing)e(orthogonal)h(pro)5 b(jections)30 b(in)f Fs(L)2987 3667 y Fp(2)3027 3700 y Ft(\()p Fs(\026)p Ft(\))978 3866 y Fs(Q)1050 3880 y Fp(1)1115 3866 y Ft(=)c Fs(\026)p Ft(\()15 b Fk(\001)g(j)g(F)1461 3880 y Fp(\003)1510 3889 y Fi(1)1550 3866 y Ft(\))91 b Fs(Q)1748 3880 y Fp(2)1813 3866 y Ft(=)25 b Fs(\026)p Ft(\()15 b Fk(\001)g(j)g(F)2159 3880 y Fp(\003)2208 3889 y Fi(2)2248 3866 y Ft(\))91 b Fs(Q)2446 3880 y Fq(A)2532 3866 y Ft(=)25 b Fs(\026)p Ft(\()15 b Fk(\001)g(j)g Fs(N)2886 3880 y Fq(A)2948 3866 y Ft(\))g Fs(:)382 4033 y Ft(Then,)30 b(since)f Fs(G)938 4047 y Fq(A)999 4033 y Fs(Q)1071 4047 y Fq(A)1157 4033 y Ft(=)c Fs(Q)1325 4047 y Fq(A)1385 4033 y Fs(G)1456 4047 y Fq(A)1543 4033 y Ft(=)g Fs(Q)1711 4047 y Fq(A)1771 4033 y Ft(,)31 b(w)m(e)f(ha)m(v)m(e)863 4192 y Fs(\026)p Ft(\()p Fs(g)999 4155 y Fp(2)996 4215 y Fr(x;t)1085 4192 y Ft(\))c(=)f Fk(k)p Fs(g)1330 4206 y Fr(x;t)1420 4192 y Fk(k)1465 4155 y Fp(2)1530 4192 y Ft(=)g Fk(k)p Ft(\()p Fs(I)j Fk(\000)20 b Fs(Q)1937 4206 y Fp(1)1976 4192 y Ft(\))p Fs(g)2054 4206 y Fr(x;t)2144 4192 y Fk(k)2189 4155 y Fp(2)2249 4192 y Ft(+)g Fk(k)p Fs(Q)2457 4206 y Fp(1)2497 4192 y Fs(g)2540 4206 y Fr(x;t)2629 4192 y Fk(k)2674 4155 y Fp(2)978 4341 y Ft(=)25 b Fk(k)p Ft(\()p Fs(I)j Fk(\000)20 b Fs(Q)1385 4355 y Fp(1)1425 4341 y Ft(\))p Fs(g)1503 4355 y Fr(x;t)1592 4341 y Fk(k)1637 4304 y Fp(2)1697 4341 y Ft(+)g Fk(k)p Fs(G)1904 4355 y Fq(A)1966 4341 y Fs(Q)2038 4355 y Fp(1)2077 4341 y Fs(g)2120 4355 y Fr(x;t)2209 4341 y Fk(k)2254 4304 y Fp(2)2315 4341 y Ft(+)f Fk(k)p Ft(\(1)j Fk(\000)e Fs(G)2714 4355 y Fq(A)2775 4341 y Ft(\))p Fs(Q)2882 4355 y Fp(1)2921 4341 y Fs(g)2964 4355 y Fr(x;t)3054 4341 y Fk(k)3099 4304 y Fp(2)978 4489 y Ft(=)25 b Fk(k)p Ft(\()p Fs(I)j Fk(\000)20 b Fs(Q)1385 4503 y Fp(1)1425 4489 y Ft(\))p Fs(g)1503 4503 y Fr(x;t)1592 4489 y Fk(k)1637 4451 y Fp(2)1697 4489 y Ft(+)g Fk(k)p Fs(Q)1905 4503 y Fq(A)1966 4489 y Fs(Q)2038 4503 y Fp(1)2077 4489 y Fs(g)2120 4503 y Fr(x;t)2210 4489 y Fk(k)2255 4451 y Fp(2)1064 4636 y Ft(+)g Fk(k)p Ft(\()p Fs(I)28 b Fk(\000)20 b Fs(Q)1466 4650 y Fq(A)1527 4636 y Ft(\))p Fs(G)1633 4650 y Fq(A)1694 4636 y Fs(Q)1766 4650 y Fp(1)1805 4636 y Fs(g)1848 4650 y Fr(x;t)1938 4636 y Fk(k)1983 4599 y Fp(2)2043 4636 y Ft(+)g Fk(k)p Ft(\()p Fs(I)28 b Fk(\000)20 b Fs(G)2444 4650 y Fq(A)2505 4636 y Ft(\))p Fs(Q)2612 4650 y Fp(1)2651 4636 y Fs(g)2694 4650 y Fr(x;t)2784 4636 y Fk(k)2829 4599 y Fp(2)2884 4636 y Fs(:)3433 4414 y Ft(\(3.9\))382 4795 y(On)29 b(the)i(other)g(side,)e(since)h Fs(Q)1430 4809 y Fq(A)1490 4795 y Fs(Q)1562 4809 y Fp(2)1627 4795 y Ft(=)25 b Fs(Q)1795 4809 y Fq(A)1886 4795 y Ft(and)k Fs(Q)2134 4809 y Fp(2)2174 4795 y Fs(Q)2246 4809 y Fp(1)2310 4795 y Ft(=)c Fs(Q)2478 4809 y Fp(1)557 4954 y Fk(k)p Fs(Q)674 4968 y Fq(A)734 4954 y Fs(Q)806 4968 y Fp(1)846 4954 y Fs(g)889 4968 y Fr(x;t)978 4954 y Fk(k)h(\024)f(k)p Fs(Q)1262 4968 y Fq(A)1323 4954 y Fs(Q)1395 4968 y Fp(2)1434 4954 y Fs(g)1477 4968 y Fr(x;t)1566 4954 y Fk(k)c Ft(+)f Fk(k)p Fs(Q)1840 4968 y Fq(A)1901 4954 y Ft(\()p Fs(Q)2008 4968 y Fp(1)2068 4954 y Fk(\000)f Fs(Q)2230 4968 y Fp(2)2270 4954 y Ft(\))p Fs(g)2348 4968 y Fr(x;t)2437 4954 y Fk(k)760 5091 y(\024)25 b(k)p Fs(Q)973 5105 y Fq(A)1034 5091 y Fs(g)1077 5105 y Fr(x;t)1166 5091 y Fk(k)20 b Ft(+)g Fk(k)p Ft(\()p Fs(Q)1474 5105 y Fp(2)1535 5091 y Fk(\000)g Fs(Q)1698 5105 y Fp(1)1737 5091 y Ft(\))p Fs(g)1815 5105 y Fr(x;t)1905 5091 y Fk(k)25 b Ft(=)g Fk(k)p Fs(Q)2188 5105 y Fq(A)2249 5091 y Fs(g)2292 5105 y Fr(x;t)2381 5091 y Fk(k)c Ft(+)f Fk(k)p Fs(Q)2655 5105 y Fp(2)2695 5091 y Ft(\()p Fs(I)27 b Fk(\000)20 b Fs(Q)2960 5105 y Fp(1)3000 5091 y Ft(\))p Fs(g)3078 5105 y Fr(x;t)3167 5091 y Fk(k)760 5229 y(\024)25 b(k)p Fs(Q)973 5243 y Fq(A)1034 5229 y Fs(g)1077 5243 y Fr(x;t)1166 5229 y Fk(k)20 b Ft(+)g Fk(k)p Ft(\()p Fs(I)29 b Fk(\000)19 b Fs(Q)1633 5243 y Fp(1)1673 5229 y Ft(\))p Fs(g)1751 5243 y Fr(x;t)1840 5229 y Fk(k)c Fs(:)3387 5092 y Ft(\(3.10\))p 382 5318 1296 4 v 486 5371 a Fi(4)520 5403 y Fv(for)27 b(those)f(readers)g(who)g(do)g(not)f(lik)n(e)h(pro)r(ceeding)g(on)g(a)g (\\need-to-kno)n(w")f(basis)1978 5652 y Ft(9)p eop %%Page: 10 11 10 10 bop 382 548 a Ft(F)-8 b(rom)31 b(\(3.9\),)h(\(3.10\))h(w)m(e)e (get)705 708 y Fs(\026)p Ft(\()p Fs(g)841 670 y Fp(2)838 730 y Fr(x;t)927 708 y Ft(\))26 b Fk(\024)f Ft(3)15 b Fs(\026)g Ft([)q(V)-8 b(ar\()p Fs(g)1459 722 y Fr(x;t)1564 708 y Fk(j)15 b(F)1669 722 y Fp(\003)1718 731 y Fi(1)1757 708 y Ft(\)])21 b(+)f(2)15 b Fs(\026)2059 634 y Fj(\002)2098 708 y Fs(\026)p Ft(\()p Fs(g)2231 722 y Fr(x;t)2335 708 y Fk(j)g Fs(N)2448 722 y Fq(A)2509 708 y Ft(\))2544 670 y Fp(2)2584 634 y Fj(\003)1074 856 y Ft(+)20 b Fs(\026)1220 782 y Fj(\002)1257 856 y Ft(V)-8 b(ar)q(\()p Fs(G)1505 870 y Fq(A)1566 856 y Fs(Q)1638 870 y Fp(1)1678 856 y Fs(g)1721 870 y Fr(x;t)1825 856 y Fk(j)15 b Fs(N)1938 870 y Fq(A)1999 856 y Ft(\))2034 782 y Fj(\003)2093 856 y Ft(+)k Fs(\026)c Ft([)h Fs(G)2365 870 y Fq(A)2426 856 y Ft(\()p Fs(Q)2533 870 y 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Fs(G)1096 5332 y Fq(A)1157 5318 y Ft(\()p Fs(Q)1264 5332 y Fp(1)1304 5318 y Fs(g)1347 5332 y Fr(x;t)1436 5318 y Fs(;)g(Q)1548 5332 y Fp(1)1588 5318 y Fs(g)1631 5332 y Fr(x;t)1720 5318 y Ft(\))g(])26 b Fk(\024)f Fs(C)1982 5332 y Fp(3)2036 5217 y Fj(h)2089 5256 y Fs(`)2127 5223 y Fp(2)p 2089 5297 78 4 v 2112 5380 a Fs(t)2192 5318 y( )s Ft(\()p Fs(t=)p Ft(4)p Fs(;)15 b(K)7 b Ft(\))22 b(+)e Fs(e)2726 5280 y Fq(\000)p Fr(L)2829 5289 y Fi(1)2864 5280 y Fr(=)2899 5226 y Fq(p)p 2958 5226 26 3 v 54 x Fr(t)3003 5318 y Fk(k)p Fs(g)s Fk(k)3139 5280 y Fp(2)3180 5217 y Fj(i)3238 5318 y Fs(:)124 b Ft(\(3.22\))1955 5652 y(11)p eop %%Page: 12 13 12 12 bop 382 548 a Fl(End)31 b(of)h(pr)-5 b(o)g(of)34 b(of)e(The)-5 b(or)g(em)33 b(2.2)p Ft(.)41 b(T)-8 b(o)29 b(conclude)g(the)g(pro)s(of)g(w)m(e)g(c)m(ho)s(ose)h(appropriate)e(v)-5 b(alues)382 661 y(for)29 b Fs(`;)15 b(L;)g(L)762 675 y Fp(1)801 661 y Fs(;)g(L)903 675 y Fp(2)972 661 y Ft(and)28 b(collect)i(the)f(v)-5 b(arious)28 b(pieces)g(together.)42 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1475 a Fu(4)109 b(Cuto\013)37 b(estimate)g(and)f(p)m(ro)s (of)h(of)g(Prop)s(osition)g(3.1)382 1678 y Ft(In)32 b(this)f(section)i (w)m(e)g(pro)m(v)m(e)g(Prop)s(osition)d(3.1.)48 b(W)-8 b(e)34 b(observ)m(e)f(that)g(the)g(factor)g(3)g(app)s(earing)382 1791 y(in)26 b(the)i(assumption)e Fs(S)1173 1805 y Fr(g)1238 1791 y Fk(\032)f Fs(B)1403 1820 y Fp(3)p Fq(b)1469 1766 y(p)p 1528 1766 26 3 v 54 x Fr(t)q Fq(c)1617 1791 y Ft(is)h(completely) h(arbitrary)-8 b(.)39 b(By)28 b(rede\014ning)e(the)h(constan)m(t)382 1904 y Fs(C)37 b Ft(one)31 b(can)g(replace)g(this)f(3)h(with)e(an)m(y)i (n)m(um)m(b)s(er.)41 b(W)-8 b(e)32 b(follo)m(w)e(the)h(strategy)h(of)f (Prop)s(osition)382 2017 y(3.1)c(in)e([JLQY99],)j(with)c(suitable)h(mo) s(di\014cations)f(required)g(b)m(y)i(the)g(fact)h(that,)h(in)d(our)g (case,)382 2130 y Fs(\026)30 b Ft(is)f(not)i(a)g(pro)s(duct)e(measure.) 382 2302 y Ff(Lemma)k(4.1.)42 b Fl(Assume)32 b(\(USM\))g(and)i(let,)e (for)i Fs(j)d Fk(2)24 b Fm(N)7 b Fl(,)835 2471 y Fs(A)903 2485 y Fr(j)965 2471 y Ft(:=)26 b Fs(\026)p Ft(\()p Fk(\001)15 b(j)g(F)1322 2485 y Fr(B)1375 2495 y Fn(j)1413 2471 y Ft(\))186 b Fs(D)1709 2485 y Fr(j)1771 2471 y Ft(:=)25 b Fm(E)1947 2485 y Fr(B)2000 2495 y Fn(j)2043 2471 y Fk(n)p Fm(E)2143 2485 y Fr(B)2196 2495 y Fn(j)s Fw(\000)p Fn(r)p 2503 2398 78 4 v 2503 2471 a Fs(D)2581 2485 y Fr(j)2643 2471 y Ft(:=)g Fs(D)2839 2485 y Fr(j)2896 2471 y Fk([)20 b Fs(\016)s(B)3089 2485 y Fr(j)3141 2471 y Fs(:)382 2640 y Fl(Then,)45 b(ther)-5 b(e)43 b(exists)g(a)g(c)-5 b(onstant)44 b Fs(C)49 b(>)43 b Ft(0)g Fl(such)g(that)g(for)g(al)5 b(l)43 b Fs(#)g(>)g Ft(0)p Fl(,)i(and)e(for)g(al)5 b(l)43 b(lo)-5 b(c)g(al)382 2753 y(functions)33 b Fs(g)j Fl(on)d Ft(\012)p Fl(,)f(we)h(have)g(\(r)-5 b(ememb)g(er)34 b(\(2.15\)\))765 2964 y Fk(j)15 b Fs(\026)p Ft(\()p Fs(A)963 2978 y Fr(j)1000 2964 y Fs(g)k Fk(L)p Fs(g)s Ft(\))c Fk(j)26 b(\024)f(E)1416 2978 y Fr(B)1469 2988 y Fn(j)s Fw(\000)p Fn(r)1585 2964 y Ft(\()p Fs(g)s Ft(\))c(+)f Fs(#)15 b(C)22 b Fk(E)p 2017 2938 60 3 v 29 x Fr(D)2076 3003 y Fn(j)2113 2964 y Ft(\()p Fs(g)s Ft(\))g(+)2352 2902 y Fs(C)p 2352 2943 72 4 v 2361 3026 a(#)2448 2964 y(\026)p Ft([)15 b(\()p Fs(A)2646 2978 y Fr(j)t Fp(+)p Fr(r)2772 2964 y Fs(g)24 b Fk(\000)c Fs(A)2998 2978 y Fr(j)3034 2964 y Fs(g)s Ft(\))3115 2926 y Fp(2)3171 2964 y Ft(])15 b Fs(:)382 3173 y Fl(Pr)-5 b(o)g(of.)32 b Ft(W)-8 b(e)31 b(let)f Fk(h)p Fs(f)5 b(;)15 b(g)s Fk(i)27 b Ft(:=)e Fs(\026)p Ft(\()p Fs(f)10 b(g)s Ft(\),)31 b Fk(k)p Fs(f)10 b Fk(k)25 b Ft(:=)h Fs(\026)p Ft(\()p Fs(f)2012 3140 y Fp(2)2050 3173 y Ft(\))2085 3140 y Fp(1)p Fr(=)p Fp(2)2196 3173 y Ft(,)k(and)g(w)m(e)h(de\014ne,)f(for)g Fs(x)25 b Fk(2)g Fm(Z)3217 3140 y Fr(d)3253 3173 y Ft(,)31 b Fs(e)25 b Fk(2)g Fm(E)3517 3195 y Fo(Z)3573 3177 y Fn(d)791 3404 y Fs(h)843 3366 y Fr(j)843 3426 y(e)906 3404 y Ft(:=)1125 3343 y Fs(e)1167 3310 y Fq(\000r)1281 3318 y Fn(e)1314 3310 y Fr(H)1372 3318 y Fn(e)p 1037 3383 461 4 v 1037 3466 a Fs(A)1105 3480 y Fr(j)1142 3466 y Ft(\()p Fs(e)1219 3440 y Fq(\000r)1333 3448 y Fn(e)1367 3440 y Fr(H)1425 3448 y Fn(e)1462 3466 y Ft(\))1689 3404 y Fs(h)1741 3366 y Fr(j)1741 3426 y(x)1811 3404 y Ft(:=)2030 3343 y Fs(e)2072 3306 y Fq(\000r)2186 3314 y Fn(x)2225 3306 y Fr(H)2283 3320 y Fw(f)p Fn(x)p Fw(g)p 1942 3383 536 4 v 1942 3476 a Fs(A)2010 3490 y Fr(j)2047 3476 y Ft(\()p Fs(e)2124 3439 y Fq(\000r)2238 3447 y Fn(x)2277 3439 y Fr(H)2335 3453 y Fw(f)p Fn(x)p Fw(g)2442 3476 y Ft(\))2669 3404 y Fs(U)2741 3366 y Fr(j)2731 3426 y(e)2803 3404 y Ft(:=)g(1)c Fk(\000)f Fs(h)3133 3366 y Fr(j)3133 3426 y(e)3185 3404 y Fs(:)382 3635 y Ft(A)35 b(straigh)m(tforw)m(ard)h(computation)f(sho)m (ws)g(that)h(if)e Fs(e)g Ft(=)g Fk(f)p Fs(x;)15 b(y)s Fk(g)34 b(\032)g Fs(B)2841 3649 y Fr(j)2913 3635 y Ft(then)h(w)m(e)g (ha)m(v)m(e)i(\(re-)382 3748 y(mem)m(b)s(er)30 b(\(2.2\)\))1346 3853 y Fk(r)1422 3867 y Fr(e)1459 3853 y Fs(A)1527 3867 y Fr(j)1563 3853 y Fs(f)35 b Ft(=)25 b Fs(A)1807 3867 y Fr(j)1843 3853 y Ft(\()p Fk(r)1954 3867 y Fr(e)1991 3853 y Fs(f)10 b 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b(and)e(let)i Fs(q)1378 4477 y Fr(e)1414 4463 y Ft(\()p Fs(\033)s Ft(\))26 b(=)f(1)-25 b(I)1714 4482 y Fq(f)p Fr(\033)r Fp(\()p Fr(x)p Fp(\))p Fq(6)p Fp(=)p Fr(\033)r Fp(\()p Fr(y)r Fp(\))p Fq(g)2116 4463 y Ft(.)41 b(Then)855 4647 y Fk(r)931 4661 y Fr(e)968 4647 y Fs(A)1036 4661 y Fr(j)1072 4647 y Fs(f)35 b Ft(=)25 b Fs(q)1289 4661 y Fr(e)1340 4574 y Fj(\010)1393 4647 y Fs(A)1461 4661 y Fr(j)1498 4647 y Ft([)p Fk(r)1599 4661 y Fr(e)1636 4647 y Fs(f)10 b Ft(\(1)20 b(+)g Fs(h)1934 4610 y Fr(j)1934 4670 y(y)1976 4647 y Ft(\)])h(+)f Fs(A)2216 4661 y Fr(j)2252 4647 y Ft([)p Fs(f)10 b(V)2405 4610 y Fr(j)2385 4670 y(xy)2466 4647 y Ft(])21 b(+)e Fs(S)2658 4661 y Fr(x)2717 4647 y Fs(A)2785 4661 y Fr(j)2822 4647 y Ft([)p Fs(f)10 b(W)3001 4610 y Fr(j)2988 4670 y(xy)3068 4647 y Ft(])3093 4574 y Fj(\011)3433 4647 y Ft(\(4.3\))382 4817 y(where)1329 4986 y Fs(V)1402 4948 y Fr(j)1382 5008 y(xy)1488 4986 y Ft(:=)25 b Fs(h)1661 4948 y Fr(j)1661 5008 y(y)1718 4986 y Fs(q)1759 5000 y Fr(xy)1860 4986 y Fk(\000)20 b Ft(\(1)h Fk(\000)f Fs(q)2184 5000 y Fr(xy)2265 4986 y Ft(\))1296 5136 y Fs(W)1395 5098 y Fr(j)1382 5158 y(xy)1488 5136 y Ft(:=)25 b(1)c Fk(\000)f Fs(h)1818 5098 y Fr(j)1818 5158 y(x)1877 5136 y Fs(q)1918 5150 y Fr(xy)2019 5136 y Fk(\000)g Fs(q)2151 5150 y Fr(xy)2247 5136 y Fs(S)2303 5150 y Fr(y)2359 5136 y Ft(\()p Fs(h)2446 5098 y Fr(j)2446 5158 y(x)2491 5136 y Fs(=h)2588 5098 y Fr(j)2588 5158 y(y)2630 5136 y Ft(\))15 b Fs(:)p 382 5226 1296 4 v 486 5280 a Fi(5)520 5312 y Fv(the)30 b(follo)n(wing)j(\(apparen)n(tly?