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Fv(These)j(lemmas)f(giv)o(e)i(useful)f(form)o (ulas)f(for)h(the)h(in)o(v)m(arian)o(t)e(measure)h(and)g(for)g(the)h(la)o(ws) f(of)h(the)0 355 y(exit)15 b(time)f(and)g(exit)h(p)q(oin)o(t)f(for)g(an)g (arbitrary)f(subset)g(of)i Fs(E)s Fv(.)20 b(These)14 b(form)o(ulas)f(are)g (rational)h(fractions)0 414 y(of)k(the)h(co)q(e\016cien)o(ts)e(of)i(the)f (transition)f(matrix)g(whose)h(n)o(umerators)e(and)h(denominators)f(are)i (most)0 474 y(con)o(v)o(enien)o(tly)d(written)i(as)f(sums)f(o)o(v)o(er)h (particular)f(t)o(yp)q(es)i(of)f(graphs.)0 580 y Fx(De\014nition)21 b(3.1.)27 b Fv(\(the)17 b(graphs)e Fs(G)p Fv(\()p Fs(W)7 b Fv(\)\))0 640 y(Let)17 b Fs(W)24 b Fv(b)q(e)17 b(an)f(arbitrary)f(non{empt)o (y)g(subset)h(of)h Fs(E)s Fv(.)0 699 y(An)g(orien)o(ted)e(graph)g(on)i Fs(E)i Fv(is)d(called)g(a)h Fs(W)7 b Fv({graph)15 b(if)h(and)g(only)h(if)60 759 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(arro)o(w)f(starting)h(from)g(a)g (p)q(oin)o(t)g(of)h Fs(W)60 819 y Fr(\017)49 b Fv(eac)o(h)17 b(p)q(oin)o(t)f(of)g Fs(W)487 801 y Fo(c)524 819 y Fv(is)g(the)h(initial)f(p) q(oin)o(t)g(of)h(exactly)g(one)g(arro)o(w)60 879 y Fr(\017)49 b Fv(for)17 b(eac)o(h)f(p)q(oin)o(t)g Fs(x)h Fv(in)f Fs(W)610 861 y Fo(c)630 879 y Fv(,)h(there)f(exists)h(a)f(path)g(in)h(the)f(graph)g (leading)f(from)h Fs(x)h Fv(to)g Fs(W)7 b Fv(.)0 938 y(The)16 b(set)h(of)g(all)f Fs(W)7 b Fv({graphs)15 b(is)h(denoted)g(b)o(y)g Fs(G)p Fv(\()p Fs(W)7 b Fv(\).)0 1044 y Fj(R)m(emark.)27 b Fv(The)16 b(third)g(condition)f(ab)q(o)o(v)o(e)h(is)g(equiv)m(alen)o(t)h(to) 60 1104 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(cycle)h(in)f(the)h(graph.)0 1210 y Fx(De\014nition)k(3.2.)27 b Fv(\(the)17 b(graphs)e Fs(G)690 1217 y Fo(x;y)749 1210 y Fv(\()p Fs(W)7 b Fv(\)\))0 1269 y(Let)17 b Fs(W)24 b Fv(b)q(e)17 b(an)f(arbitrary)f(non{empt)o(y)g(subset)h(of)h Fs(E)s Fv(,)f(let)h Fs(x)g Fv(b)q(elong)f(to)h Fs(E)i Fv(and)d Fs(y)j Fv(to)e Fs(W)7 b Fv(.)0 1329 y(If)17 b Fs(x)g Fv(b)q(elongs)f(to)h Fs(W)389 1311 y Fo(c)409 1329 y Fv(,)f(the)h(set)g Fs(G)642 1336 y Fo(x;y)700 1329 y Fv(\()p Fs(W)7 b Fv(\))18 b(is)e(the)h(set)f(of)h (all)f(orien)o(ted)g(graphs)f(on)h Fs(E)j Fv(suc)o(h)c(that)60 1389 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(arro)o(w)f(starting)h(from)g(a)g (p)q(oin)o(t)g(of)h Fs(W)60 1449 y Fr(\017)49 b Fv(eac)o(h)17 b(p)q(oin)o(t)f(of)g Fs(W)487 1431 y Fo(c)524 1449 y Fv(is)g(the)h(initial)f (p)q(oin)o(t)g(of)h(exactly)g(one)g(arro)o(w)60 1509 y Fr(\017)49 b Fv(for)17 b(eac)o(h)f(p)q(oin)o(t)g Fs(z)j Fv(in)d Fs(W)607 1490 y Fo(c)627 1509 y Fv(,)h(there)f(exists)h(a)f(path)g(in)g(the)h(graph)f (leading)f(from)h Fs(z)j Fv(to)e Fs(W)60 1568 y Fr(\017)49 b Fv(there)17 b(exists)f(a)h(path)f(in)g(the)h(graph)e(leading)h(from)g Fs(x)h Fv(to)g Fs(y)r Fv(.)0 1628 y(More)f(concisely)l(,)g(they)h(are)f(the)h (graphs)e(of)h Fs(G)p Fv(\()p Fs(W)7 b Fv(\))18 b(whic)o(h)d(con)o(tain)h(a)h (path)f(leading)f(from)h Fs(x)h Fv(to)g Fs(y)r Fv(.)0 1688 y(If)g Fs(x)g Fv(b)q(elongs)f(to)h Fs(W)7 b Fv(,)16 b(the)h(set)g Fs(G)622 1695 y Fo(x;y)680 1688 y Fv(\()p Fs(W)7 b Fv(\))17 b(is)f(empt)o(y)h(if)f Fs(x)f Fr(6)p Fv(=)e Fs(y)19 b Fv(and)d(is)g(equal)g (to)h Fs(G)p Fv(\()p Fs(W)7 b Fv(\))18 b(if)e Fs(x)f Fv(=)e Fs(y)r Fv(.)0 1794 y Fj(R)m(emark.)27 b Fv(The)e(graphs)g(in)h Fs(G)597 1801 y Fo(x;y)655 1794 y Fv(\()p Fs(W)7 b Fv(\))27 b(ha)o(v)o(e)e(no)h(cycles.)51 b(F)l(or)25 b(an)o(y)g Fs(x)i Fv(in)e Fs(E)k Fv(and)d Fs(y)i Fv(in)e Fs(W)7 b Fv(,)28 b(the)0 1853 y(set)17 b Fs(G)117 1860 y Fo(x;y)175 1853 y Fv(\()p Fs(W)7 b Fv(\))18 b(is)e(included)f(in)h Fs(G)p Fv(\()p Fs(W)7 b Fv(\).)0 1959 y Fx(De\014nition)21 b(3.3.)27 b Fv(\(the)17 b(graphs)e Fs(G)p Fv(\()p Fs(x)g Fr(6!)f Fs(W)7 b Fv(\)\))0 2019 y(Let)17 b Fs(W)24 b Fv(b)q(e)17 b(an)f(arbitrary)f(non{empt)o(y)g(subset)h(of)h Fs(E)i Fv(and)d(let)h Fs(x)g Fv(b)q(e)g(a)f(p)q(oin)o(t)g(of)h Fs(E)s Fv(.)0 2079 y(If)g Fs(x)g Fv(b)q(elongs)f(to)h Fs(W)23 b Fv(the)17 b(set)g Fs(G)p Fv(\()p Fs(x)e Fr(6!)e Fs(W)7 b Fv(\))17 b(is)f(empt)o(y)l(.)0 2138 y(If)h Fs(x)g Fv(b)q(elongs)f(to)h Fs(W)389 2120 y Fo(c)425 2138 y Fv(the)g(set)g Fs(G)p Fv(\()p Fs(x)e Fr(6!)e Fs(W)7 b Fv(\))17 b(is)f(the)h(set)g(of)g(all)f(orien)o(ted)f (graphs)g(on)i Fs(E)i Fv(suc)o(h)c(that)60 2198 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(arro)o(w)f(starting)h(from)g(a)g(p)q(oin)o(t)g(of)h Fs(W)60 2258 y Fr(\017)49 b Fv(eac)o(h)17 b(p)q(oin)o(t)f(of)g Fs(W)487 2240 y Fo(c)524 2258 y Fv(except)i(one,)e(sa)o(y)g Fs(y)r Fv(,)g(is)h(the)f(initial)g(p)q(oin)o(t)g(of)h(exactly)h(one)e(arro)o (w)60 2318 