\))48 b(paranoic)32 b(de\014nitions)e(are)h(due)f(to)h(the)f(fact)h(that)f Fh(`)g Fv(m)n(ust)f(b)r(e)i(an)f Fb(o)l(dd)382 5403 y Fv(in)n(teger)c(whic)n(h)g(divides)f(2)p Fh(L)1204 5411 y Fi(2)1257 5403 y Fv(+)16 b(1.)1955 5652 y Ft(12)p eop %%Page: 13 14 13 13 bop 382 548 a Ft(It)30 b(is)g(easy)h(to)g(v)m(erify)f(that)1581 653 y Fs(A)1649 667 y Fr(j)1686 653 y Fs(V)1759 615 y Fr(j)1739 675 y(xy)1846 653 y Ft(=)24 b Fs(A)2009 667 y Fr(j)2046 653 y Fs(W)2145 615 y Fr(j)2132 675 y(xy)2238 653 y Ft(=)h(0)15 b Fs(:)382 860 y Ft(By)31 b(de\014nition)d(w)m(e)i (ha)m(v)m(e)844 1080 y Fs(\026)p Ft(\()p Fs(A)1002 1094 y Fr(j)1038 1080 y Fs(g)19 b Ft(\()p Fk(\000L)p Fs(g)s Ft(\)\))27 b(=)e Fk(E)8 b Ft(\()p Fs(A)1667 1094 y Fr(j)1704 1080 y Fs(g)s(;)15 b(g)s Ft(\))27 b(=)2004 1019 y(1)p 2004 1059 46 4 v 2004 1142 a(2)2169 994 y Fj(X)2074 1195 y Fr(e)p Fp(:)12 b Fr(e)p Fq(\\)p Fr(B)2272 1205 y Fn(j)2304 1195 y Fq(6)p Fp(=)p Fq(;)2409 1080 y Fs(\026)2464 1006 y Fj(\000)2506 1080 y Fs(c)2545 1094 y Fr(e)2582 1080 y Fk(r)2658 1094 y Fr(e)2695 1080 y Ft(\()p Fs(A)2798 1094 y Fr(j)2835 1080 y Fs(g)s Ft(\))p Fk(r)2992 1094 y Fr(e)3029 1080 y Fs(g)3075 1006 y Fj(\001)3133 1080 y Fs(;)382 1383 y Ft(so,)31 b(letting,)f Fs(Y)885 1397 y Fr(e)947 1383 y Ft(:=)25 b Fs(\026)1123 1310 y Fj(\000)1164 1383 y Fs(c)1203 1397 y Fr(e)1241 1383 y Fk(r)1317 1397 y Fr(e)1353 1383 y Ft(\()p Fs(A)1456 1397 y Fr(j)1493 1383 y Fs(g)s Ft(\))p Fk(r)1650 1397 y Fr(e)1688 1383 y Fs(g)1734 1310 y Fj(\001)1776 1383 y Ft(,)31 b(w)m(e)g(can)f(write) 1360 1570 y Fk(j)p Fs(\026)p Ft(\()p Fs(A)1543 1584 y Fr(j)1580 1570 y Fs(g)19 b Ft(\()p Fk(\000L)p Fs(g)s Ft(\)\))p Fk(j)27 b(\024)e Fs(X)2150 1584 y Fp(1)2210 1570 y Ft(+)20 b Fs(X)2376 1584 y Fp(2)2436 1570 y Ft(+)g Fs(X)2602 1584 y Fp(3)382 1757 y Ft(where)727 1959 y Fs(X)802 1973 y Fp(1)867 1959 y Ft(:=)999 1897 y(1)p 999 1938 V 999 2021 a(2)1069 1854 y Fj(\014)1069 1909 y(\014)1069 1964 y(\014)1160 1873 y(X)1099 2069 y Fr(e)p Fq(\032)p Fr(B)1240 2079 y Fn(j)s Fw(\000)p Fn(r)1366 1959 y Fs(Y)1419 1973 y Fr(e)1456 1854 y Fj(\014)1456 1909 y(\014)1456 1964 y(\014)1668 1959 y Fs(X)1743 1973 y Fp(2)1808 1959 y Ft(:=)1939 1897 y(1)p 1939 1938 V 1939 2021 a(2)2010 1854 y Fj(\014)2010 1909 y(\014)2010 1964 y(\014)2060 1873 y(X)2040 2069 y Fr(e)p Fq(2)p Fr(D)2178 2079 y Fn(j)2226 1959 y Fs(Y)2279 1973 y Fr(e)2315 1854 y Fj(\014)2315 1909 y(\014)2315 1964 y(\014)2527 1959 y Fs(X)2602 1973 y Fp(3)2667 1959 y Ft(:=)2799 1897 y(1)p 2799 1938 V 2799 2021 a(2)2869 1854 y Fj(\014)2869 1909 y(\014)2869 1964 y(\014)2933 1873 y(X)2899 2070 y Fr(e)p Fq(2)p Fr(\016)r(B)3065 2080 y Fn(j)3114 1959 y Fs(Y)3167 1973 y Fr(e)3204 1854 y Fj(\014)3204 1909 y(\014)3204 1964 y(\014)3249 1959 y Fs(:)382 2221 y Ft(F)-8 b(or)33 b(what)f(concerns)h(those)g(edges)g Fs(e)c Fk(\032)f Fs(B)1872 2235 y Fr(j)t Fq(\000)p Fr(r)2030 2221 y Ft(whic)m(h)j(con)m (tribute)h(to)h Fs(X)2919 2235 y Fp(1)2991 2221 y Ft(w)m(e)g(observ)m (e)g(that)382 2334 y Fs(c)421 2348 y Fr(e)458 2334 y Fs(A)526 2348 y Fr(j)563 2334 y Ft(\()p Fs(f)10 b Ft(\))39 b(=)f Fs(A)904 2348 y Fr(j)941 2334 y Ft(\()p Fs(c)1015 2348 y Fr(e)1052 2334 y Fs(f)10 b Ft(\).)65 b(F)-8 b(rom)39 b(this)e(equalit)m(y)-8 b(,)41 b(from)d(the)h(fact)g(that)g Fs(A)2886 2348 y Fr(j)2961 2334 y Ft(is)f(an)g(orthogonal)382 2447 y(pro)5 b(jection)30 b(in)f Fs(L)982 2414 y Fp(2)1021 2447 y Ft(\()p Fs(\026)p Ft(\),)i(and)f(from)g(\(4.2\),)i(it)e(follo)m (ws)g(that,)h(for)f(eac)m(h)i Fs(e)25 b Fk(\032)g Fs(B)3024 2461 y Fr(j)t Fq(\000)p Fr(r)582 2633 y Fs(Y)635 2647 y Fr(e)697 2633 y Ft(=)g Fs(\026)848 2560 y Fj(\000)889 2633 y Fs(c)928 2647 y Fr(e)965 2633 y Fs(A)1033 2647 y Fr(j)1070 2633 y Ft(\()p Fk(r)1181 2647 y Fr(e)1218 2633 y Fs(g)s Ft(\))p Fk(r)1375 2647 y Fr(e)1413 2633 y Fs(g)1459 2560 y Fj(\001)1526 2633 y Ft(=)g Fk(k)p Fs(A)1735 2647 y Fr(j)1772 2633 y Ft(\()1807 2570 y Fk(p)p 1884 2570 77 4 v 1884 2633 a Fs(c)1923 2647 y Fr(e)1975 2633 y Fk(r)2051 2647 y Fr(e)2088 2633 y Fs(g)s Ft(\))p Fk(k)2214 2596 y Fp(2)2280 2633 y Fk(\024)g(k)2421 2570 y(p)p 2497 2570 V 63 x Fs(c)2536 2647 y Fr(e)2588 2633 y Fk(r)2664 2647 y Fr(e)2701 2633 y Fs(g)s Fk(k)2792 2596 y Fp(2)2858 2633 y Ft(=)g Fs(\026)p Ft([)p Fs(c)3073 2647 y Fr(e)3125 2633 y Ft(\()p Fk(r)3236 2647 y Fr(e)3273 2633 y Fs(g)s Ft(\))3354 2596 y Fp(2)3394 2633 y Ft(])382 2820 y(hence)1696 2925 y Fs(X)1771 2939 y Fp(1)1836 2925 y Fk(\024)g(E)1980 2939 y Fr(B)2033 2949 y Fn(j)s Fw(\000)p Fn(r)2148 2925 y Ft(\()p 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3364 y Fr(j)3025 3424 y(e)3073 3401 y Ft(\))p Fk(j)3133 3328 y Fj(\003)3186 3401 y Fs(:)382 3675 y Ft(Using)29 b Fs(xy)f Fk(\024)d Ft(\()p Fs(#x)999 3642 y Fp(2)1059 3675 y Ft(+)20 b Fs(#)1204 3642 y Fq(\000)p Fp(1)1298 3675 y Fs(y)1346 3642 y Fp(2)1385 3675 y Ft(\))p Fs(=)p Ft(2)32 b(in)d(the)i(second)f(term,)h(w)m(e)g(get)908 3904 y Fs(X)983 3918 y Fp(2)1048 3904 y Fk(\024)1154 3843 y Fs(c)1193 3857 y Fr(M)p 1154 3883 V 1160 3967 a Fs(c)1199 3981 y Fr(m)1283 3803 y Fj(\020)1337 3904 y Ft(1)20 b(+)1503 3843 y Fs(#)p 1503 3883 54 4 v 1507 3967 a Ft(2)1567 3803 y Fj(\021)1637 3904 y Fk(E)1685 3918 y Fr(D)1743 3928 y Fn(j)1779 3904 y Ft(\()p Fs(g)s Ft(\))i(+)2017 3843 y Fs(c)2056 3857 y Fr(M)p 2017 3883 119 4 v 2027 3967 a Ft(4)p Fs(#)2180 3818 y Fj(X)2161 4015 y Fr(e)p Fq(2)p Fr(D)2299 4025 y Fn(j)2346 3904 y Fs(\026)2401 3831 y Fj(\002)2439 3904 y Fs(T)2492 3918 y Fr(e)2529 3831 y Fj(\000)2586 3904 y Fs(A)2654 3918 y Fr(j)2690 3904 y Ft(\()p Fs(g)e(U)2860 3867 y Fr(j)2850 3927 y(e)2896 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Ft(Cho)s(osing)c(no)m(w)i Fs(`)i Ft(=)g(3\()p Fk(b)1232 2323 y(p)p 1309 2323 33 4 v 1309 2397 a Fs(t=r)s Fk(c)5 b Ft(+)g(1\))23 b(and)f Fs(#)j Ft(=)1999 2323 y Fk(p)p 2075 2323 V 74 x Fs(t=)p Ft(\(2)p Fs(r)s(C)7 b Ft(\),)25 b(w)m(e)e(obtain)f(Prop)s(osition)f(3.1.)p 3554 2397 4 62 v 3558 2338 55 4 v 3558 2397 V 3612 2397 4 62 v 1955 5652 a(16)p eop %%Page: 17 18 17 17 bop 382 548 a Fu(5)109 b(In\015uence)36 b(of)h(the)g(b)s(ounda)m (ry)f(condition)h(on)f(multicanonical)h(exp)s(ectations)382 751 y Ft(In)26 b(this)f(section)i(w)m(e)g(study)f(ho)m(w)g(the)h(m)m (ulticanonical)e(exp)s(ectation)i Fs(\027)2822 718 y Fr(\034)2816 778 y Fq(A)p Fr(;M)2971 751 y Ft(\()p Fs(f)10 b Ft(\))26 b(of)h(a)g(function)382 864 y Fs(f)41 b Ft(on)32 b(\012)f(is)g(a\013ected)i(b)m(y)f(a)g(v)-5 b(ariation)32 b(of)g(the)g(b)s(oundary)d(condition)i Fs(\034)10 b Ft(.)45 b(More)33 b(precisely)d(w)m(e)382 977 y(w)m(an)m(t)h(to)g(\014nd)e(an)h (upp)s(er)f(b)s(ound)f(to)j(the)g(quan)m(tit)m(y)1575 1149 y Fk(j)p Fs(\027)1651 1111 y Fr(\034)1645 1171 y Fq(A)p Fr(;M)1800 1149 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)g Fs(\027)2087 1111 y Fr(\033)2081 1171 y Fq(A)p Fr(;M)2236 1149 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)15 b Fs(:)1007 b Ft(\(5.1\))382 1321 y(This)35 b(problem)h(has)h(b)s(een)f (studied,)i(in)e(a)h(particular)f(geometrical)i(setting,)h(in)d([CM00a) q(].)382 1434 y(F)-8 b(ollo)m(wing)24 b(a)h(similar)d(approac)m(h)j(w)m (e)g(are)g(going)g(to)g(sho)m(w)f(ho)m(w)h(to)g(deal)g(with)e(a)i(more) g(general)382 1547 y(geometry)-8 b(.)382 1682 y(Let)27 b(then)f(\(\003)p Fs(;)15 b(`;)g Fk(A)p Ft(\))28 b(b)s(e)e(a)h(p)s (olycub)s(e)e(and)h(let)h Fs(M)35 b Fk(2)25 b Fm(M)2261 1649 y Fq(A)2261 1709 y Fr(`)2354 1682 y Ft(a)i(p)s(ossible)d(c)m (hoice)k(for)e(the)h(n)m(um)m(b)s(er)382 1795 y(of)39 b(particles)f(in)f(eac)m(h)j(elemen)m(t)f(of)g Fk(A)p Ft(.)65 b(In)38 b(order)h(to)g(study)f(ho)m(w)h(the)g(quan)m(tit)m(y)g Fs(\027)3346 1762 y Fr(\034)3340 1822 y Fq(A)p Fr(;M)3495 1795 y Ft(\()p Fs(f)10 b Ft(\))382 1908 y(dep)s(ends)28 b(on)h Fs(\034)10 b Ft(,)30 b(w)m(e)g(\014rst)f(appro)m(ximate)g(this)g (m)m(ulticanonical)f(exp)s(ectation)i(with)e(a)i(grand-)382 2021 y(canonical)45 b(exp)s(ectation)h Fs(\026)1356 1988 y Fr(\034)1356 2048 y Fq(A)p Fr(;\025)1477 2021 y Ft(\()p Fs(f)10 b Ft(\))45 b(in)f(whic)m(h)g Fs(\025)i Ft(is)e(a)i(suitable)e (c)m(hemical)h(p)s(oten)m(tial)g(\(re-)382 2133 y(mem)m(b)s(er)37 b(\(2.4\)\))j(whic)m(h)c(w)m(e)j(assume)e(constan)m(t)i(in)d(eac)m(h)j (cub)s(e)e(of)h Fk(A)p Ft(.)63 b(The)37 b(v)-5 b(alue)37 b(of)h Fs(\025)g Ft(is)382 2246 y(determined)29 b(b)m(y)h(the)g (requiremen)m(t)f(that)i(the)f(exp)s(ectation)g(of)h(the)f(n)m(um)m(b)s (er)f(of)h(particles)f(in)382 2359 y(eac)m(h)40 b(cub)s(e)f(is)g(equal) g(to)h(the)f(n)m(um)m(b)s(er)f(of)i(particles)f(\014xed)f(b)m(y)i(the)f (m)m(ulticanonical)f(mea-)382 2472 y(sure.)57 b(In)35 b(other)h(w)m(ords)f(w)m(e)i(w)m(an)m(t)f Fs(\026)1677 2439 y Fr(\034)1677 2500 y Fq(A)p Fr(;\025)1798 2472 y Ft(\()p Fs(N)1906 2486 y Fr(V)1968 2472 y Ft(\))f(=)f Fs(M)2231 2486 y Fr(V)2328 2472 y Ft(for)h(all)g Fs(V)55 b Fk(2)34 b(A)p Ft(.)57 b(The)35 b(existence)i(of)382 2585 y(this)i Fl(tilting)i(\014eld)50 b Fs(\025)40 b Ft(is)e(pro)m(v)m(ed)j(in)d(the)i(app)s(endix)d(of)j([CM00a)r(].)69 b(Th)m(us)39 b(there)h(is)f(a)h(map)386 2674 y(^)382 2698 y Fs(\025)25 b Ft(:)h(\()p Fk(A)p Fs(;)15 b(M)5 b(;)15 b(\034)10 b Ft(\))26 b Fk(!)f Fs(\025)31 b Ft(suc)m(h)f(that) 1273 2870 y Fs(\026)1328 2832 y Fr(\034)1328 2911 y Fq(A)p Fr(;)1407 2893 y Fp(^)1404 2911 y Fr(\025)p Fp(\()p Fq(A)p Fr(;M)s(;\034)8 b Fp(\))1709 2870 y Ft(\()p Fs(N)1817 2884 y Fr(V)1879 2870 y Ft(\))25 b(=)g Fs(M)2123 2884 y Fr(V)2366 2870 y Fk(8)15 b Fs(V)45 b Fk(2)25 b(A)15 b Fs(:)704 b Ft(\(5.2\))382 3058 y(F)-8 b(or)31 b(brevit)m(y)f(w)m(e)h (de\014ne)1591 3162 y(~)-52 b Fs(\026)1639 3125 y Fr(\034)1639 3185 y Fq(A)p Fr(;M)1819 3162 y Ft(:=)25 b Fs(\026)1995 3125 y Fr(\034)1995 3203 y Fq(A)p Fr(;)2074 3186 y Fp(^)2071 3203 y Fr(\025)p Fp(\()p Fq(A)p Fr(;M)s(;\034)8 b Fp(\))2392 3162 y Fs(:)382 3416 y Fe(5.1)100 b(The)33 b(basic)g(estimate)382 3587 y Ft(The)d(idea)g(for)g(estimating)g(\(5.1\))i(is)d(to)i(write)479 3757 y Fk(j)p Fs(\027)555 3719 y Fr(\034)549 3779 y Fq(A)p Fr(;M)704 3757 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)g Fs(\027)991 3719 y Fr(\033)985 3779 y Fq(A)p Fr(;M)1140 3757 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)504 3904 y(\024)25 b(j)p Fs(\027)676 3867 y Fr(\034)670 3927 y Fq(A)p Fr(;M)825 3904 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)27 b Ft(~)-52 b Fs(\026)1116 3867 y Fr(\034)1116 3927 y Fq(A)p Fr(;M)1271 3904 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)20 b Ft(+)g Fk(j)7 b Ft(~)-52 b Fs(\026)1612 3867 y Fr(\034)1612 3927 y Fq(A)p Fr(;M)1767 3904 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)28 b Ft(~)-53 b Fs(\026)2058 3867 y Fr(\033)2058 3927 y Fq(A)p Fr(;M)2213 3904 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)21 b Ft(+)f Fk(j)7 b Ft(~)-52 b Fs(\026)2555 3867 y Fr(\033)2555 3927 y Fq(A)p Fr(;M)2710 3904 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)g Fs(\027)2997 3867 y Fr(\033)2991 3927 y Fq(A)p Fr(;M)3146 3904 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)15 b Fs(:)3433 3832 y Ft(\(5.3\))382 4070 y(The)36 b(\014rst)g(and)g(third)e(term)j(can)g(b)s(e)e(estimated)i(using)e (Prop)s(osition)g(5.1)i(b)s(elo)m(w,)h(a)f(result)382 4183 y(concerning)24 b(the)g(\\equiv)-5 b(alence)25 b(of)f(the)h(ensem) m(bles",)h(while)c(the)j(second)f(term)h(will)d(b)s(e)i(tak)m(en)382 4296 y(care)31 b(of)g(in)e(Prop)s(osition)f(5.2.)382 4471 y Ff(Prop)s(osition)34 b(5.1.)41 b Fl(Assume)31 b(\(USM\).)f(Ther)-5 b(e)32 b(exists)f Fs(C)h(>)25 b Ft(0)32 b Fl(such)f(that)h(for)g(al)5 b(l)31 b(p)-5 b(olycub)g(es)382 4584 y Ft(\(\003)p Fs(;)15 b(`;)g Fk(A)p Ft(\))p Fl(,)33 b(for)f(al)5 b(l)33 b Fs(M)i Fk(2)25 b Fm(M)1330 4551 y Fq(A)1330 4611 y Fr(`)1396 4584 y Fl(,)32 b(for)g(al)5 b(l)33 b(functions)f Fs(f)41 b Fl(on)32 b Ft(\012)g Fl(such)g(that)h Fk(j)p Fs(S)2909 4599 y Fr(f)2954 4584 y Fk(j)26 b(\024)f Fs(`)3139 4551 y Fr(d=)p Fp(2)3249 4584 y Fl(,)32 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Fr(\033)1589 875 y Fq(A)p Fr(;M)1744 853 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)26 b(\024)f Fs(C)c Fk(k)p Fs(f)10 b Fk(k)2247 867 y Fr(u)2308 853 y Fk(j)p Fs(S)2389 868 y Fr(f)2434 853 y Fk(j)15 b Ft(\()p Fs(\020)7 b(`)p Ft(\))2629 815 y Fp(2+)p Fr(d)p Fq(\000)p Fr(d)p Fp(\()p Fr(S)2916 827 y Fn(f)2957 815 y Fr(;W)j Fp(\))p Fr(=`)3433 853 y Ft(\(5.5\))382 1103 y Fl(wher)-5 b(e)34 b Fs(W)j Ft(:=)26 b Fk(f)p Fs(x)f Fk(2)g Fs(@)1145 1070 y Fp(+)1140 1126 y Fr(r)1204 1103 y Ft(\003)h(:)f Fs(\034)10 b Ft(\()p Fs(x)p Ft(\))26 b Fk(6)p Ft(=)f Fs(\033)s Ft(\()p Fs(x)p Ft(\))p Fk(g)p Fl(.)382 1286 y(Pr)-5 b(o)g(of.)43 b Ft(The)f(pro)s(of)f(of)i(this)e (statemen)m(t)j(requires)d(some)h(mo)s(di\014cations)e(of)j(the)f(pro)s (of)g(of)382 1399 y(Prop)s(osition)c(7.1)j(in)d([CM00a)r(],)k(where)d (a)i(di\013eren)m(t)e(geometry)i(is)e(considered.)67 b(W)-8 b(e)41 b(\014rst)382 1512 y(observ)m(e)31 b(that)g(w)m(e)g(can)f (assume)850 1694 y Fs(`)25 b Fk(\025)g Fs(\020)1056 1656 y Fq(\000)p Fp(1)1150 1694 y 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4309 y(whic)m(h)33 b(is)h(trivially)e(true)i(also)h(when)e Fs(i)g Fk(2)f Fs(I)7 b Ft(.)53 b(F)-8 b(or)35 b(the)g(upp)s(er)e(b)s(ound)f(w)m(e)j (observ)m(e)g(that,)i(if)382 4422 y Fs(x)25 b Fk(2)g Ft(\003)608 4436 y Fr(j)675 4422 y Ft(for)30 b(some)h Fs(j)g Fk(2)25 b Fs(I)7 b Ft(,)30 b(then)631 4608 y Fs(d)p Ft(\()p Fs(x;)15 b(W)891 4622 y Fp(0)931 4608 y Ft(\))26 b Fk(\024)f Fs(d)p Ft(\(\003)1233 4622 y Fr(j)1270 4608 y Fs(;)15 b(W)1396 4622 y Fp(0)1436 4608 y Ft(\))20 b(+)g(diam)14 b(\003)1857 4622 y Fr(j)1919 4608 y Ft(=)25 b Fs(d)p Ft(\(\003)2160 4622 y Fr(j)2198 4608 y Fs(;)15 b(W)2324 4622 y Fp(0)2363 4608 y Ft(\))21 b(+)f Fs(`)g Fk(\000)g Ft(1)25 b Fk(\024)g Fs(`=)p Ft(3)c(+)f(\()p Fs(`)h Fk(\000)f Ft(1\))15 b Fs(:)382 4795 y Ft(Th)m(us,)29 b(if)h(w)m(e)h(let)f(\003) 1049 4809 y Fr(I)1114 4795 y Ft(:=)25 b Fk([)1296 4809 y Fr(j)t Fq(2)p Fr(I)1415 4795 y Ft(\003)1478 4809 y Fr(j)1515 4795 y Ft(,)31 b(w)m(e)f(ha)m(v)m(e)i(that)f(\003)2174 4809 y Fr(I)2239 4795 y Fk(\032)25 b 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Fr(d)1968 5403 y Ft(\()p Fs(\020)7 b(`)p Ft(\))2123 5366 y Fr(\024)p Fp(\()p Fr(i;I)e Fp(\))2317 5403 y Fk(j)p Fs(\026)2397 5366 y Fr(\034)2397 5426 y Fq(A)p Fr(;\025)2514 5435 y Fi(1)2553 5403 y Ft(\()p Fs(N)2661 5417 y Fp(\003)2710 5427 y Fn(i)2741 5403 y Fs(;)15 b(h)p Ft(\))p Fk(j)g Fs(:)454 b Ft(\(5.33\))1955 5652 y(21)p eop %%Page: 22 23 22 22 bop 382 548 a Ft(If)30 b Fs(i)25 b Fk(2)g Fs(I)38 b Ft(w)m(e)30 b(ha)m(v)m(e,)i(since)e Fs(\026)1339 515 y Fr(\034)1339 576 y Fq(A)p Fr(;\025)1456 585 y Fi(1)1495 548 y Ft(\()p Fs(h)p Ft(\))c(=)f(1)826 744 y Fk(j)p Fs(\026)906 707 y Fr(\034)906 767 y Fq(A)p Fr(;\025)1023 776 y Fi(1)1062 744 y Ft(\()p Fs(N)1170 758 y Fp(\003)1219 768 y Fn(i)1250 744 y Fs(;)15 b(h)p Ft(\))p Fk(j)27 b(\024)e Fs(\026)1580 707 y Fr(\034)1580 767 y Fq(A)p Fr(;\025)1697 776 y Fi(1)1735 744 y Ft(\()p Fs(N)1843 758 y Fp(\003)1892 768 y Fn(i)1923 744 y Fs(h)p Ft(\))c(+)f Fs(\026)2177 707 y Fr(\034)2177 767 y Fq(A)p Fr(;\025)2294 776 y Fi(1)2333 744 y Ft(\()p Fs(N)2441 758 y Fp(\003)2490 768 y Fn(i)2521 744 y Ft(\))p Fs(\026)2611 707 y Fr(\034)2611 767 y Fq(A)p Fr(;\025)2728 776 y Fi(1)2767 744 y Ft(\()p Fs(h)p Ft(\))26 b Fk(\024)f Ft(2)p Fs(`)3094 707 y Fr(d)3150 744 y Fs(;)382 921 y Ft(so,)31 b(b)m(y)f(\(5.33\),)1378 1026 y Fk(j)p Fs(D)1481 987 y Fq(\000)p Fp(1)1478 1053 y Fr(i)1576 1026 y Fs(G)1647 1040 y Fr(i)1690 1026 y Fs( )1749 1040 y Fr(i)1778 1026 y Ft(\()p Fs(h)p Ft(\))p Fk(j)c(\024)f Ft(2)p Fs(A)183 b Fk(8)p Fs(i)25 b Fk(2)g Fs(I)d(:)764 b Ft(\(5.34\))382 1179 y(If)30 b(instead)f Fs(i)d Fk(62)f Fs(I)37 b Ft(\(and)30 b Fs(`)h Ft(is)e(large)h(enough\))h(w)m(e)g(can)f(use)h(\(2.10\))h(and) e(w)m(e)h(get)1324 1355 y Fk(j)p Fs(\026)1404 1318 y Fr(\034)1404 1378 y Fq(A)p Fr(;\025)1521 1387 y Fi(1)1560 1355 y Ft(\()p Fs(N)1668 1369 y Fp(\003)1717 1379 y Fn(i)1748 1355 y Fs(;)15 b(h)p Ft(\))p Fk(j)26 b(\024)f Fs(`)2060 1318 y Fr(d)2116 1355 y Fs(e)2158 1318 y Fq(\000)p Fr(md)p Fp(\(\003)2387 1328 y Fn(i)2414 1318 y Fr(;W)2501 1327 y Fi(0)2535 1318 y Fp(\))p Fr(=)p Fp(2)2653 1355 y Fs(;)382 1532 y Ft(whic)m(h,)k(together)j(with)d(\(5.31\))k(yields,)c(for)h Fs(`)g Ft(large)g(enough,)888 1709 y Fk(j)p Fs(D)991 1670 y Fq(\000)p Fp(1)988 1736 y Fr(i)1085 1709 y Fs(G)1156 1723 y Fr(i)1200 1709 y Fs( )1259 1723 y Fr(i)1287 1709 y Ft(\()p Fs(h)p Ft(\))p Fk(j)d(\024)e Fs(A)15 b Ft(\()p Fs(\020)7 b(`)p Ft(\))1795 1671 y Fr(\024)p Fp(\()p Fr(i;I)e Fp(\))1989 1709 y Fs(e)2031 1671 y Fq(\000)p Fr(m\024)p Fp(\()p Fr(i;I)g Fp(\))p Fr(`=)p Fp(6)2453 1709 y Fk(\024)25 b Fs(A)182 b Fk(8)p Fs(i)25 b Fk(2)g Fs(I)3039 1671 y Fr(c)3089 1709 y Fs(:)273 b Ft(\(5.35\))382 1885 y(F)-8 b(rom)31 b(\(5.34\),)i(\(5.35\),)g(w)m(e)d(\014nally)f(get)1578 2062 y Fk(j)p Fs(D)1681 2025 y Fq(\000)p Fp(1)1776 2062 y Fs(G)15 b( )s Ft(\()p Fs(h)p Ft(\))p Fk(j)2071 2076 y Fq(1)2173 2062 y Fk(\024)25 b Ft(2)p Fs(A)15 b(:)965 b Ft(\(5.36\))382 2285 y Fl(Estimate)30 b(of)g Fk(j)p Fs(G)961 2252 y Fq(\000)p Fp(1)1056 2285 y Fs(')p Ft(\()p Fs(f)10 b Ft(\))p Fk(j)1265 2299 y Fp(1)1335 2285 y Fl(in)30 b Ft(\(5)p Fs(:)p Ft(24\))i Fl(and)e(end)g(of)g(the)h(pr)-5 b(o)g(of)p Ft(.)41 b(In)27 b(order)g(to)h(pro)m(v)m(e)g(\(5.12\))h(w)m (e)382 2398 y(ha)m(v)m(e)i(to)h(b)s(ound)c(the)i(last)h(factor)g(in)e (\(5.24\),)k(namely)d Fk(j)p Fs(G)2378 2365 y Fq(\000)p Fp(1)2473 2398 y Fs(')p Ft(\()p Fs(f)10 b Ft(\))p Fk(j)2682 2412 y Fp(1)2722 2398 y Ft(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)491 2644 y Fk(j)p Fs(G)587 2606 y Fq(\000)p Fp(1)682 2644 y Fs(')p Ft(\()p Fs(f)10 b Ft(\))p Fk(j)891 2658 y Fp(1)957 2644 y Ft(=)1097 2530 y Fr(n)1053 2558 y Fj(X)1061 2753 y Fr(i)p Fp(=1)1209 2582 y Fk(j)p Fs(')1293 2596 y Fr(i)1322 2582 y Ft(\()p Fs(f)g Ft(\))p Fk(j)p 1209 2623 264 4 v 1291 2706 a Fs(G)1362 2720 y Fr(i)1507 2644 y Fk(\024)1648 2530 y Fr(n)1603 2558 y Fj(X)1612 2753 y Fr(i)p Fp(=1)1735 2644 y Ft(\()e(\026)-53 b Fs(\032)1817 2658 y Fr(i)1845 2644 y Ft(\))1880 2606 y Fq(\000)p Fp(1)1990 2644 y Ft(\()p Fs(\020)7 b(`)p Ft(\))2145 2606 y Fq(\000)p Fr(\024)p Fp(\()p Fr(i;I)e 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y(If)30 b Fs(i)g Ft(is)g(suc)m(h)g(that)h Fs(d)p Ft(\(\003)1173 3328 y Fr(i)1202 3314 y Fs(;)15 b(S)1298 3329 y Fr(f)1343 3314 y Ft(\))26 b Fk(\024)f Fs(`=)p Ft(3)31 b(w)m(e)g(use)f(inequalit)m(y)f(\(4\))i(of)g(Lemma)f (5.3)i(and)d(w)m(e)i(\014nd)1084 3482 y Fk(j)p Fs(')1168 3496 y Fr(i)1197 3482 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)p 1084 3523 264 4 v 1166 3606 a Fs(G)1237 3620 y Fr(i)1383 3544 y Fk(\024)25 b Fs(A)15 b Fk(k)p Fs(f)10 b Fk(k)1707 3558 y Fr(u)1767 3544 y Fs(`)1805 3506 y Fr(d)1861 3544 y Ft(\()p Fs(\020)d(`)p Ft(\))2016 3506 y Fp(4)p Fr(=)p Fp(3)p Fq(\000)p Fr(d)p Fp(\()p Fr(S)2282 3518 y Fn(f)2323 3506 y Fr(;W)2410 3515 y Fi(0)2444 3506 y Fp(\))p Fr(=`)p Fp(+)p Fr(d)p Fp(\()p Fr(S)2696 3518 y Fn(f)2737 3506 y Fr(;)p Fp(\003)2806 3516 y Fn(i)2832 3506 y Fp(\))p Fr(=`)1383 3740 y Fk(\024)25 b Fs(A)1547 3754 y Fp(1)1601 3740 y Fk(k)p Fs(f)10 b Fk(k)1746 3754 y Fr(u)1807 3740 y Ft(\()p Fs(\020)d(`)p Ft(\))1962 3703 y Fp(2+)p Fr(d)p Fq(\000)p Fr(d)p Fp(\()p Fr(S)2249 3715 y 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2227 y Fr(`)2887 2199 y Fl(,)33 b(we)f(have)980 2386 y Fk(j)p Fs(\027)1056 2348 y Fr(\034)1050 2408 y Fq(A)p Fr(;M)1205 2386 y Ft(\()p Fs(f)10 b Ft(\))20 b Fk(\000)g Fs(\027)1492 2348 y Fr(\033)1486 2408 y Fq(A)p Fr(;M)1641 2386 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)26 b(\024)f Fs(C)c Fk(k)p Fs(f)10 b Fk(k)2144 2400 y Fr(u)2205 2386 y Fk(j)p Fs(S)2286 2401 y Fr(f)2331 2386 y Fk(j)15 b(j)p Fs(I)2436 2401 y Fr(f)2482 2386 y Fk(j)g Fs(`)2560 2348 y Fq(\000)p Fr(d)2671 2386 y Ft(\(log)i Fs(`)p Ft(\))2912 2348 y Fp(3)p Fr(=)p Fp(2)3387 2386 y Ft(\(5.44\))382 2562 y Fl(wher)-5 b(e)34 b Fs(I)679 2577 y Fr(f)749 2562 y Ft(:=)25 b Fk(f)p Fs(V)46 b Fk(2)25 b(A)40 b Ft(:)h Fs(S)1335 2577 y Fr(f)1400 2562 y Fk(\\)20 b Fs(V)45 b Fk(6)p Ft(=)25 b Fk(;g)p Fl(.)382 2800 y Fe(5.2)100 b(Imp)m(roving)32 b(the)h(basic)g(estimate)382 3018 y Ft(Inequalit)m(y)i(\(5.44\))k(can)e(b)s(e)f(iterated)g(and)g(consequen) m(tly)h(impro)m(v)m(ed.)58 b(What)37 b(follo)m(ws)f(is)f(a)382 3131 y(generalization)30 b(of)g(the)h(strategy)h(adopted)e(in)f([CM00a) r(].)382 3309 y Ff(Prop)s(osition)47 b(5.6.)g Fl(Assume)41 b(\(USM\).)g(Ther)-5 b(e)42 b(exists)f Fs(C)48 b(>)40 b Ft(0)i Fl(such)f(that)i(the)f(fol)5 b(lowing)382 3422 y(holds:)47 b(let)34 b Ft(\(\003)p Fs(;)15 b(`;)g Fk(A)p Ft(\))35 b Fl(b)-5 b(e)34 b(a)h(p)-5 b(olycub)g(e,)35 b(and)h(let)e Fs(f)43 b Fl(b)-5 b(e)34 b(a)h(function)f(on)h Ft(\012)f Fl(such)g(that)i Fs(S)3357 3437 y Fr(f)3430 3422 y Fk(\032)28 b Ft(\003)p Fl(.)382 3535 y(Given)44 b(a)h(p)-5 b(air)46 b(of)f(b)-5 b(oundary)47 b(c)-5 b(onditions)46 b Fs(\034)5 b(;)15 b(\033)51 b Fk(2)46 b Ft(\012)p Fl(,)h(let)e Fs(W)60 b Ft(:=)47 b Fk(f)p Fs(x)g Fk(2)g Fs(@)3088 3502 y Fp(+)3083 3558 y Fr(r)3147 3535 y Ft(\003)g(:)g Fs(\034)10 b Ft(\()p Fs(x)p Ft(\))48 b Fk(6)p Ft(=)382 3648 y Fs(\033)s Ft(\()p Fs(x)p Ft(\))p Fk(g)p Fl(.)64 b(Assume)40 b(that)h(ther)-5 b(e)41 b(exists)f(an)g(incr)-5 b(e)g(asing)41 b(se)-5 b(quenc)g(e)39 b(of)h(p)-5 b(olycub)g(es)41 b Ft(\()p Fs(T)3279 3663 y Fr(k)3323 3648 y Fs(;)15 b(`;)g Fk(A)3514 3663 y Fr(k)3557 3648 y Ft(\))p Fl(,)382 3761 y Fs(k)28 b Ft(=)d(0)p Fs(;)15 b(:)g(:)g(:)i(;)e(n)p Fl(,)33 b(such)g(that)434 3909 y Ft(\()p Fs(i)p Ft(\))47 b Fs(S)638 3924 y Fr(f)708 3909 y Fk(\032)25 b Fs(T)857 3923 y Fp(0)922 3909 y Fk(\032)g Fs(T)1071 3923 y Fp(1)1136 3909 y Fk(\032)g(\001)15 b(\001)g(\001)26 b(\032)f Fs(T)1512 3923 y Fr(n)1584 3909 y Fk(\032)g Ft(\003)403 4035 y(\()p Fs(ii)p Ft(\))47 