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(cycle)h(in)f(the)h(graph)60 2377 y Fr(\017)49 b Fv(there)17 b(is)f(no)g(path)g(in)h(the)f(graph)g (leading)f(from)h Fs(x)h Fv(to)g Fs(W)7 b Fv(.)0 2437 y(The)16 b(third)g(condition)g(\(no)g(cycle\))i(is)e(equiv)m(alen)o(t)g(to)60 2497 y Fr(\017)49 b Fv(for)17 b(eac)o(h)f Fs(z)j Fv(in)d Fs(W)477 2479 y Fo(c)508 2497 y Fr(n)11 b(f)p Fs(y)r Fr(g)p Fv(,)17 b(there)f(is)g(a)h(path)f(in)g(the)h(graph)e(leading)h(from)f Fs(z)k Fv(to)e Fs(W)h Fr([)12 b(f)p Fs(y)r Fr(g)p Fv(.)964 2670 y Ft(4)p eop %%Page: 5 5 5 4 bop 0 200 a Fx(Lemma)17 b(3.4.)28 b Fi(Let)17 b Fs(W)24 b Fi(b)q(e)16 b(an)h(arbitrary)e(non{empt)o(y)g(subset)h(of)h Fs(E)i Fi(and)d(let)h Fs(x)g Fi(b)q(e)g(a)f(p)q(oin)o(t)g(of)h Fs(E)s Fi(.)0 260 y(The)f(set)h Fs(G)p Fv(\()p Fs(x)e Fr(6!)f Fs(W)7 b Fv(\))17 b Fi(is)f(the)g(union)g(of)h(all)f(the)h(sets)f Fs(G)1057 267 y Fo(x;y)1115 260 y Fv(\()p Fs(W)j Fr([)11 b(f)p Fs(y)r Fr(g)p Fv(\))p Fs(;)17 b(y)f Fr(2)e Fs(W)1511 242 y Fo(c)1531 260 y Fi(.)0 347 y Fj(R)m(emark.)27 b Fv(In)22 b(the)h(case)f Fs(x)j Fr(2)f Fs(W)643 329 y Fo(c)663 347 y Fs(;)8 b(y)27 b Fr(2)d Fs(W)7 b Fv(,)24 b(the)f(de\014nitions)e(of)i Fs(G)1326 354 y Fo(x;y)1384 347 y Fv(\()p Fs(W)7 b Fv(\))24 b(and)e Fs(G)p Fv(\()p Fs(x)j Fr(6!)e Fs(W)7 b Fv(\))23 b(are)0 407 y(those)16 b(giv)o(en)g(b)o(y)h(W)l(en)o(tzell)f(and)g(F)l(reidlin)e([10].)22 b(W)l(e)17 b(ha)o(v)o(e)e(extended)i(these)g(de\014nitions)e(to)i(co)o(v)o (er)e(all)0 466 y(p)q(ossible)f(v)m(alues)i(of)g Fs(x)p Fv(.)22 b(With)16 b(our)e(c)o(hoice)h(for)h(the)g(de\014nition)e(of)i(the)g(time)f (of)h(exit)g Fs(\034)6 b Fv(\()p Fs(W)1704 448 y Fo(c)1724 466 y Fv(\))17 b(\(the)f(\014rst)0 526 y(time)f(greater)g(than)f(or)h(equal)g (to)g(zero)g(when)g(the)g(c)o(hain)f(is)h(outside)f Fs(W)1363 508 y Fo(c)1384 526 y Fv(\),)h(the)h(form)o(ulas)d(for)h(the)i(la)o(w)0 586 y(of)h Fs(X)98 595 y Fo(\034)t Fl(\()p Fo(W)178 585 y Fh(c)197 595 y Fl(\))232 586 y Fv(and)f(for)g(the)h(exp)q(ectation)g(of)g Fs(\034)6 b Fv(\()p Fs(W)917 568 y Fo(c)937 586 y Fv(\))17 b(will)f(remain)f(v)m(alid)i(in)f(all)g(cases.)60 673 y(Let)h Fs(g)h Fv(b)q(e)f(a)f(graph)g(on)g Fs(E)s Fv(,)g(w)o(e)g(de\014ne)g(its)h (probabilit)o(y)d Fs(p)1147 680 y Fo(\014)1174 673 y Fv(\()p Fs(g)r Fv(\))j(b)o(y)547 794 y Fs(p)572 801 y Fo(\014)599 794 y Fv(\()p