b Fs(d)p Ft(\()p Fs(T)717 4049 y Fr(n)765 4035 y Fs(;)15 b(W)e Ft(\))25 b Fs(>)g(r)372 4161 y Ft(\()p Fs(iii)p Ft(\))47 b Fs(d)p Ft(\(\003)p Fk(n)p Fs(T)825 4176 y Fr(k)869 4161 y Fs(;)15 b(T)962 4176 y Fr(k)r Fq(\000)p Fp(1)1095 4161 y Ft(\))26 b Fk(\025)f Ft(\(3)p Fs(d)c Ft(+)f(4\))p Fs(`)p Fl(.)382 4309 y(Then,)33 b(for)g(al)5 b(l)33 b Fs(M)i Fk(2)25 b Fm(M)1211 4276 y Fq(A)1211 4337 y Fr(`)1278 4309 y Fl(,)704 4560 y Fk(j)p Fs(\027)780 4522 y Fr(\034)774 4582 y Fq(A)p Fr(;M)929 4560 y Ft(\()p Fs(f)10 b Ft(\))21 b Fk(\000)f Fs(\027)1217 4522 y Fr(\033)1211 4582 y Fq(A)p Fr(;M)1365 4560 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)26 b(\024)f(k)p Fs(f)10 b Fk(k)1782 4574 y Fr(u)1842 4432 y Fj(\024)1890 4560 y Fs(C)1987 4498 y Ft(\(log)17 b Fs(`)p Ft(\))2228 4465 y Fp(3)p Fr(=)p Fp(2)p 1987 4539 352 4 v 2123 4622 a Fs(`)2161 4596 y Fr(d)2348 4432 y Fj(\025)2396 4454 y Fr(n)2501 4446 y(n)2464 4474 y Fj(Y)2458 4671 y Fr(k)r Fp(=1)2602 4560 y Fk(j)p Fs(@)2680 4522 y Fp(+)2675 4582 y Fr(r)2740 4560 y Fs(T)2793 4575 y Fr(k)2856 4560 y Fk(\\)i Ft(\003)p Fk(j)c Fs(:)323 b Ft(\(5.45\))382 4815 y Fl(Pr)-5 b(o)g(of.)32 b Ft(Let)1229 4920 y Fs(f)1274 4935 y Fr(k)1342 4920 y Ft(:=)25 b Fs(\027)1514 4882 y Fr(\034)1508 4942 y Fq(A)p Fr(;M)1663 4920 y Ft(\()p Fs(f)g Fk(j)15 b(F)1873 4938 y Fp(\003)p Fq(n)p Fr(T)1998 4950 y Fn(k)2041 4920 y Ft(\))182 b Fs(k)29 b Ft(=)c(0)p Fs(;)15 b(:)g(:)g(:)i(;)e(n)g(:)382 5073 y Ft(W)-8 b(e)38 b(denote)g(with)e Fs(M)1147 5088 y Fr(k)1226 5073 y Ft(the)i (restriction)e(of)h Fs(M)47 b Ft(to)38 b Fk(A)2268 5088 y Fr(k)2310 5073 y Ft(.)61 b(The)37 b(function)e Fs(f)2997 5088 y Fr(k)3076 5073 y Ft(is)i(measurable)382 5186 y(w.r.t.)k Fk(F)700 5216 y Fr(@)741 5186 y Fi(+)737 5226 y Fn(r)792 5216 y Fr(T)833 5228 y Fn(k)871 5216 y Fq(\\)p Fp(\003)971 5186 y Ft(.)g(Denoting)30 b(with)1647 5163 y(~)1636 5186 y(\012)g(the)h(set)g(of)f(all)f Fs(\021)g Fk(2)c Ft(\012)30 b(suc)m(h)g(that)h Fs(\021)2963 5200 y Fp(\003)3012 5181 y Fn(c)3073 5186 y Ft(=)25 b Fs(\034)3209 5200 y Fp(\003)3258 5181 y Fn(c)3294 5186 y Ft(,)31 b(w)m(e)f(can)382 5298 y(write)1391 5403 y Fs(f)1436 5418 y Fr(k)1478 5403 y Ft(\()p Fs(\021)s Ft(\))d(=)e Fs(\027)1770 5359 y Fr(\021)1764 5432 y Fq(A)1820 5444 y Fn(k)1858 5432 y Fr(;M)1946 5444 y Fn(k)1988 5403 y Ft(\()p Fs(f)10 b Ft(\))181 b Fk(8)p Fs(\021)29 b Fk(2)2515 5380 y Ft(~)2504 5403 y(\012)15 b Fs(:)1955 5652 y Ft(23)p eop %%Page: 24 25 24 24 bop 382 548 a Ft(Since,)30 b(b)m(y)g(h)m(yp)s(othesis,)f Fs(d)p Ft(\()p Fs(T)1379 563 y Fr(k)1422 548 y Fs(;)15 b(W)e Ft(\))26 b Fk(\025)f Fs(d)p Ft(\()p Fs(T)1853 562 y Fr(n)1901 548 y Fs(;)15 b(W)e Ft(\))25 b Fs(>)g(r)s Ft(,)30 b(w)m(e)h(ha)m(v)m(e)537 723 y Fs(\027)588 685 y Fr(\033)582 745 y Fq(A)p Fr(;M)737 723 y Ft(\()p Fs(f)25 b Fk(j)15 b(F)947 741 y Fp(\003)p Fq(n)p Fr(T)1072 753 y Fn(k)1115 723 y Ft(\)\()p Fs(\021)s Ft(\))27 b(=)e Fs(\027)1442 685 y Fr(\034)1436 745 y Fq(A)p Fr(;M)1591 723 y Ft(\()p Fs(f)g Fk(j)15 b(F)1801 741 y Fp(\003)p Fq(n)p Fr(T)1926 753 y Fn(k)1969 723 y Ft(\)\()p Fs(\021)s Ft(\))27 b(=)e Fs(\027)2296 679 y Fr(\021)2290 751 y Fq(A)2346 763 y Fn(k)2384 751 y Fr(;M)2472 763 y Fn(k)2514 723 y Ft(\()p Fs(f)10 b Ft(\))25 b(=)g Fs(f)2805 738 y Fr(k)2847 723 y Ft(\()p Fs(\021)s Ft(\))183 b Fk(8)p Fs(\021)28 b Fk(2)3368 700 y Ft(~)3358 723 y(\012)15 b Fs(;)382 898 y Ft(By)31 b(consequence)g(\(remem)m(b)s(er)f(that)h (osc\()p Fs(f)10 b Ft(\))26 b(:=)f(sup)14 b Fs(f)29 b Fk(\000)20 b Ft(inf)h Fs(f)10 b Ft(\))869 1073 y Fk(j)p Fs(\027)945 1035 y Fr(\034)939 1095 y Fq(A)p Fr(;M)1094 1073 y Ft(\()p Fs(f)g Ft(\))20 b Fk(\000)g Fs(\027)1381 1035 y Fr(\033)1375 1095 y Fq(A)p Fr(;M)1530 1073 y Ft(\()p Fs(f)10 b Ft(\))p Fk(j)26 b Ft(=)e Fk(j)p Fs(\027)1877 1035 y Fr(\034)1871 1095 y Fq(A)p Fr(;M)2027 1073 y Ft(\()p Fs(f)2107 1087 y Fr(n)2154 1073 y Ft(\))c Fk(\000)g Fs(\027)2351 1035 y Fr(\033)2345 1095 y Fq(A)p Fr(;M)2500 1073 y Ft(\()p Fs(f)2580 1087 y Fr(n)2627 1073 y Ft(\))p Fk(j)26 b(\024)f Ft(osc\()p Fs(f)3010 1087 y Fr(n)3057 1073 y Ft(\))15 b Fs(:)255 b Ft(\(5.46\))382 1248 y(Since)29 b Fs(f)664 1263 y Fr(k)737 1248 y Ft(is)g(measurable)h(w.r.t.)40 b Fk(F)1621 1278 y Fr(@)1662 1248 y Fi(+)1658 1288 y Fn(r)1714 1278 y Fr(T)1755 1290 y Fn(k)1793 1278 y Fq(\\)p Fp(\003)1062 1438 y Ft(osc)q(\()p Fs(f)1264 1453 y Fr(k)1306 1438 y Ft(\))26 b Fk(\024)f(j)p Fs(@)1541 1400 y Fp(+)1536 1460 y Fr(r)1600 1438 y Fs(T)1653 1453 y Fr(k)1716 1438 y Fk(\\)20 b Ft(\003)p Fk(j)139 b Ft(sup)1916 1532 y Fr(x)p Fq(2)p Fr(@)2044 1502 y Fi(+)2040 1543 y Fn(r)2095 1532 y Fr(T)2136 1544 y Fn(k)2174 1532 y Fq(\\)p Fp(\003)2300 1438 y Ft(sup)2301 1532 y Fr(\021)r Fq(2)2393 1515 y Fp(~)2385 1532 y(\012)2452 1438 y Fk(jr)2553 1452 y Fr(x)2597 1438 y Ft(\()p Fs(f)2677 1453 y Fr(k)2719 1438 y Ft(\)\()p Fs(\021)s Ft(\))p Fk(j)15 b Fs(:)450 b Ft(\(5.47\))382 1687 y(On)29 b(the)i(other)g(side,)e(if)h(w)m(e)g(let)1469 1872 y Fs(h)1521 1835 y Fr(x)1521 1895 y(k)1565 1872 y Ft(\()p Fs(\021)s Ft(\))c(:=)1964 1811 y Fs(e)2006 1778 y Fq(\000r)2120 1786 y Fn(x)2159 1778 y Fr(H)2217 1789 y Fi(\003)2262 1778 y Fp(\()p Fr(\021)r Fp(\))p 1840 1851 643 4 v 1840 1935 a Fs(\027)1891 1891 y Fr(\021)1885 1964 y Fq(A)1941 1976 y Fn(k)1980 1964 y Fr(;M)2068 1976 y Fn(k)2109 1935 y Ft(\()p Fs(e)2186 1909 y Fq(\000r)2300 1917 y Fn(x)2340 1909 y Fr(H)2398 1920 y Fi(\003)2447 1935 y Ft(\))2508 1872 y Fs(;)382 2107 y Ft(w)m(e)31 b(ha)m(v)m(e,)h(for)e Fs(x)25 b Fk(2)g Fs(@)1106 2074 y Fp(+)1101 2129 y Fr(r)1165 2107 y Fs(T)1218 2122 y Fr(k)1281 2107 y Fk(\\)20 b Ft(\003)30 b(and)g Fs(\021)f Fk(2)1802 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Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Fs(\026)p Ft([)15 b Fs(G)2227 4464 y Fq(A)2283 4473 y Fi(0)2323 4450 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])855 b(\(6.1\))382 4614 y(and,)33 b(in)f(particular,)g Fs(\026)p Ft([)15 b Fs(G)1310 4628 y Fq(A)1371 4614 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])29 b Fk(\024)g Fs(\026)p Ft([)15 b Fs(G)1921 4628 y Fr(B)1974 4639 y Fn(L)2026 4614 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(].)48 b(On)32 b(the)h(other)g(side)f(the)h(canonical)382 4727 y(measure)d(satis\014es)g(\(see)h([L)-8 b(Y93)r(])31 b(and)e([CM00b)q(]\))i(a)g(P)m(oincar)m(\023)-43 b(e)31 b(inequalit)m(y)e(whic)m(h)g(sa)m(ys)1329 4892 y Fs(\027)1380 4854 y Fr(\034)1374 4914 y(B)1427 4925 y Fn(L)1473 4914 y Fr(;N)1560 4892 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))25 b Fk(\024)g Fs(C)1961 4906 y Fp(0)2016 4892 y Fs(L)2078 4854 y Fp(2)2132 4892 y Fk(E)2180 4906 y Fr(\027)2219 4883 y Fn(\034)2215 4929 y(B)2261 4946 y(L)2307 4929 y(;N)2384 4906 y Fr(;B)2457 4917 y 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y(\(1\))47 b(\003)c(is)e(a)i(rectangle,)k(\003)e(=)1534 1610 y Fj(\000)1591 1684 y Ft([)p Fs(a)1664 1698 y Fp(1)1703 1684 y Fs(;)15 b(b)1782 1698 y Fp(1)1822 1684 y Ft(\))29 b Fk(\002)f(\001)15 b(\001)g(\001)29 b(\002)f Ft([)p Fs(a)2291 1699 y Fr(d)2332 1684 y Fs(;)15 b(b)2411 1699 y Fr(d)2452 1684 y Ft(\))2502 1610 y Fj(\001)2572 1684 y Fk(\\)28 b Fm(Z)2726 1651 y Fr(d)2805 1684 y Ft(and)42 b Fs(a)3042 1698 y Fr(i)3070 1684 y Fs(;)15 b(b)3149 1698 y Fr(i)3223 1684 y Fk(2)45 b Fs(`)p Fm(Z)3432 1651 y Fr(d)3511 1684 y Ft(for)582 1797 y Fs(i)25 b Ft(=)g(1)p Fs(;)15 b(:)g(:)g(:)j(;)d(n)420 1922 y Ft(\(2\))47 b(\003)25 b Fk(\032)766 1848 y Fj(\000)823 1922 y Ft([0)p Fs(;)15 b(w)998 1937 y Fr(k)r Fp(+1)1132 1922 y Ft(\))h Fk(\002)g(\001)f(\001)g (\001)i(\002)e Ft([0)p Fs(;)g(w)1653 1937 y Fr(k)r Fp(+)p Fr(d)1788 1922 y Ft(\))1838 1848 y Fj(\001)1896 1922 y Fk(\\)g Fm(Z)2037 1889 y Fr(d)2102 1922 y Ft(mo)s(dulo)26 b(translations)h(and)h(p)s(erm)m(utations)582 2035 y(of)i(the)h(co)s 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b(Consider)35 b(the)i(m)m (ulticanonical)e(measure)i Fs(\027)3022 3729 y Fr(\034)3016 3789 y Fq(A)p Fr(;M)3171 3762 y Ft(,)i(let)e Fs(M)3461 3776 y Fr(i)3526 3762 y Ft(b)s(e)382 3875 y(the)30 b(restriction)g(of)g Fs(M)41 b Ft(to)31 b Fk(B)1377 3889 y Fr(i)1435 3875 y Ft(and)f(de\014ne)1106 4026 y(~)1096 4049 y(\012)1162 4063 y Fp(\003)1211 4044 y Fn(c)1243 4063 y Fr(;\034)1331 4049 y Ft(:=)25 b Fk(f)p Fs(\021)k Fk(2)c Ft(\012)g(:)g Fs(\021)s Ft(\()p Fs(x)p Ft(\))i(=)d Fs(\034)10 b Ft(\()p Fs(x)p Ft(\))31 b(for)g(all)e Fs(x)c Fk(2)g Ft(\003)2785 4011 y Fr(c)2820 4049 y Fk(g)15 b Fs(:)382 4222 y Ft(Then)29 b(for)h(eac)m(h)i(lo)s(cal)e(function)f Fs(g)34 b Ft(measurable)29 b(w.r.t.)41 b Fk(F)2404 4241 y Fp(\003)p Fq(n)p Fp(\003)2537 4250 y Fi(2)2606 4222 y Ft(w)m(e)31 b(ha)m(v)m(e)656 4422 y Fk(k)p Fs(\027)752 4385 y Fr(\034)746 4445 y Fq(A)p Fr(;M)902 4422 y Ft(\()p Fs(g)s Ft(\))21 b Fk(\000)f Fs(\027)1181 4385 y Fr(\034)1175 4445 y Fq(A)p Fr(;M)1330 4422 y Ft(\()p Fs(g)f Fk(j)c(F)1532 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Ft(\()p Fs(g)s Ft(\))c Fk(\000)f Fs(\027)1245 4884 y Fr(\034)1239 4944 y Fq(A)p Fr(;M)1394 4922 y Ft(\()p Fs(g)f Fk(j)c(F)1596 4940 y Fp(\003)p Fq(n)p Fp(\003)1729 4949 y Fi(1)1769 4922 y Ft(\))p Fk(k)1849 4936 y Fr(u)1920 4922 y Fk(\024)25 b(k)p Fs(g)s Fk(k)2152 4936 y Fr(u)2213 4922 y Fs(C)2310 4860 y Ft(\(log)17 b Fs(`)p Ft(\))2551 4827 y Fp(3)p Fr(=)p Fp(2)p 2310 4901 352 4 v 2446 4984 a Fs(`)2484 4958 y Fr(d)2686 4922 y Fs(L)2748 4884 y Fr(d)p Fq(\000)p Fp(1)2904 4922 y Ft(=:)25 b Fs(\013)16 b Fk(k)p Fs(g)s Fk(k)3235 4936 y Fr(u)382 5116 y Ft(where)26 b(the)h(last)f(equalit)m (y)h(represen)m(ts)f(a)h(de\014nition)e(of)h Fs(\013)p Ft(.)40 b(W)-8 b(e)28 b(can)f(th)m(us)g(apply)e(Lemma)i(3.1)382 5229 y(in)i([BCC02)q(])h(and,)g(letting)1483 5403 y(1)21 b(+)f Fs(\013)1698 5417 y Fp(1)1763 5403 y Ft(:=)25 b(\(1)c Fk(\000)f Ft(2)p Fs(\013)h Fk(\000)f Fs(\013)2349 5366 y Fp(2)2389 5403 y Ft(\))2424 5366 y Fq(\000)p Fp(1)1955 5652 y Ft(26)p eop %%Page: 27 28 27 27 bop 382 548 a Ft(w)m(e)31 b(obtain,)538 708 y Fs(\027)589 671 y Fr(\034)583 731 y Fq(A)p Fr(;M)738 708 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))25 b Fk(\024)g Ft(\(1)c(+)f Fs(\013)1324 722 y Fp(1)1364 708 y Ft(\))15 b Fs(\027)1465 671 y Fr(\034)1459 731 y Fq(A)p Fr(;M)1629 708 y Ft([)p Fs(\027)1705 671 y Fr(\034)1699 731 y Fq(A)p Fr(;M)1854 708 y Ft(\()p Fs(f)5 b(;)15 b(f)40 b Fk(j)15 b(F)2169 727 y Fp(\003)p Fq(n)p Fp(\003)2302 736 y Fi(1)2342 708 y Ft(\))20 b(+)g Fs(\027)2539 671 y Fr(\034)2533 731 y Fq(A)p Fr(;M)2688 708 y Ft(\()p Fs(f)5 b(;)15 b(f)40 b Fk(j)15 b(F)3003 727 y Fp(\003)p Fq(n)p Fp(\003)3136 736 y Fi(2)3175 708 y Ft(\)])g Fs(:)158 b Ft(\(6.7\))382 869 y(After)31 b(taking)g(the)h(exp)s(ectation)f(of)g(this)f(inequalit) m(y)g(w.r.t.)43 b Fs(\026)p Ft(,)31 b(recalling)f(the)h(de\014nition)e (of)382 981 y(\010)448 996 y Fr(k)521 981 y Ft(\(6.6\),)j(w)m(e)f(get) 1017 1142 y Fs(\026)p Ft([)15 b Fs(G)1183 1156 y Fq(B)1236 1142 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Ft(\(1)c(+)f Fs(\013)1862 1156 y Fp(1)1901 1142 y Ft(\))15 b(\010)2017 1157 y Fr(k)r Fq(\000)p Fp(1)2165 1142 y Ft([)p Fk(E)2238 1156 y Fp(\003)2292 1142 y Ft(\()p Fs(f)10 b Ft(\))20 b(+)g Fk(E)2576 1156 y Fp(\003)2625 1165 y Fi(1)2659 1156 y