Fs(g)r Fv(\))22 b(=)771 746 y Fg(Y)721 855 y Fl(\()p Fo(x)p Fn(!)p Fo(y)q Fl(\))p Fn(2)p Fo(g)892 794 y Fs(P)924 801 y Fo(\014)951 794 y Fv(\()p Fs(X)1011 801 y Fo(n)p Fl(+1)1103 794 y Fv(=)13 b Fs(y)r(=X)1247 801 y Fo(n)1289 794 y Fv(=)g Fs(x)p Fv(\))p Fs(:)0 960 y Fx(Lemma)k(3.5.)28 b Fi(\(exp)q(ected)18 b(n)o(um)o(b)q(er)c(of)j(visits)f(b)q(efore)h(exit\))0 1020 y(F)l(or)e(an)o(y)h(non{empt)o(y)g(subset)g Fs(W)23 b Fi(of)17 b Fs(E)s Fi(,)f Fs(y)j Fi(in)d Fs(W)946 1002 y Fo(c)983 1020 y Fi(and)g Fs(x)h Fi(in)f Fs(E)s Fi(,)324 1169 y Fn(1)308 1184 y Fg(X)307 1290 y Fo(n)p Fl(=0)390 1232 y Fs(P)422 1239 y Fo(\014)449 1232 y Fv(\()p Fs(X)509 1239 y Fo(n)550 1232 y Fv(=)e Fs(y)r(;)j(\034)6 b Fv(\()p Fs(W)761 1211 y Fo(c)781 1232 y Fv(\))14 b Fs(>)f(n=X)962 1239 y Fl(0)999 1232 y Fv(=)g Fs(x)p Fv(\))32 b(=)1316 1088 y Fg(X)1205 1197 y Fo(g)q Fn(2)p Fo(G)1283 1202 y Fh(x;y)1336 1197 y Fl(\()p Fo(W)5 b Fn([f)p Fo(y)q Fn(g)p Fl(\))1507 1135 y Fs(p)1532 1142 y Fo(\014)1559 1135 y Fv(\()p Fs(g)r Fv(\))p 1205 1220 419 2 v 1316 1231 a Fg(X)1276 1340 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(W)g Fl(\))1436 1278 y Fs(p)1461 1285 y Fo(\014)1488 1278 y Fv(\()p Fs(g)r Fv(\))1630 1232 y Fs(:)0 1445 y Fv(This)11 b(lemma,)h(due)f(to)i(Catoni)e([5],)i(ma)o(y)e(b)q(e)h(used)g(to)g(pro)o(v)o (e)f(the)h(follo)o(wing)f(three)g(lemmas)g(of)h(W)l(en)o(tzell)0 1505 y(and)k(F)l(reidlin.)0 1592 y Fx(Lemma)h(3.6.)28 b Fi(\(stationary)16 b(measure\))0 1652 y(The)g(stationary)g(measure)g Fs(\026)564 1659 y Fo(\014)607 1652 y Fi(of)h(the)g(Mark)o(o)o(v)e(c)o(hain)g Fv(\(\()p Fs(X)1141 1659 y Fo(n)1170 1652 y Fv(\))1189 1659 y Fo(n)p Fn(2)p Fm(N)1268 1652 y Fs(;)8 b(P)1322 1659 y Fo(\014)1349 1652 y Fv(\))17 b Fi(is)550 1867 y Fr(8)p Fs(x)c Fr(2)h Fs(E)103 b(\026)837 1874 y Fo(\014)864 1867 y Fv(\()p Fs(x)p Fv(\))31 b(=)1110 1724 y Fg(X)1080 1834 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(x)p Fl(\))1220 1772 y Fs(p)1245 1779 y Fo(\014)1272 1772 y Fv(\()p Fs(g)r Fv(\))p 1036 1855 344 2 v 1040 1866 a Fg(X)1036 1973 y Fo(y)q Fn(2)p Fo(E)1154 1866 y Fg(X)1124 1975 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(y)q Fl(\))1264 1914 y Fs(p)1289 1921 y Fo(\014)1316 1914 y Fv(\()p Fs(g)r Fv(\))1386 1867 y Fs(:)0 2108 y Fx(Lemma)17 b(3.7.)28 b Fi(\(exit)17 b(p)q(oin)o(t\))0 2167