Fq(\\)p Fp(\003)2755 1165 y Fi(2)2794 1142 y Ft(\()p Fs(f)10 b Ft(\)])15 b Fs(:)449 b Ft(\(6.8\))382 1302 y(If,)40 b(at)e(this)f(p)s(oin)m(t,)j(w)m(e)e(estimate)h(the)f(o)m (v)m(erlap)h(term)f Fk(E)2358 1316 y Fp(\003)2407 1325 y Fi(1)2441 1316 y Fq(\\)p Fp(\003)2537 1325 y Fi(2)2576 1302 y Ft(\()p Fs(f)10 b Ft(\))38 b(simply)d(with)i Fk(E)3300 1316 y Fp(\003)3353 1302 y Ft(\()p Fs(f)10 b Ft(\))38 b(w)m(e)382 1415 y(run)25 b(in)m(to)h(troubles,)g(since)g(w)m(e)g(w)m (ould)f(\014nd)g(the)h(iterativ)m(e)h(inequalit)m(y)e(\010)2874 1430 y Fr(k)2941 1415 y Fk(\024)g Ft(2)15 b(\(1)d(+)g Fs(\013)3330 1429 y Fp(1)3371 1415 y Ft(\))j(\010)3487 1430 y Fr(k)r Fq(\000)p Fp(1)382 1528 y Ft(whic)m(h)28 b(do)s(es)g(not)h(lo)s(ok)g(v)m(ery)g(promising.)38 b(The)28 b(idea)h([Mar99)q(])g(is)f(to)i(write)e(sev)m(eral)h(di\013eren)m(t)382 1641 y(\\copies")35 b(of)f(inequalit)m(y)e(\(6.8\),)37 b(where)c(eac)m(h)i(cop)m(y)g(corresp)s(onds)d(to)j(a)f(di\013eren)m(t) g(c)m(hoice)h(of)382 1754 y(the)30 b(subsets)g(\003)916 1768 y Fp(1)956 1754 y Ft(,)g(\003)1074 1768 y Fp(2)1114 1754 y Ft(.)40 b(F)-8 b(or)32 b(this)d(purp)s(ose)g(w)m(e)h(need)g(the) h(follo)m(wing)e(result:)382 1919 y Ff(Lemma)42 b(6.1.)k Fl(F)-7 b(or)41 b(al)5 b(l)40 b Fs(k)i Fk(2)c Fm(Z)1537 1933 y Fp(+)1591 1919 y Fl(,)k(for)e(al)5 b(l)40 b Ft(\003)f Fk(2)f(R)2228 1934 y Fr(k)2270 1919 y Fk(nR)2392 1934 y Fr(k)r Fq(\000)p Fp(1)2565 1919 y Fl(ther)-5 b(e)41 b(esists)f(a)g(c)-5 b(ol)5 b(le)-5 b(ction)41 b(of)382 2047 y(p)-5 b(olycub)g(es)39 b Ft(\(\003)880 1999 y Fp(\()p Fr(i)p Fp(\))880 2059 y Fr(n)964 2047 y Fs(;)15 b(`;)g Fk(B)1145 1999 y Fp(\()p Fr(i)p Fp(\))1142 2059 y Fr(n)1228 2047 y Ft(\))39 b Fl(wher)-5 b(e)39 b Fs(n)c Ft(=)g(1)p Fs(;)15 b Ft(2)40 b Fl(and)f Fs(i)d Ft(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(s)2575 2062 y Fr(k)2653 2047 y Ft(:=)35 b Fk(b)p Ft(\(3)p Fs(=)p Ft(2\))3029 2014 y Fr(k)r(=d)3146 2047 y Fk(c)p Fl(,)40 b(such)e(that)382 2160 y(for)33 b(al)5 b(l)33 b Fs(i;)15 b(j)32 b Ft(=)25 b(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(s)1183 2175 y Fr(k)1258 2160 y Fl(we)33 b(have)420 2317 y Ft(\(1\))47 b(\003)25 b(=)g(\003)829 2269 y Fp(\()p Fr(i)p Fp(\))829 2343 y(1)933 2317 y Fk([)19 b Ft(\003)1076 2269 y Fp(\()p Fr(i)p Fp(\))1076 2343 y(2)1192 2317 y Fl(and)34 b Ft(\003)1432 2269 y Fp(\()p Fr(i)p Fp(\))1432 2329 y Fr(n)1540 2317 y Fk(2)25 b(R)1703 2332 y Fr(k)r Fq(\000)p Fp(1)1868 2317 y Fl(for)34 b(al)5 b(l)33 b Fs(n)25 b Ft(=)g(1)p Fs(;)15 b Ft(2)420 2456 y(\(2\))47 b Fs(d)p Ft(\(\003)p Fk(n)p Ft(\003)835 2408 y Fp(\()p Fr(i)p Fp(\))835 2482 y(1)919 2456 y Fs(;)15 b Ft(\003)p Fk(n)p Ft(\003)1130 2408 y Fp(\()p Fr(i)p Fp(\))1130 2482 y(2)1215 2456 y Ft(\))25 b Fk(\025)g Fs(\016)420 2596 y Ft(\(3\))47 b Fl(If)32 b Fs(i)26 b Fk(6)p Ft(=)f Fs(j)38 b Fl(then)33 b Ft(\003)1170 2548 y Fp(\()p Fr(i)p Fp(\))1170 2622 y(1)1274 2596 y Fk(\\)19 b Ft(\003)1417 2548 y Fp(\()p Fr(i)p Fp(\))1417 2622 y(2)1521 2596 y Fk(\\)h Ft(\003)1665 2548 y Fp(\()p Fr(j)t Fp(\))1665 2622 y(1)1776 2596 y Fk(\\)g Ft(\003)1920 2548 y Fp(\()p Fr(j)t Fp(\))1920 2622 y(2)2037 2596 y Ft(=)25 b Fk(;)p Fl(.)382 2761 y(Pr)-5 b(o)g(of.)42 b Ft(Since)d(\003)k Fk(2)e(R)1194 2776 y Fr(k)1278 2761 y Ft(w)m(e)g(can)f(assume)h(that)g(\003)h(=)2321 2687 y Fj(\000)2378 2761 y Ft([0)p Fs(;)15 b(b)2527 2775 y Fp(1)2567 2761 y Ft(\))28 b Fk(\002)e(\001)15 b(\001)g(\001)28 b(\002)f Ft([0)p Fs(;)15 b(b)3107 2776 y Fr(d)3149 2761 y Ft(\))3199 2687 y Fj(\001)3241 2761 y Fk(\\)p Fm(Z)3367 2728 y Fr(d)3443 2761 y Ft(with)382 2874 y Fs(b)421 2888 y Fr(j)483 2874 y Fk(\024)25 b Fs(w)644 2889 y Fr(k)r Fp(+)p Fr(j)804 2874 y Ft(for)30 b Fs(j)h Ft(=)25 b(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(d)p Ft(.)41 b(De\014ne)645 3082 y(\003)708 3034 y Fp(\()p Fr(i)p Fp(\))708 3108 y(1)817 3082 y Ft(:=)938 2981 y Fj(\020)992 3082 y Ft([0)p Fs(;)15 b(b)1141 3096 y Fp(1)1182 3082 y Ft(\))20 b Fk(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Ft([0)p Fs(;)15 b(b)1694 3097 y Fr(d)p Fq(\000)p Fp(1)1826 3082 y Ft(\))21 b Fk(\002)1972 2981 y Fj(h)2015 3082 y Ft(0)15 b Fs(;)2115 2981 y Fj(j)2176 3020 y Fs(b)2215 3035 y Fr(d)p 2174 3061 84 4 v 2174 3144 a Ft(2)p Fs(`)2268 2981 y Fj(k)2316 3082 y Fs(`)20 b Ft(+)g Fs(i\016)2539 2981 y Fj(\021)c(\021)2684 3082 y Fk(\\)j Ft(\003)606 b(\(6.9\))645 3302 y(\003)708 3254 y Fp(\()p Fr(i)p Fp(\))708 3328 y(2)817 3302 y Ft(:=)938 3201 y Fj(\020)992 3302 y Ft([0)p Fs(;)15 b(b)1141 3316 y Fp(1)1182 3302 y Ft(\))20 b Fk(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Ft([0)p Fs(;)15 b(b)1694 3317 y Fr(d)p Fq(\000)p Fp(1)1826 3302 y Ft(\))21 b Fk(\002)1972 3201 y Fj(hj)2075 3240 y Fs(b)2114 3255 y Fr(d)p 2073 3281 V 2073 3364 a Ft(2)p Fs(`)2167 3201 y Fj(k)2215 3302 y Fs(`)f Ft(+)g(\()p Fs(i)g Fk(\000)g Ft(1\))p Fs(\016)n(;)33 b(b)2756 3317 y Fr(d)2796 3201 y Fj(\021)15 b(\021)2940 3302 y Fk(\\)20 b Ft(\003)15 b Fs(:)263 b Ft(\(6.10\))382 3518 y(It)27 b(is)f(straigh)m(tforw)m(ard)h(to)h(c)m(hec)m(k)g(that)g(\003)1808 3470 y Fp(\()p Fr(i)p Fp(\))1808 3544 y(1)1918 3518 y Ft(and)e(\003)2154 3470 y Fp(\()p Fr(i)p Fp(\))2154 3544 y(1)2264 3518 y Ft(satisfy)h(the)g(required)e(prop)s(erties)h(\(for)382 3631 y(more)k(details)g(see)h([BCC02)q(]\).)p 3554 3631 4 62 v 3558 3572 55 4 v 3558 3631 V 3612 3631 4 62 v 382 3784 a(F)-8 b(rom)31 b(Lemma)f(6.1)i(and)e(inequalit)m(y)e(\(6.8\)) k(w)m(e)f(get)527 3944 y Fs(\026)p Ft([)15 b Fs(G)693 3958 y Fq(B)746 3944 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Ft(\(1)c(+)f Fs(\013)1372 3958 y Fp(1)1412 3944 y Ft(\))15 b(\010)1528 3959 y Fr(k)r Fq(\000)p Fp(1)1676 3944 y Ft([)p Fk(E)1749 3958 y Fp(\003)1802 3944 y Ft(\()p Fs(f)10 b Ft(\))21 b(+)e Fk(E)2086 3986 y Fp(\003)2135 3950 y Fi(\()p Fn(i)p Fi(\))2135 4007 y(1)2210 3986 y Fq(\\)p Fp(\003)2306 3950 y Fi(\()p Fn(i)p Fi(\))2306 4007 y(2)2384 3944 y Ft(\()p Fs(f)10 b Ft(\)])182 b Fs(i)26 b Ft(=)f(1)p Fs(;)15 b(:)g(:)g(:)i(;)e(s)3159 3959 y Fr(k)3217 3944 y Fs(:)145 b Ft(\(6.11\))382 4134 y(Thanks)26 b(to)j(\(3\))f(of)g(Lemma)f(6.1)i(w)m(e)f(ha)m(v)m(e)h (that)2055 4066 y Fj(P)2151 4091 y Fr(s)2184 4103 y Fn(k)2151 4161 y Fr(i)p Fp(=1)2284 4134 y Fk(E)2332 4176 y Fp(\003)2381 4140 y Fi(\()p Fn(i)p Fi(\))2381 4197 y(1)2455 4176 y Fq(\\)p Fp(\003)2551 4140 y Fi(\()p Fn(i)p Fi(\))2551 4197 y(2)2630 4134 y Ft(\()p Fs(f)10 b Ft(\))25 b Fk(\024)g(E)2924 4148 y Fp(\003)2977 4134 y Ft(\()p Fs(f)10 b Ft(\),)28 b(th)m(us)f(w)m(e)h(can)382 4273 y(a)m(v)m(erge)k(\(6.11\))h(o)m(v)m (er)f Fs(i)e Ft(and)g(w)m(e)h(\014nd)1387 4434 y(\010)1453 4449 y Fr(k)1521 4434 y Fk(\024)25 b Ft(\010)1683 4449 y Fr(k)r Fq(\000)p Fp(1)1830 4434 y Ft(\(1)c(+)f Fs(\013)2080 4448 y Fp(1)2120 4434 y Ft(\))15 b(\(1)22 b(+)d(1)p Fs(=s)2495 4449 y Fr(k)2539 4434 y Ft(\))c Fs(:)773 b Ft(\(6.12\))382 4594 y(F)-8 b(rom)34 b(our)e(assumption)g(on)h Fs(L)p Ft(,)g(it)g(follo)m(ws)f(that)i Fs(B)2185 4608 y Fr(L)2267 4594 y Fk(\032)29 b(R)2444 4609 y Fr(k)2481 4618 y Fi(1)2553 4594 y Ft(with)j(\(sa)m(y\))i Fs(k)3040 4608 y Fp(1)3110 4594 y Ft(:=)29 b Fk(b)p Ft(3)p Fs(d)15 b Ft(log)j Fs(`)p Fk(c)p Ft(,)382 4707 y(since)1324 4811 y Fs(w)1389 4826 y Fr(k)1426 4835 y Fi(1)1489 4811 y Fk(\025)25 b Fs(\016)19 b Ft(\(3)p Fs(=)p Ft(2\))1849 4774 y Fp(3)c(log)d Fr(`)2060 4811 y Fk(\025)25 b Fs(\016)19 b(`)25 b Fk(\025)g Ft(2)p Fs(L)20 b Ft(+)g(1)15 b Fs(:)382 4956 y Ft(By)28 b(consequence)h(w)m(e) f(can)g(iterate)h(\(6)p Fs(:)p Ft(12\))h(up)d(to)h Fs(k)h Ft(=)c Fs(k)2330 4970 y Fp(1)2397 4956 y Ft(and)i(obtain)h(an)f(upp)s (er)f(b)s(ound)g(for)382 5069 y(the)k(P)m(oincar)m(\023)-43 b(e)32 b(constan)m(t)g(of)e(the)h(p)s(olycub)s(e)d(\()p Fs(B)2028 5083 y Fr(L)2081 5069 y Fs(;)15 b(`;)g Fk(A)p Ft(\))31 b(as)560 5318 y(\010\()p Fk(A)p Ft(\))26 b Fk(\024)f Ft(\010)957 5333 y Fr(k)994 5342 y Fi(1)1057 5318 y Fk(\024)g Ft(\010)1219 5332 y Fp(0)1311 5203 y Fr(k)1348 5212 y Fi(1)1288 5231 y Fj(Y)1289 5427 y Fr(i)p Fp(=1)1404 5318 y Ft(\(1)c(+)f Fs(\013)1654 5332 y Fp(1)1694 5318 y Ft(\))1744 5189 y Fj(\022)1811 5318 y Ft(1)h(+)1998 5256 y(1)p 1978 5297 86 4 v 1978 5380 a Fs(s)2021 5395 y Fr(k)2073 5189 y Fj(\023)2165 5318 y Fk(\024)k Ft(\010)2327 5332 y Fp(0)2397 5318 y Ft(exp)2536 5217 y Fj(h)2578 5318 y Fs(k)2625 5332 y Fp(1)2665 5318 y Fs(\013)2723 5332 y Fp(1)2783 5318 y Ft(+)2904 5204 y Fq(1)2874 5231 y Fj(X)2878 5427 y Fr(j)t Fp(=1)3050 5256 y Ft(1)p 3030 5297 V 3030 5380 a Fs(s)3073 5395 y Fr(k)3126 5217 y Fj(i)3184 5318 y Fs(:)178 b Ft(\(6.13\))1955 5652 y(27)p eop %%Page: 28 29 28 28 bop 382 548 a Ft(Since)31 b Fs(\015)i(<)27 b Ft(\()p Fs(d)22 b Fk(\000)f Ft(1\))1075 515 y Fq(\000)p Fp(1)1202 548 y Ft(from)32 b(the)g(de\014nition)d(of)j Fs(\013)h Ft(if)e(follo)m(ws)g(that)h(there)g(exists)g Fs(`)3286 562 y Fp(0)3325 548 y Ft(\()p Fs(\015)5 b Ft(\))29 b Fs(>)e Ft(0)382 661 y(suc)m(h)g(that)h(if)e Fs(`)f Fk(\025)g Fs(`)1055 675 y Fp(0)1094 661 y Ft(\()p Fs(\015)5 b Ft(\))28 b(then)f Fs(\013)g Ft(is)g(b)s(ounded)e(b)m(y)i(a)g(negativ)m(e)i(p)s (o)m(w)m(er)e(of)g Fs(`)p Ft(,)h(hence)f Fs(k)3290 675 y Fp(1)3330 661 y Fs(\013)3388 675 y Fp(1)3453 661 y Fk(\024)e Ft(1.)382 774 y(On)41 b(the)h(other)h(side)d(there)j(exists)e Fs(K)7 b Ft(\()p Fs(d)p Ft(\))43 b(suc)m(h)f(that)2332 706 y Fj(P)2428 732 y Fq(1)2428 801 y Fr(j)t Fp(=1)2570 774 y Fs(s)2613 736 y Fq(\000)p Fp(1)2613 803 y Fr(k)2752 774 y Fs(<)i(K)7 b Ft(\()p Fs(d)p Ft(\).)76 b(Finally)-8 b(,)44 b(for)382 887 y(what)30 b(concerns)g(\010)1044 901 y Fp(0)1083 887 y Ft(,)h(w)m(e)f(observ)m(e)h(that)g(if)e(\(\003)p Fs(;)15 b(`;)g Fk(B)s Ft(\))26 b Fk(2)f(R)2379 901 y Fp(0)2448 887 y Ft(then,)30 b(since)g Fk(G)2957 901 y Fp(\003)3006 882 y Fn(c)3038 901 y Fr(;)p Fq(B)3140 887 y Ft(is)f(\014ner)g(than)382 1000 y Fk(G)436 1018 y Fp(\003)485 999 y Fn(c)517 1018 y Fr(;)p Fq(f)p Fp(\003)p Fq(g)660 1000 y Ft(,)i(b)m(y)f(\(6.2\))i(w)m(e)f(get)888 1198 y Fs(\026)p Ft([)15 b Fs(G)1054 1212 y Fq(B)1107 1198 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Fs(\026)p Ft([)15 b Fs(G)1649 1212 y Fp(\003)1703 1198 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Fs(C)2144 1212 y Fp(0)2199 1198 y Fs(w)2266 1161 y Fp(2)2264 1221 y Fr(d)2321 1198 y Fk(E)2369 1212 y Fp(\003)2422 1198 y Ft(\()p Fs(f)10 b Ft(\))25 b Fk(\024)g Fs(C)d(`)2793 1161 y Fp(2)2847 1198 y Fk(E)2895 1212 y Fp(\003)2948 1198 y Ft(\()p Fs(f)10 b Ft(\))15 b Fs(:)274 b Ft(\(6.14\))382 1386 y(F)-8 b(rom)29 b(\(6.13\),)j(\(6.14\))f(and)d(what)h(w)m(e)g (said)f(in)g(b)s(et)m(w)m(een)h(them,)h(it)e(follo)m(ws)g(that)i(if)e Fs(`)d Fk(\025)g Fs(`)3458 1400 y Fp(0)3497 1386 y Ft(\()p Fs(\015)5 b Ft(\))382 1499 y(then)1580 1603 y(\010\()p Fk(A)p Ft(\))25 b Fk(\024)g Fs(C)d(e)2039 1566 y Fp(1+)p Fr(K)5 b Fp(\()p Fr(d)p Fp(\))2304 1603 y Fs(`)2342 1566 y Fp(2)2396 1603 y Fs(:)382 1762 y Ft(On)29 b(the)i(other)g(side)e(if)g Fs(`)c(<)g(`)1390 1776 y Fp(0)1430 1762 y Ft(\()p Fs(\015)5 b Ft(\))31 b(w)m(e)g(can)f(simply)e(use)i(\(6.2\))i(and)e(obtain)670 1949 y Fs(\026)p Ft([)15 b Fs(G)836 1963 y Fq(B)889 1949 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Fs(\026)p Ft([)15 b Fs(G)1431 1963 y Fp(\003)1485 1949 y Ft(\()p Fs(f)5 b(;)15 b(f)10 b Ft(\))15 b(])25 b Fk(\024)g Fs(C)1926 1963 y Fp(0)1981 1949 y Fs(L)2043 1912 y Fp(2)2097 1949 y Fk(E)2145 1963 y Fr(B)2198 1974 y Fn(L)2249 1949 y Ft(\()p Fs(f)10 b Ft(\))25 b Fk(\024)g Fs(C)2560 1963 y Fp(0)2615 1949 y Fs(`)2653 1963 y Fp(0)2692 1949 y Ft(\()p Fs(\015)5 b Ft(\))2814 1912 y Fp(2\(1+)p Fr(\015)t Fp(\))3055 1949 y Fk(E)3103 1963 y Fr(B)3156 1974 y Fn(L)3206 1949 y Ft(\()p Fs(f)10 b Ft(\))382 2137 y(hence)30 b(\(3.18\))j(holds)c (if)g(w)m(e)i(rede\014ne)f Fs(C)1756 2151 y Fr(\015)1830 2137 y Ft(suitably)-8 b(.)p 3554 2137 4 62 v 3558 2079 55 4 v 3558 2137 V 3612 2137 4 62 v 1955 5652 a(28)p eop %%Page: 29 30 29 29 bop 382 548 a Fu(7)109 b(Fluctuations)38 b(of)e(the)h(numb)s(er)f (of)h(pa)m(rticles)382 751 y Ft(W)-8 b(e)34 b(pro)m(v)m(e)h(here)e (Prop)s(osition)e(3.2.)51 b(Consider)31 b(a)j(p)s(olycub)s(e)d(\(\003)p Fs(;)15 b(`;)g Fk(A)p Ft(\))35 b(and)e(\014x)f Fs(")f(>)f Ft(0.)50 b(F)-8 b(or)382 864 y(all)29 b Fs(M)36 b Fk(2)24 b Fm(M)797 831 y Fq(A)797 891 y Fr(`)864 864 y Ft(,)30 b(all)g Fs(x)25 b Fk(2)g Fs(B)1278 878 y Fr(L)1360 864 y Ft(and)30 b(all)f Fs(t)c Fk(\025)g Ft(0,)31 b(let)962 1104 y Fs(h)1014 1067 y Fr(M)1093 1104 y Ft(\()p Fs(\033)s Ft(\))c(:=)1394 1043 y(1)-25 b(I)1447 1057 y Fr(M)1526 1043 y Ft(\()p Fs(N)1634 1057 y Fq(A)1695 1043 y Ft(\()p Fs(\033)s Ft(\)\))p 1376 1083 499 4 v 1376 1167 a Fs(\026)p Fk(f)p Fs(N)1549 1181 y Fq(A)1635 1167 y Ft(=)25 b Fs(M)10 b Fk(g)2066 1104 y Ft(and)181 b Fs(h)2446 1067 y Fr(M)2446 1127 y(x;t)2561 1104 y Ft(:=)25 b Fs(#)2736 1118 y Fr(x)2780 1104 y Fs(P)2838 1118 y Fr(t)2868 1104 y Fs(h)2920 1067 y Fr(M)3014 1104 y Fs(:)382 1335 y Ft(Then,)30 b(b)m(y)g(rev)m (ersibilit)m(y)e(and)i(translation)f(in)m(v)-5 b(ariance,)30 b(if)f Fs(s)c Fk(\025)g Ft(0)756 1523 y Fs(\026)p Ft(\()p Fs(g)889 1537 y Fr(x;s)986 1523 y Fs(h)1038 1486 y Fr(M)1117 1523 y Ft(\))h(=)f Fs(\026)1344 1449 y Fj(\002)1382 1523 y Fs(P)1440 1537 y Fr(s)1477 1523 y Fs(#)1531 1537 y Fr(x)1575 1523 y Ft(\()p Fs(f)k Fk(\000)20 b Fs(R)1844 1537 y Fr(K)1913 1523 y Fs(f)10 b Ft(\))p Fs(h)2055 1486 y Fr(M)2134 1449 y Fj(\003)2197 1523 y Ft(=)25 b Fs(\026)2363 1449 y Fj(\002)2401 1523 y Ft(\()p Fs(f)k Fk(\000)20 b Fs(R)2670 1537 y Fr(K)2739 1523 y Fs(f)10 b Ft(\))p Fs(#)2883 1537 y Fq(\000)p Fr(x)2981 1523 y Fs(P)3039 1537 y Fr(s)3076 1523 y Fs(h)3128 1486 y Fr(M)3207 1449 y Fj(\003)1178 1710 y Ft(=)1355 1649 y(1)p 1284 1689 188 4 v 1284 1773 a Fk(j)p Fs(B)1378 1787 y Fr(K)1446 1773 y Fk(j)1529 1624 y Fj(X)1497 1821 y Fr(y)r Fq(2)p Fr(B)1634 1832 y Fn(K)1708 1710 y Fs(\026)1778 1637 y Fj(\002)1816 1710 y Ft(\()p Fs(f)30 b Fk(\000)20 b Fs(#)2071 1724 y Fr(y)2112 1710 y Fs(f)10 b Ft(\))p Fs(h)2254 1673 y Fr(M)2254 1733 y Fq(\000)p Fr(x;s)2405 1637 y Fj(\003)2473 1710 y Fs(:)382 1989 y Ft(Th)m(us,)39 b(using)e(the)h(Cauc)m(h)m(y-Sc)m (h)m(w)m(arz)i(inequalit)m(y)c(and)i(the)g(in)m(v)-5 b(ariance)37 b(of)i Fs(B)3138 2003 y Fr(L)3228 1989 y Ft(under)d(the)382 2102 y(mapping)29 b Fs(x)c Fk(!)g(\000)p Fs(x)p Ft(,)30 b(w)m(e)h(can)g(write)657 2261 y(1)p 594 2302 172 4 v 594 2385 a Fk(j)p Fs(B)688 2399 y Fr(L)740 2385 y Fk(j)818 2236 y Fj(X)791 2433 y Fr(x)p Fq(2)p Fr(B)931 2444 y Fn(L)992 2323 y Fs(\026)1062 2249 y Fj(\002)1100 2323 y Fs(\026)p Ft(\()p Fs(g)1233 2337 y Fr(x;s)1345 2323 y Fk(j)15 b Fs(N)1458 2337 y Fq(A)1519 2323 y Ft(\))1554 2285 y Fp(2)1594 2249 y Fj(\003)1657 2323 y Ft(=)1826 2261 y(1)p 1763 2302 V 1763 2385 a Fk(j)p Fs(B)1857 2399 y Fr(L)1909 2385 y Fk(j)1987 2236 y Fj(X)1959 2433 y Fr(x)p Fq(2)p Fr(B)2099 2444 y Fn(L)2214 2236 y Fj(X)2161 2445 y Fr(M)7 b Fq(2)p Fo(M)2339 2422 y Fw(A)2339 2468 y Fn(`)2414 2323 y Fs(\026)p Fk(f)p Fs(N)2587 2337 y Fq(A)2673 2323 y Ft(=)25 b Fs(M)10 b Fk(g)15 b Fs(\026)p Ft(\()p 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Fk(g)1547 2851 y Fj(\024)1669 2917 y Ft(1)p 1606 2958 172 4 v 1606 3041 a Fk(j)p Fs(B)1700 3055 y Fr(L)1752 3041 y Fk(j)1830 2892 y Fj(X)1803 3089 y Fr(x)p Fq(2)p Fr(B)1943 3100 y Fn(L)2085 2917 y Ft(1)p 2014 2958 188 4 v 2014 3041 a Fk(j)p Fs(B)2108 3055 y Fr(K)2177 3041 y Fk(j)2259 2892 y Fj(X)2227 3089 y Fr(y)r Fq(2)p Fr(B)2364 3100 y Fn(K)2438 2979 y Fs(\026)p Ft([\()p Fs(f)30 b Fk(\000)20 b Fs(#)2773 2993 y Fr(y)2814 2979 y Fs(f)10 b Ft(\))15 b Fs(h)2971 2941 y Fr(M)2971 3001 y(x;s)3068 2979 y Ft(])3093 2941 y Fp(2)3132 2851 y Fj(\025)3206 2979 y Fs(:)202 b Ft(\(7.1\))382 3295 y(No)m(w)25 b(w)m(e)f(deal)g(with)f(the)i(terms)f Fs(\026)p Ft([\()p Fs(f)17 b Fk(\000)8 b Fs(#)1805 3309 y Fr(y)1846 3295 y Fs(f)i Ft(\))15 b Fs(h)2003 3262 y Fr(M)2003 3318 y(x;s)2099 3295 y Ft(])2124 3262 y Fp(2)2164 3295 y Ft(.)39 b(F)-8 b(or)25 b(an)m(y)f Fs(y)k Fk(2)d Fs(B)2778 3309 y Fr(K)2846 3295 y Ft(,)h(there)e(exists)g(a)h(path)382 3408 y(\(0)p Fs(;)15 b(y)547 3422 y Fp(1)587 3408 y Fs(;)g(:)g(:)g(:)i (;)e(y)834 3423 y Fr(k)910 3408 y Ft(=)34 b Fs(y)s Ft(\))h(going)h (from)f(0)h(to)g Fs(y)i Ft(whic)m(h)d(consists)g(of)g Fs(k)i Ft(=)d Fk(j)p Fs(y)s Fk(j)2882 3422 y Fp(1)2957 3408 y Ft(nearest)i(neigh)m(b)s(or)382 3521 y(steps.)67 b(Hence)40 b(w)m(e)g(can)f(de\014ne)g Fs(\015)1588 3535 y Fr(y)1669 3521 y Ft(=)h(\()p Fs(e)1857 3535 y Fp(1)1897 3521 y Fs(;)15 b(:)g(:)g(:)i(;)e(e)2141 3539 y Fq(j)p Fr(y)r Fq(j)2218 3548 y Fi(1)2257 3521 y Ft(\))39 b(where)g(eac)m(h)h Fs(e)2858 3535 y Fr(i)2927 3521 y Ft(=)g(\()p Fs(y)3118 3535 y Fr(i)p Fq(\000)p Fp(1)3236 3521 y Fs(;)15 b(y)3321 3535 y Fr(i)3349 3521 y Ft(\))40 b(is)e(an)382 3634 y(\(orien)m(ted\)) 24 b(edge)g(in)f Fs(B)1166 3648 y Fr(K)1234 3634 y Ft(.)38 b(Finally)-8 b(,)24 b(for)f(an)m(y)h(edge)h Fs(e)g Ft(=)g(\()p Fs(u;)15 b(v)s Ft(\),)27 b(w)m(e)d(de\014ne)f Fs(d)2971 3648 y Fr(e)3008 3634 y Fs(f)34 b Ft(:=)26 b Fs(#)3263 3648 y Fr(v)3303 3634 y Fs(f)16 b Fk(\000)7 b Fs(#)3496 3648 y Fr(u)3540 3634 y Fs(f)j Ft(.)382 3747 y(By)31 b(the)f(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)i(inequalit)m(y)d(w)m(e)i(ha)m(v) m(e)787 3968 y Fs(\026)p Ft([\()p Fs(f)f Fk(\000)20 b Fs(#)1122 3982 y Fr(y)1163 3968 y Fs(f)10 b Ft(\))15 b Fs(h)1320 3931 y Fr(M)1320 3991 y(x;s)1416 3968 y Ft(])1441 3931 y Fp(2)1506 3968 y Ft(=)25 b Fs(\026)1657 3868 y Fj(h)1711 3882 y(X)1700 4073 y Fr(e)p Fq(2)p Fr(\015)1816 4081 y Fn(y)1869 3968 y Fs(d)1916 3982 y Fr(e)1953 3968 y Fs(f)f(h)2074 3931 y Fr(M)2074 3991 y(x;s)2171 3868 y Fj(i)2214 3890 y Fp(2)2278 3968 y Fk(\024)h(j)p Fs(\015)2446 3982 y Fr(y)2488 3968 y Fk(j)2540 3882 y Fj(X)2528 4073 y Fr(e)p Fq(2)p Fr(\015)2644 4081 y Fn(y)2697 3968 y Fs(\026)p Ft(\()p Fs(d)2834 3982 y Fr(e)2871 3968 y Fs(f)g(h)2993 3931 y Fr(M)2993 3991 y(x;s)3089 3968 y Ft(\))3124 3931 y Fp(2)3189 3968 y Fs(:)219 b Ft(\(7.2\))382 4261 y(In)32 b(the)i(next)f(t)m(w)m(o)i(Lemmas)e(w)m(e)h(deals)e(with)g Fs(\026)p Ft(\()p Fs(d)2117 4275 y Fr(e)2154 4261 y Fs(f)25 b(h)2276 4228 y Fr(M)2276 4284 y(x;s)2372 4261 y Ft(\))2407 4228 y Fp(2)2447 4261 y Ft(.)49 b(In)33 b(the)g(pro)s(of)g(it)f(will)f (b)s(e)i(clear)382 4374 y(wh)m(y)-8 b(,)39 b(at)f(the)f(v)m(ery)h(b)s (egining,)e(w)m(e)i(ha)m(v)m(e)g(substracted)f Fs(R)2406 4388 y Fr(K)2475 4374 y Fs(f)10 b Ft(.)60 b(This)35 b(leads)i(to)h(ha)m (ving)e Fs(d)3528 4388 y Fr(e)3565 4374 y Fs(f)382 4487 y Ft(instead)30 b(of)g Fs(f)40 b Ft(in)29 b(\(7.2\))r(.)382 4675 y Ff(Lemma)38 b(7.1.)45 b Fl(Assume)36 b(\(USM\).)g(F)-7 b(or)38 b(any)g Fs(\013)33 b(>)g Ft(0)p Fl(,)38 b(ther)-5 b(e)38 b(is)e Fs(C)2703 4689 y Fr(\013)2786 4675 y Fs(>)c Ft(0)38 b Fl(such)f(that)h(for)f(al)5 b(l)382 4788 y(lo)-5 b(c)g(al)29 b(functions)f Fs(f)36 b Fl(on)28 b Ft(\012)f Fl(with)h Fs(S)1518 4803 y Fr(f)1588 4788 y Fk(3)d Ft(0)p Fl(,)k(for)f(al)5 b(l)27 b Fs(u)f Fk(2)f Fm(Z)2269 4755 y Fr(d)2332 4788 y Fl(with)j Fk(j)p Fs(u)p Fk(j)2626 4802 y Fp(1)2691 4788 y Ft(=)d(1)p Fl(,)k(and)f(for)g(al)5 b(l)28 b(p)-5 b(ositive)382 4901 y(inte)g(gers)33 b Fs(L)p Fl(,)f(we)h(have)1286 5133 y Ft(sup)1296 5211 y Fr(\034)t(;N)1438 5133 y Fk(j)p Fs(\027)1514 5095 y Fr(\034)1508 5155 y(B)1561 5166 y Fn(L)1608 5155 y Fr(;N)1695 5133 y Ft(\()p Fs(d)1777 5151 y Fp(\(0)p Fr(;u)p Fp(\))1932 5133 y Fs(f)10 b Ft(\))p Fk(j)25 b(\024)2178 5070 y Fs(C)2243 5084 y Fr(\013)2308 5070 y Fk(k)p Fs(f)10 b Fk(k)2453 5084 y Fr(u)2513 5070 y Fk(j)p Fs(S)2594 5085 y Fr(f)2640 5070 y Fk(j)p 2178 5112 487 4 v 2366 5195 a Fs(L)2428 5169 y Fr(\013)2690 5133 y Fs(:)718 b Ft(\(7.3\))1955 5652 y(29)p eop %%Page: 30 31 30 30 bop 382 548 a Fl(Pr)-5 b(o)g(of.)57 b Ft(Since)919 708 y Fk(j)p Fs(\027)995 671 y Fr(\034)989 731 y(B)1042 742 y Fn(L)1089 731 y Fr(;N)1176 708 y Ft(\()p Fs(d)1258 727 y Fp(\(0)p Fr(;u)p Fp(\))1413 708 y Fs(f)10 b Ft(\))p Fk(j)25 b(\024)40 b Ft(2)p Fk(k)p Fs(f)10 b Fk(k)1854 722 y Fr(u)1934 622 y Fj(X)1915 819 y Fr(x)p Fq(2)p Fr(S)2045 831 y Fn(f)2100 708 y Fs(\027)2151 671 y Fr(\034)2145 731 y(B)2198 742 y Fn(L)2244 731 y Fr(;N)2331 708 y Ft(\()p Fk(j)p Fs(\033)s Ft(\()p Fs(x)p Ft(\))22 b Fk(\000)e Fs(\033)s Ft(\()p Fs(x)h Ft(+)e Fs(u)p Ft(\))p Fk(j)p Ft(\))382 965 y(one)34 b(can)g(use)f(Lemma)h(10.1)h(of)f([VY97)q(])g (where)f(estimate)h(\(7.3\))i(is)c(pro)m(v)m(ed)i(when)f Fs(f)43 b Ft(is)32 b(the)382 1078 y(particular)d(function)g Fs(\033)s Ft(\(0\))i(and)f(the)h(result)e(follo)m(ws.)p 3554 1078 4 62 v 3558 1020 55 4 v 3558 1078 V 3612 1078 4 62 v 382 1235 a Ff(Lemma)35 b(7.2.)43 b Fl(Assume)35 b(\(USM\).)f(F)-7 b(or)36 b(al)5 b(l)35 b(lo)-5 b(c)g(al)36 b(functions)f Fs(f)44 b Fl(on)35 b Ft(\012)p Fl(,)g(for)g(al)5 b(l)35 b Fs(")29 b(>)g Ft(0)35 b Fl(ther)-5 b(e)382 1348 y(exists)37 b Fs(A)d Ft(=)g Fs(A)p Ft(\()p Fs(f)5 b(;)15 b(")p Ft(\))34 b Fs(>)g Ft(0)j Fl(such)h(that)g(if)f Fs(u;)15 b(v)41 b Fl(ar)-5 b(e)38 b(ne)-5 b(ar)g(est)39 b(neighb)-5 b(ors)38 b(in)f Fs(B)3069 1362 y Fr(K)3138 1348 y Fl(,)h Fs(e)c Ft(:=)f(\()p Fs(u;)15 b(v)s Ft(\))382 1461 y Fl(and)32 b Fs(W)643 1475 y Fr(e)705 1461 y Ft(:=)25 b Fs(u)17 b Ft(+)g Fs(B)1052 1480 y Fq(b)p Fr(L)1131 1461 y Fn(")1165 1480 y Fq(c)1200 1461 y Fl(,)31 b(then)h(for)g(al)5 b(l)31 b(non)h(ne)-5 b(gative)31 b(functions)h Fs(h)f Fl(with)h Fs(\026)p Ft(\()p Fs(h)p Ft(\))26 b(=)f(1)32 b Fl(we)f(have)1253 1664 y Fs(\026)p Ft(\()p Fs(d)1390 1678 y Fr(e)1428 1664 y Fs(f)24 b(h)p Ft(\))1584 1626 y Fp(2)1649 1664 y Fk(\024)h Fs(A)1813 1563 y Fj(h)1857 1664 y Fs(L)1919 1626 y Fp(4)p Fr(")1990 1664 y Fk(E)2038 1678 y Fr(W)2105 1686 y Fn(e)2143 1664 y Ft(\()2178 1582 y Fk(p)p 2254 1582 53 4 v 82 x Fs(h)p Ft(\))c(+)2536 1602 y(1)p 2463 1643 193 4 v 2463 1726 a Fs(L)2525 1700 y Fr(d)p Fp(+2)2665 1563 y Fj(i)2723 1664 y Fs(:)685 b Ft(\(7.4\))382 1846 y Fl(Pr)-5 b(o)g(of.)84 b Ft(First)39 b(w)m(e)h(write)f Fs(\026)p Ft(\()p Fs(d)1457 1860 y Fr(e)1494 1846 y Fs(f)25 b(h)p Ft(\))41 b(=)1803 1773 y Fj(R)1879 1846 y Fs(\026)p Ft(\()p Fs(d\034)10 b Ft(\))15 b Fs(\027)2167 1813 y Fr(\034)2161 1878 y(W)2228 1886 y Fn(e)2262 1878 y Fr(;N)2338 1889 y Fn(W)2395 1897 y(e)2432 1878 y Fp(\()p Fr(\034)8 b Fp(\))2530 1846 y Ft(\()p Fs(d)2612 1860 y Fr(e)2650 1846 y Fs(f)24 b(h)p Ft(\).)69 b(F)-8 b(or)40 b(simplicit)m(y)-8 b(,)40 b(w)m(e)382 1974 y(let)g Fs(\027)574 1941 y Fr(\034)568 1996 y(e)658 1974 y Ft(:=)h Fs(\027)846 1941 y Fr(\034)840 2005 y(W)907 2013 y Fn(e)940 2005 y Fr(;N)1016 2016 y Fn(W)1073 2024 y(e)1111 2005 y Fp(\()p Fr(\034)8 b Fp(\))1209 1974 y Ft(.)69 b(By)40 b(the)h(en)m(trop)m(y)f(inequalit)m(y)f(\(see)h(for)g (instance)g(Chapter)f(1)i(of)382 2106 y([ABC)605 2073 y Fp(+)664 2106 y Ft(00)q(]\),)31 b(for)g(an)m(y)f Fs(s)25 b(>)g Ft(0,)1077 2306 y Fs(\027)1128 2268 y Fr(\034)1122 2328 y(e)1171 2306 y Ft(\()p Fs(d)1253 2320 y Fr(e)1290 2306 y Fs(f)g(h)p Ft(\))h Fk(\024)1579 2244 y Fs(\027)1630 2211 y Fr(\034)1624 2267 y(e)1672 2244 y Ft(\()p Fs(h)p Ft(\))p 1579 2285 217 4 v 1666 2368 a Fs(s)1821 2306 y Ft(log)16 b Fs(\027)2004 2268 y Fr(\034)1998 2328 y(e)2047 2232 y Fj(\000)2089 2306 y Fs(e)2131 2268 y Fr(s)11 b(d)2211 2276 y Fn(e)2245 2268 y Fr(f)2291 2232 y Fj(\001)2352 2306 y Ft(+)2453 2244 y(1)p 2453 2285 46 4 v 2454 2368 a Fs(s)2524 2306 y Ft(En)m(t)2669 2320 y Fr(\027)2708 2300 y Fn(\034)2704 2336 y(e)2751 2306 y Ft(\()p Fs(h)p Ft(\))27 b Fs(;)382 2491 y Ft(where,)47 b(for)d(an)g(arbitrary)e (probabilit)m(y)g(measure)i Fs(\032)f Ft(on)h(\(\012)p Fs(;)15 b Fk(F)9 b Ft(\),)49 b(and)43 b Fs(h)49 b Fk(2)e Fs(L)3242 2458 y Fp(1)3281 2491 y Ft(\()p Fs(\032)p Ft(\))e(with)382 2604 y Fs(\032)p Ft(\()p Fs(h)p Ft(\))26 b(=)f(1,)31 b(w)m(e)g(denote)g(b)m(y)f(En)m(t)1472 2618 y Fr(\032)1513 2604 y Ft(\()p Fs(h)p Ft(\))h(the)g(en)m(trop)m(y)g(of)f Fs(h\032)h Ft(with)e(resp)s(ect)h(to)i Fs(\032)p Ft(,)e Fl(i.e.)1298 2756 y Ft(En)m(t)1444 2770 y Fr(\032)1484 2756 y Ft(\()p Fs(h)p Ft(\))c(:=)g(En)m(t\()p Fs(h\032)15 b Fk(j)g Fs(\032)p Ft(\))27 b(=)e Fs(\032)p Ft(\()p Fs(h)15 b Ft(log)i Fs(h)p Ft(\))e Fs(:)731 b Ft(\(7.5\))382 2907 y(The)40 b(probabilit)m(y)f(measure)i Fs(\027)1471 2874 y Fr(\034)1465 2930 y(e)1555 2907 y Ft(is)f(kno)m(wn)g(to)i(satisfy)f ([Y)-8 b(au96)q(,)42 b(CMR02])g(a)f(logarithmic)382 3020 y(Sob)s(olev)29 b(inequalit)m(y)g(whic)m(h)g(states)j(that)f(for)f(all) f(functions)g Fs(g)34 b Ft(on)c(\012)1530 3172 y(En)m(t)1676 3186 y Fr(\027)1715 3167 y Fn(\034)1711 3202 y(e)1758 3172 y Ft(\()p Fs(g)s Ft(\))c Fk(\024)f Fs(C)7 b(L)2130 3135 y Fp(2)p Fr(")2202 3172 y Fk(E)2250 3186 y Fr(W)2317 3194 y Fn(e)2354 3172 y Ft(\()p Fs(g)s Ft(\))963 b(\(7.6\))382 3324 y(for)34 b(some)h(constan)m(t)g Fs(C)7 b Ft(.)52 b(Consequen)m(tly)33 b(it)h(follo)m(ws)g(from)f(the)i(Herbst)f(argumen) m(t)h([Led99)q(,)382 3437 y(ABC)580 3404 y Fp(+)639 3437 y Ft(00)q(])c(that)1058 3611 y Fs(\027)1109 3574 y Fr(\034)1103 3634 y(e)1167 3510 y Fj(\020)1221 3611 y Fs(e)1263 3574 y Fr(s)12 b(d)1344 3582 y Fn(e)1377 3574 y Fr(f)1423 3510 y Fj(\021)1502 3611 y Fk(\024)25 b Ft(exp[)15 b Fs(C)7 b(s)1892 3574 y Fp(2)1946 3611 y Fk(k)p Fs(d)2038 3625 y Fr(e)2076 3611 y Fs(f)j Fk(k)2176 3574 y Fp(2)2176 3634 y(Lip)2298 3611 y Fs(L)2360 3574 y Fp(2)p Fr(")2452 3611 y Ft(+)20 b Fs(s)15 b(\027)2652 3574 y Fr(\034)2646 3634 y(e)2694 3611 y Ft(\()p Fs(d)2776 3625 y Fr(e)2814 3611 y Fs(f)10 b Ft(\))15 b(])382 3793 y(with)29 b Fk(k)p Fs(d)681 3807 y Fr(e)719 3793 y Fs(f)10 b Fk(k)819 3760 y Fp(2)819 3820 y(Lip)951 3793 y Ft(:=)1072 3725 y Fj(P)1168 3820 y Fr(x)p Fq(2)p Fr(W)1322 3828 y Fn(e)1374 3793 y Fk(kr)1495 3807 y Fr(x)1539 3793 y Fs(d)1586 3807 y Fr(e)1623 3793 y Fs(f)g Fk(k)1723 3760 y Fp(2)1723 3816 y Fr(u)1793 3793 y Fk(\024)25 b Ft(4)p Fk(j)p Fs(S)2015 3808 y Fr(f)2061 3793 y Fk(j)15 b(k)p Fs(f)10 b Fk(k)2246 3807 y Fr(u)2317 3793 y Ft(=:)25 b Fs(A)2506 3807 y Fp(1)2546 3793 y Ft(\()p Fs(f)10 b Ft(\).)40 b(Th)m(us,)834 4001 y Fs(\027)885 3963 y Fr(\034)879 4023 y(e)928 4001 y Ft(\()p Fs(d)1010 4015 y Fr(e)1048 4001 y Fs(f)24 b(h)p Ft(\))i Fk(\024)f Fs(\027)1377 3963 y Fr(\034)1371 4023 y(e)1420 4001 y Ft(\()p Fs(h)p Ft(\))p Fs(\027)1593 3963 y Fr(\034)1587 4023 y(e)1637 4001 y Ft(\()p Fs(d)1719 4015 y Fr(e)1756 4001 y Fs(f)10 b Ft(\))20 b(+)g Fs(s)15 b(\027)2066 3963 y Fr(\034)2060 4023 y(e)2109 4001 y Ft(\()p Fs(h)p Ft(\))g Fs(C)22 b(A)2401 4015 y Fp(1)2441 4001 y Fs(L)2503 3963 y Fp(2)p Fr(")2595 4001 y Ft(+)2696 3939 y(1)p 2696 3980 V 2697 4063 a Fs(s)2767 4001 y Ft(En)m(t)2912 4015 y Fr(\027)2951 3995 y Fn(\034)2947 4031 y(e)2994 4001 y Ft(\()p Fs(h)p Ft(\))k Fs(:)266 b Ft(\(7.7\))382 4178 y(Optimizing)28 b(o)m(v)m(er)j(the)g(free)f(parameter)h Fs(s)f Ft(and)g(using)f(\(7.6\))j(once)f(again)f(w)m(e)h(get)874 4330 y Fs(\027)925 4293 y Fr(\034)919 4353 y(e)967 4330 y Ft(\()p Fs(d)1049 4344 y Fr(e)1087 4330 y Fs(f)24 b(h)p Ft(\))i Fk(\024)f Ft(2[)p Fs(C)7 b(A)1575 4344 y Fp(1)1615 4330 y Fs(L)1677 4293 y Fp(2)p Fr(")1749 4330 y Fs(\027)1800 4293 y Fr(\034)1794 4353 y(e)1843 4330 y Ft(\()p Fs(h)p Ft(\))31 b(En)m(t)2141 4344 y Fr(\027)2180 4325 y Fn(\034)2176 4360 y(e)2224 4330 y Ft(\()p Fs(h)p Ft(\)])2371 4293 y Fp(1)p Fr(=)p Fp(2)2502 4330 y Ft(+)20 b Fs(\027)2644 4293 y Fr(\034)2638 4353 y(e)2687 4330 y Ft(\()p Fs(h)p Ft(\))15 b Fs(\027)2875 4293 y Fr(\034)2869 4353 y(e)2919 4330 y Ft(\()p Fs(d)3001 4344 y Fr(e)3038 4330 y Fs(f)10 b Ft(\))1269 4486 y Fk(\024)25 b Ft([)p Fs(A)1458 4500 y Fp(2)1513 4486 y Fs(L)1575 4449 y Fp(4)p Fr(")1647 4486 y Fs(\027)1698 4449 y Fr(\034)1692 4509 y(e)1741 4486 y Ft(\()p Fs(h)p Ft(\))15 b Fk(E)1926 4500 y Fr(\027)1965 4481 y Fn(\034)1961 4517 y(e)2009 4486 y Ft(\()2044 4404 y Fk(p)p 2121 4404 53 4 v 2121 4486 a Fs(h)p Ft(\)])2233 4449 y Fp(1)p Fr(=)p Fp(2)2364 4486 y Ft(+)20 b Fs(\027)2506 4449 y Fr(\034)2500 4509 y(e)2548 4486 y Ft(\()p Fs(h)p Ft(\))15 b Fs(\027)2736 4449 y Fr(\034)2730 4509 y(e)2781 4486 y Ft(\()p Fs(d)2863 4500 y Fr(e)2900 4486 y Fs(f)10 b Ft(\))25 b Fs(:)382 4649 y Ft(No)m(w,)53 b(b)m(y)48 b(Lemma)g(7.1)g(\(with)f Fs(\013)55 b Ft(:=)1818 4614 y Fr(d)p Fp(+2)p 1818 4629 127 4 v 1848 4681 a(2)p Fr(")1955 4649 y Ft(\),)d(there)c(exists)g Fs(A)2654 4663 y Fp(3)2748 4649 y Ft(=)54 b Fs(A)2941 4663 y Fp(3)2980 4649 y Ft(\()p Fs(f)5 b(;)15 b(")p Ft(\))49 b(suc)m(h)e(that)382 4772 y Fs(\027)433 4739 y Fr(\034)427 4794 y(e)476 4772 y Ft(\()p Fs(d)558 4786 y Fr(e)595 4772 y Fs(f)10 b Ft(\))45 b Fk(\024)f Fs(A)913 4786 y Fp(3)953 4772 y Fs(L)1015 4739 y Fq(\000)p Fp(\()p Fr(d)p Fp(+2\))p Fr(=)p Fp(2)1326 4772 y Ft(.)76 b(Th)m(us,)44 b(since)d Fs(\026)p Ft(\()p Fs(h)p Ft(\))46 b(=)e(1,)i(an)c(in)m(tegration)g(with)f(resp)s(ect)h (to)382 4885 y Fs(\026)p Ft(\()p Fs(d\034)10 b Ft(\))31 b(giv)m(es)497 5092 y Fs(\026)p Ft(\()p Fs(d)634 5106 y Fr(e)671 5092 y Fs(f)25 b(h)p Ft(\))828 5054 y Fp(2)893 5092 y Fk(\024)g Ft(2)p Fs(A)1102 5054 y Fp(2)1102 5114 y(2)1157 5092 y Fs(L)1219 5054 y Fp(4)p Fr(")1291 4968 y Fj(Z)1367 5092 y Fs(\026)p Ft(\()p Fs(d\034)10 b Ft(\))15 b Fs(\027)1655 5054 y Fr(\034)1649 5114 y(e)1699 5092 y Ft(\()p Fs(h)p Ft(\))1836 4968 y Fj(Z)1913 5092 y Fs(\026)p Ft(\()p Fs(d\034)10 b Ft(\))15 b Fk(E)2198 5106 y Fr(\027)2237 5087 y Fn(\034)2233 5122 y(e)2281 5092 y Ft(\()2316 5010 y Fk(p)p 2392 5010 53 4 v 82 x Fs(h)p Ft(\))21 b(+)f(2)2689 5030 y Fs(A)2757 4997 y Fp(2)2757 5055 y(3)p 2646 5071 193 4 v 2646 5154 a Fs(L)2708 5128 y Fr(d)p Fp(+2)2849 4991 y Fj(h)2892 4968 y(Z)2967 5092 y Fs(\026)p Ft(\()p Fs(d\034)10 b Ft(\))15 b Fs(\027)3255 5054 y Fr(\034)3249 5114 y(e)3299 5092 y Ft(\()p Fs(h)p Ft(\))3421 4991 y Fj(i)3465 5014 y Fp(2)893 5341 y Ft(=)25 b(2)p Fs(A)1102 5303 y Fp(2)1102 5363 y(2)1157 5341 y Fs(L)1219 5303 y Fp(4)p Fr(")1291 5341 y Fk(E)1339 5355 y Fr(W)1406 5363 y Fn(e)1443 5341 y Ft(\()1478 5259 y Fk(p)p 1555 5259 53 4 v 1555 5341 a Fs(h)p Ft(\))c(+)e(2)1851 5279 y Fs(A)1919 5246 y Fp(2)1919 5304 y(3)p 1808 5320 193 4 v 1808 5403 a Fs(L)1870 5377 y Fr(d)p Fp(+2)2037 5341 y Fs(:)1955 5652 y Ft(30)p eop %%Page: 31 32 31 31 bop 382 548 a Ft(And)29 b(the)i(result)e(of)i(the)f(Lemma)h (follo)m(ws.)p 3554 548 4 62 v 3558 490 55 4 v 3558 548 V 3612 548 4 62 v 382 753 a(Bac)m(k)37 b(to)f(the)g(inequalit)m(y)e (\(7.1\).)58 b(Using)35 b(Lemma)h(7.2)g(together)h(with)e(\(7.2\))i (and)e(the)h(fact)382 866 y(that)31 b Fk(j)p Fs(\015)651 880 y Fr(y)693 866 y Fk(j)25 b Ft(=)g Fk(j)p Fs(y)s Fk(j)937 880 y Fp(1)1002 866 y Fk(\024)g Fs(dK)37 b Ft(for)31 b(an)m(y)f Fs(y)e Fk(2)d Fs(B)1798 880 y Fr(K)1866 866 y Ft(,)31 b(w)m(e)g(get)g(that)g(for)g(an)m(y)f Fs(M)36 b Fk(2)24 b Fm(M)3004 833 y Fq(A)3004 894 y Fr(`)3071 866 y Ft(,)30 b(an)m(y)h Fs(s)25 b Fk(\025)g Ft(0,)739 1024 y(1)p 676 1065 172 4 v 676 1148 a Fk(j)p Fs(B)770 1162 y Fr(L)822 1148 y Fk(j)900 999 y Fj(X)872 1196 y Fr(x)p Fq(2)p Fr(B)1012 1207 y Fn(L)1155 1024 y Ft(1)p 1084 1065 188 4 v 1084 1148 a Fk(j)p Fs(B)1178 1162 y Fr(K)1246 1148 y Fk(j)1329 999 y Fj(X)1297 1196 y Fr(y)r Fq(2)p Fr(B)1434 1207 y Fn(K)1508 1086 y Fs(\026)p Ft([\()p Fs(f)30 b Fk(\000)20 b Fs(#)1843 1100 y Fr(y)1884 1086 y Fs(f)10 b Ft(\))15 b Fs(h)2041 1048 y Fr(M)2041 1108 y(x;s)2137 1086 y Ft(])2162 1048 y Fp(2)873 1383 y Fk(\024)25 b Fs(A)1110 1322 y Ft(1)p 1047 1362 172 4 v 1047 1446 a Fk(j)p Fs(B)1141 1460 y Fr(L)1193 1446 y Fk(j)1271 1297 y Fj(X)1243 1494 y Fr(x)p Fq(2)p Fr(B)1383 1505 y Fn(L)1526 1322 y Ft(1)p 1455 1362 188 4 v 1455 1446 a Fk(j)p Fs(B)1549 1460 y Fr(K)1617 1446 y Fk(j)1700 1297 y Fj(X)1668 1494 y Fr(y)r Fq(2)p Fr(B)1805 1505 y Fn(K)1879 1383 y Fk(j)p Fs(\015)1951 1397 y Fr(y)1993 1383 y Fk(j)2044 1297 y Fj(X)2033 1488 y Fr(e)p Fq(2)p Fr(\015)2149 1496 y Fn(y)2187 1255 y Fj(\024)2235 1383 y Fs(L)2297 1346 y Fp(4)p Fr(")2369 1383 y Fk(E)2417 1397 y Fr(W)2484 1405 y Fn(e)2521 1310 y Fj(\000)2563 1278 y(q)p 2654 1278 149 4 v 105 x Fs(h)2706 1357 y Fr(M)2706 1406 y(x;s)2802 1310 y Fj(\001)2864 1383 y Ft(+)3038 1322 y(1)p 2965 1362 193 4 v 2965 1446 a Fs(L)3027 1419 y Fr(d)p Fp(+2)3167 1255 y Fj(\025)873 1686 y Fk(\024)g Fs(A)1037 1558 y Fj(\024)1085 1686 y Fs(d)1132 1649 y Fp(2)1216 1625 y Fs(K)1300 1592 y Fp(2)p 1181 1665 V 1181 1749 a Fs(L)1243 1722 y Fr(d)p Fp(+2)1404 1686 y Ft(+)1625 1625 y Fs(L)1687 1592 y Fp(4)p Fr(")p 1505 1665 375 4 v 1505 1749 a Fk(j)p Fs(B)1599 1763 y Fr(L)1651 1749 y Fk(j)15 b(j)p Fs(B)1785 1763 y Fr(K)1854 1749 y Fk(j)1937 1600 y Fj(X)1905 1797 y Fr(y)r Fq(2)p Fr(B)2042 1808 y Fn(K)2116 1686 y Fk(j)p Fs(\015)2188 1700 y Fr(y)2230 1686 y Fk(j)2281 1600 y Fj(X)2270 1791 y Fr(e)p Fq(2)p Fr(\015)2386 1799 y Fn(y)2466 1600 y Fj(X)2439 1797 y Fr(x)p Fq(2)p Fr(B)2579 1808 y Fn(L)2640 1686 y Fk(E)2688 1700 y Fq(\000)p Fr(x)p Fp(+)p Fr(W)2905 1708 y Fn(e)2942 1613 y Fj(\000)2984 1575 y(q)p 3075 1575 132 4 v 111 x Fs(h)3127 1660 y Fr(M)3127 1709 y(s)3206 1613 y Fj(\001)3248 1558 y(\025)3311 1686 y Fs(:)382 1959 y Ft(In)26 b(the)g(last)h (inequalit)m(y)e(w)m(e)h(used)g(the)h(fact)g(that)g Fk(E)2133 1973 y Fr(W)2200 1981 y Fn(e)2238 1959 y Ft(\()p Fs(#)2327 1973 y Fr(x)2371 1959 y Fs(H)7 b Ft(\))25 b(=)g Fk(E)2658 1973 y Fq(\000)p Fr(x)p Fp(+)p Fr(W)2875 1981 y Fn(e)2912 1959 y Ft(\()p Fs(H)7 b Ft(\))27 b(for)f(an)m(y)h Fs(x)f Ft(and)382 2072 y(an)m(y)37 b Fs(H)7 b Ft(,)39 b(due)d(to)h(the)g (translation)f(in)m(v)-5 b(ariance)36 b(prop)s(ert)m(y)-8 b(.)60 b(Then,)37 b(the)g(b)s(ound)e Fk(j)p Fs(\015)3279 2086 y Fr(y)3321 2072 y Fk(j)h(\024)g Fs(dK)382 2185 y Ft(and)30 b(an)g(explicit)f(computation)h(giv)m(es)504 2350 y(1)p 441 2391 172 4 v 441 2474 a Fk(j)p Fs(B)535 2488 y Fr(L)588 2474 y Fk(j)665 2325 y Fj(X)638 2522 y Fr(x)p Fq(2)p Fr(B)778 2533 y Fn(L)921 2350 y Ft(1)p 849 2391 188 4 v 849 2474 a Fk(j)p Fs(B)943 2488 y Fr(K)1012 2474 y Fk(j)1095 2325 y Fj(X)1062 2522 y Fr(y)r Fq(2)p Fr(B)1199 2533 y Fn(K)1274 2412 y Fs(\026)p Ft([\()p Fs(f)g Fk(\000)19 b Fs(#)1608 2426 y Fr(y)1650 2412 y Fs(f)10 b Ft(\))15 b Fs(h)1807 2374 y Fr(M)1807 2434 y(x;s)1903 2412 y Ft(])1928 2374 y Fp(2)1993 2412 y Fk(\024)2104 2350 y Fs(A)2172 2317 y Fq(0)p 2099 2391 103 4 v 2099 2474 a Fs(L)2161 2448 y Fr(d)2211 2283 y Fj(\024)2259 2412 y Fs(L)2321 2374 y Fp(4)p Fr(")2393 2412 y Fs(K)2477 2374 y Fp(2)2531 2412 y Fk(j)p Fs(W)2642 2426 y Fr(e)2679 2412 y Fk(j)g(E)2775 2338 y Fj(\000)2817 2300 y(q)p 2908 2300 132 4 v 112 x Fs(h)2960 2385 y Fr(M)2960 2434 y(s)3040 2338 y Fj(\001)3102 2412 y Ft(+)3202 2350 y Fs(K)3286 2317 y Fp(2)p 3202 2391 124 4 v 3213 2474 a Fs(L)3275 2448 y Fp(2)3336 2283 y Fj(\025)3433 2412 y Ft(\(7.8\))382 2689 y(for)21 b(some)h(other)g(constan)m(t)g Fs(A)1385 2656 y Fq(0)1409 2689 y Ft(.)38 b(It)21 b(is)g(w)m(ell)f(kno)m(wn)h (\(see)i([ABC)2495 2656 y Fp(+)2554 2689 y Ft(00)q(,)g(Chapter)e(2])h (for)g(instance\))382 2825 y(that)34 b(for)f(an)m(y)h Fs(f)10 b Ft(,)34 b Fs(@)1061 2839 y Fr(s)1113 2825 y Ft(En)m(t\()p Fs(P)1351 2839 y Fr(s)1389 2825 y Fs(f)10 b Ft(\))30 b Fk(\024)g(\000)p Ft(4)p Fk(E)8 b Ft(\()1817 2757 y Fk(p)p 1893 2757 150 4 v 68 x Fs(P)1951 2839 y Fr(s)1988 2825 y Fs(f)h Ft(\).)51 b(Th)m(us,)33 b(as)h(En)m(t\()p Fs(P)2765 2839 y Fr(s)2802 2825 y Fs(f)10 b Ft(\))33 b(is)g(non)g(increasing,)382 2938 y(w)m(e)e(ha)m(v)m(e)452 3025 y Fj(Z)543 3051 y Fr(t)502 3231 y Fp(0)573 3148 y Fk(E)629 3075 y Fj(\000)670 3037 y(q)p 761 3037 132 4 v 111 x Fs(h)813 3122 y Fr(M)813 3171 y(s)908 3075 y Fj(\001)965 3148 y Fs(ds)25 b Fk(\024)1186 3087 y Ft(1)p 1186 3127 46 4 v 1186 3211 a(4)1241 3148 y([En)m(t)q(\()p Fs(h)1499 3111 y Fr(M)1578 3148 y Ft(\))c Fk(\000)f Ft(En)m(t\()p Fs(h)1957 3111 y Fr(M)1957 3171 y(t)2037 3148 y Ft(\)])26 b Fk(\024)2228 3087 y Ft(1)p 2228 3127 V 2228 3211 a(4)2299 3148 y(En)m(t\()p Fs(h)2531 3111 y Fr(M)2611 3148 y Ft(\))g(=)2777 3087 y(1)p 2777 3127 V 2777 3211 a(4)2848 3148 y(log)3217 3087 y(1)p 2991 3127 499 4 v 2991 3211 a Fs(\026)p Fk(f)p Fs(N)3164 3225 y Fq(A)3250 3211 y Ft(=)f Fs(M)10 b Fk(g)3524 3148 y Fs(;)3433 3310 y Ft(\(7.9\))382 3423 y(where,)28 b(in)f(last)h(equalit)m(y)-8 b(,)29 b(w)m(e)g(ha)m(v)m(e)g(used)f(the)g (de\014nition)e(of)j(en)m(trop)m(y)-8 b(.)41 b(By)28 b(consequence)h(w)m(e)382 3536 y(ha)m(v)m(e)539 3660 y Fj(X)486 3868 y Fr(M)7 b Fq(2)p Fo(M)665 3845 y Fw(A)665 3892 y Fn(`)739 3746 y Fs(\026)p Fk(f)p Fs(N)912 3760 y Fq(A)998 3746 y Ft(=)25 b Fs(M)10 b Fk(g)1252 3622 y Fj(Z)1344 3648 y Fr(t)1303 3828 y Fp(0)1373 3746 y Fk(E)1429 3672 y Fj(\000)1471 3635 y(q)p 1562 3635 132 4 v 111 x Fs(h)1614 3720 y Fr(M)1614 3768 y(s)1709 3672 y Fj(\001)1766 3746 y Fs(ds)25 b Fk(\024)1987 3684 y Ft(1)p 1987 3725 46 4 v 1987 3808 a(4)2111 3660 y Fj(X)2057 3868 y Fr(M)7 b Fq(2)p Fo(M)2236 3845 y Fw(A)2236 3892 y Fn(`)2310 3746 y Fs(\026)p Fk(f)p Fs(N)2483 3760 y Fq(A)2570 3746 y Ft(=)24 b Fs(M)10 b Fk(g)15 b Ft(log)3193 3684 y(1)p 2967 3725 499 4 v 2967 3808 a Fs(\026)p Fk(f)p Fs(N)3140 3822 y Fq(A)3226 3808 y Ft(=)25 b Fs(M)10 b Fk(g)3490 3746 y Fs(:)3387 3977 y Ft(\(7.10\))382 4090 y(On)32 b(the)i(other)f(side,)g(since)f Fs(x)e Fk(!)g Fs(x)15 b Ft(log)r(\(1)p Fs(=x)p Ft(\))34 b(is)e(conca)m(v)m(e,)37 b(w)m(e)d(can)f(use)g(Jensen)f(inequalit)m(y)382 4203 y(and)e(obtain)745 4309 y Fj(X)691 4517 y Fr(M)7 b Fq(2)p Fo(M)870 4494 y Fw(A)870 4541 y Fn(`)945 4395 y Fs(\026)p Fk(f)p Fs(N)1118 4409 y Fq(A)1204 4395 y Ft(=)25 b Fs(M)10 b Fk(g)15 b Ft(log)1828 4334 y(1)p 1601 4374 V 1601 4457 a Fs(\026)p Fk(f)p Fs(N)1774 4471 y Fq(A)1860 4457 y Ft(=)25 b Fs(M)10 b Fk(g)2135 4395 y(\024)25 b Ft(log)16 b Fk(j)p Fm(M)2468 4357 y Fq(A)2468 4418 y Fr(`)2535 4395 y Fk(j)26 b Ft(=)e Fs(d)p Fk(jAj)15 b Ft(log)j Fs(`)d(:)309 b Ft(\(7.11\))382 4690 y(Prop)s(osition)20 b(3.2)j(then)f(follo)m(ws,)h (after)f(a)h(rede\014nition)c(of)j Fs(")p Ft(,)j(from)c(\(7.1\),)26 b(\(7.8\),)f(\(7.9\),)h(\(7.10\),)382 4803 y(and)k(from)g(the)g(fact)h (that)g Fk(j)p Fs(W)1417 4817 y Fr(e)1454 4803 y Fk(j)26 b(\024)f Ft(\(2)p Fs(L)c Ft(+)f(1\))1935 4770 y Fr(d")2009 4803 y Ft(.)p 3554 4803 4 62 v 3558 4745 55 4 v 3558 4803 V 3612 4803 4 62 v 382 4981 a Ff(Remark)33 b(7.3.)41 b Ft(Let)29 b(us)g(brie\015y)e(explain)h(the)h(di\013erence)f(with)g (the)h(pro)s(duct)f(case.)42 b(In)28 b(that)382 5094 y(case,)46 b(the)c(\014rst)f(term)h(in)f(\(7.7\))i(is)e(n)m(ull.)74 b(By)42 b(consequence,)k(on)41 b(can)i(c)m(ho)s(ose)f(the)h(b)s(o)m (xes)382 5207 y Fk(j)p Fs(W)493 5221 y Fr(e)530 5207 y Fk(j)27 b Ft(indep)s(enden)m(t)e(of)i Fs(L)g Ft(and)g(so)g(the)g (logarithmic)f(Sob)s(olev)g(constan)m(t)j(used)d(in)g(the)h(Herbst)382 5320 y(argumen)m(t)k(is)e(also)h(constan)m(t.)1955 5652 y(31)p eop %%Page: 32 33 32 32 bop 382 548 a Fa(References)382 751 y Ft([ABC)605 718 y Fp(+)664 751 y Ft(00])47 b(C.)d(An)m(\023)-43 b(e,)49 b(S.)44 b(Blac)m(h)m(\022)-43 b(ere,)50 b(D.)44 b(Chafa)-10 b(\177)-35 b(\020,)47 b(P)-8 b(.)45 b(F)-8 b(oug)m(\022)-43 b(eres,)50 b(I.)45 b(Gen)m(til,)i(F.)e(Malrieu,)826 864 y(C.)g(Rob)s(erto,)k(and)c(G.)h(Sc)m(he\013er,)j Fl(Sur)d(les)h(in)n (\023)-44 b(egalit)n(\023)g(es)46 b(de)h(Sob)-5 b(olev)47 b(lo)-5 b(garith-)826 977 y(miques)p Ft(,)33 b(So)s(ci)m(\023)-43 b(et)m(\023)g(e)35 b(Math)m(\023)-43 b(ematique)35 b(de)d(F)-8 b(rance,)35 b(P)m(aris,)e(2000,)i(With)d(a)h(preface)826 1090 y(b)m(y)d(Dominique)f(Bakry)i(and)f(Mic)m(hel)g(Ledoux.)382 1233 y([BCC02])108 b(L.)23 b(Bertini,)g(N.)h(Cancrini,)e(and)g(F.)i (Cesi,)g Fl(The)i(sp)-5 b(e)g(ctr)g(al)28 b(gap)f(for)f(a)g(Glaub)-5 b(er-typ)g(e)826 1346 y(dynamics)34 b(in)e(a)h(c)-5 b(ontinuous)33 b(gas)p Ft(,)e(Ann.)f(Inst.)g(H.)g(P)m(oincar)m(\023)-43 b(e)32 b(Probab.)d(Statist.)826 1458 y Ff(38)i Ft(\(2002\),)i(no.)d(1,) h(91{108.)382 1602 y([BCO99])103 b(L.)43 b(Bertini,)h(E.)f(N.)g(M.)g (Cirillo,)g(and)f(E.)h(Olivieri,)f Fl(R)-5 b(enormalization-gr)g(oup) 826 1714 y(tr)g(ansformations)28 b(under)d(str)-5 b(ong)26 b(mixing)f(c)-5 b(onditions:)39 b(Gibbsianness)25 b(and)g(c)-5 b(on-)826 1827 y(ver)g(genc)g(e)25 b(of)h(r)-5 b(enormalize)g(d)28 b(inter)-5 b(actions)p Ft(,)26 b(J.)c(Statist.)h(Ph)m(ys.)f Ff(97)h Ft(\(1999\),)k(no.)22 b(5-)826 1940 y(6,)31 b(831{915.)382 2083 y([BZ99a])139 b(L.)27 b(Bertini)e(and)h(B.)h(Zegarlinski,)f Fl(Co)-5 b(er)g(cive)29 b(ine)-5 b(qualities)30 b(for)g(Gibbs)f(me)-5 b(asur)g(es)p Ft(,)826 2196 y(J.)30 b(F)-8 b(unct.)31 b(Anal.)f Ff(162)h Ft(\(1999\),)j(no.)c(2,)h(257{286.)382 2339 y([BZ99b])133 b(L.)38 b(Bertini)f(and)h(B.)g(Zegarlinski,)g Fl(Co)-5 b(er)g(cive)41 b(ine)-5 b(qualities)40 b(for)g(Kawasaki)i(dy-) 826 2452 y(namics.)g(The)f(pr)-5 b(o)g(duct)43 b(c)-5 b(ase)p Ft(,)43 b(Mark)m(o)m(v)e(Pro)s(cess.)f(Related)f(Fields)f Ff(5)i Ft(\(1999\),)826 2565 y(no.)31 b(2,)g(125{162.)382 2708 y([CM00a])110 b(N.)35 b(Cancrini)d(and)i(F.)h(Martinelli,)e Fl(Comp)-5 b(arison)40 b(of)c(\014nite)h(volume)g(c)-5 b(anonic)g(al)826 2821 y(and)30 b(gr)-5 b(and)30 b(c)-5 b(anonic)g(al)31 b(Gibbs)e(me)-5 b(asur)g(es)31 b(under)e(a)h(mixing)f (c)-5 b(ondition)p Ft(,)29 b(Mark)m(o)m(v)826 2934 y(Pro)s(cess.)i (Related)f(Fields)f Ff(6)i Ft(\(2000\),)i(no.)d(1,)h(23{72.)382 3077 y([CM00b])104 b(N.)26 b(Cancrini)e(and)i(F.)g(Martinelli,)f Fl(On)j(the)h(sp)-5 b(e)g(ctr)g(al)31 b(gap)f(of)f(Kawasaki)h(dynam-) 826 3190 y(ics)f(under)g(a)g(mixing)g(c)-5 b(ondition)31 b(r)-5 b(evisite)g(d)p Ft(,)28 b(J.)f(Math.)g(Ph)m(ys.)f Ff(41)h Ft(\(2000\),)j(no.)c(3,)826 3303 y(1391{1423.)382 3446 y([CMR02])88 b(N.)40 b(Cancrini,)g(F.)h(Martinelli,)f(and)f(C.)h (Rob)s(erto,)j Fl(The)e(lo)-5 b(garithmic)44 b(Sob)-5 b(olev)826 3559 y(c)g(onstant)51 b(of)e(Kawasaki)i(dynamics)f(under)f (a)h(mixing)f(c)-5 b(ondition)50 b(r)-5 b(evisite)g(d)p Ft(,)826 3672 y(Ann.)30 b(Inst.)g(H.)h(P)m(oincar)m(\023)-43 b(e)31 b(Probab.)f(Statist.)h Ff(38)g Ft(\(2002\),)i(no.)d(4,)h (385{436.)382 3815 y([Da)m(v89])145 b(E.)28 b(B.)g(Da)m(vies,)h Fl(He)-5 b(at)30 b(kernels)g(and)h(sp)-5 b(e)g(ctr)g(al)33 b(the)-5 b(ory)p Ft(,)30 b(Cam)m(bridge)c(Univ.)h(Press,)826 3928 y(1989.)382 4071 y([DS87])184 b(R.)41 b(L.)f(Dobrushin)f(and)h(S.) g(B.)h(Shlosman,)h Fl(Completely)i(analytic)-5 b(al)44 b(inter)-5 b(ac-)826 4184 y(tions:)65 b(c)-5 b(onstructive)44 b(description)p Ft(,)j(J.)c(Statist.)f(Ph)m(ys.)h Ff(46)g Ft(\(1987\),)48 b(no.)42 b(5-6,)826 4296 y(983{1014.)382 4440 y([JLQY99])61 b(E.)42 b(Jan)m(vresse,)k(C.)c(Landim,)i(J.)e (Quastel,)j(and)d(H.)g(T.)g(Y)-8 b(au,)46 b Fl(R)-5 b(elaxation)46 b(to)826 4552 y(e)-5 b(quilibrium)48 b(of)h(c)-5 b(onservative)49 b(dynamics.)g(I.)f(Zer)-5 b(o-r)g(ange)49 b(pr)-5 b(o)g(c)g(esses)p Ft(,)54 b(Ann.)826 4665 y(Probab.)30 b Ff(27)h Ft(\(1999\),)i(no.)d(1,) h(325{360.)382 4808 y([Led99])156 b(M.)24 b(Ledoux,)h Fl(Conc)-5 b(entr)g(ation)29 b(of)e(me)-5 b(asur)g(e)28 b(and)f(lo)-5 b(garithmic)28 b(Sob)-5 b(olev)28 b(ine)-5 b(quali-)826 4921 y(ties)p Ft(,)27 b(S)m(\023)-43 b(eminaire)25 b(de)h(Probabilit)m(\023)-43 b(es.)25 b(XXXIII.)h(Lecture)g(Notes)h(in) e(Mathematics)826 5034 y(1709)32 b(\(Berlin\),)e(Springer,)e(Berlin,)h (1999,)k(pp.)c(120{216.)382 5177 y([L)-8 b(Y93])187 b(S.)26 b(L.)h(Lu)e(and)h(H.)h(T.)f(Y)-8 b(au,)28 b Fl(Sp)-5 b(e)g(ctr)g(al)31 b(Gap)f(and)g(L)-5 b(o)g(garithmic)32 b(Sob)-5 b(olev)30 b(Ine)-5 b(qual-)826 5290 y(ity)38 b(for)g(Kawasaki)h(and)g(Glaub)-5 b(er)39 b(Dynamics)p Ft(,)f(Comm)m(un.)d(Math.)i(Ph)m(ys.)f Ff(156)826 5403 y Ft(\(1993\),)d(399{433.)1955 5652 y(32)p eop %%Page: 33 34 33 33 bop 382 548 a Ft([L)-8 b(Y03])187 b(C.)31 b(Landim)f(and)g(H.)i (T.)f(Y)-8 b(au,)33 b Fl(Conver)-5 b(genc)g(e)34 b(to)g(e)-5 b(quilibrium)33 b(of)h(c)-5 b(onservative)826 661 y(p)g(article)34 b(systems)g(on)f Fm(Z)1680 628 y Fr(d)1716 661 y Ft(,)d(Ann.)g(Probab.) g Ff(31)h Ft(\(2003\),)i(no.)e(1,)g(115{147.)382 807 y([Mar99])140 b(F.)47 b(Martinelli,)h Fl(L)-5 b(e)g(ctur)g(es)48 b(on)f(Glaub)-5 b(er)48 b(dynamics)h(for)e(discr)-5 b(ete)48 b(spin)g(mo)-5 b(d-)826 920 y(els)p Ft(,)38 b(Lectures)f(on)g (probabilit)m(y)d(theory)i(and)g(statistics)h(\(Sain)m(t-Flour,)g (1997\).)826 1033 y(Lecture)k(Notes)g(in)e(Mathematics)i(1717)h (\(Berlin\),)g(Springer,)e(Berlin,)i(1999,)826 1146 y(pp.)30 b(93{191.)382 1292 y([Sp)s(o91])154 b(H.)53 b(Sp)s(ohn,)k Fl(L)-5 b(ar)g(ge)55 b(sc)-5 b(ale)54 b(dynamics)h(of)f(inter)-5 b(acting)54 b(p)-5 b(articles)p Ft(,)61 b(Springer-)826 1405 y(V)-8 b(erlag,)31 b(1991.)382 1551 y([VY97])168 b(S.)36 b(R.)h(S.)f(V)-8 b(aradhan)37 b(and)f(Horng-Tzer)h(Y)-8 b(au,)39 b Fl(Di\013usive)e(limit)i(of)f(lattic)-5 b(e)39 b(gas)826 1664 y(with)34 b(mixing)e(c)-5 b(onditions)p Ft(,)33 b(Asian)c(J.)i(Math.)g Ff(1)f Ft(\(1997\),)j(no.)e(4,)g (623{678.)382 1810 y([Y)-8 b(au96])148 b(H.)30 b(T.)f(Y)-8 b(au,)30 b Fl(L)-5 b(o)g(garithmic)33 b(Sob)-5 b(olev)33 b(ine)-5 b(quality)32 b(for)g(lattic)-5 b(e)32 b(gases)g(with)h(mixing) 826 1923 y(c)-5 b(onditions)p Ft(,)33 b(Comm)m(un.)c(Math.)i(Ph)m(ys.)g Ff(181)g Ft(\(1996\),)i(367{408.)1955 5652 y(33)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF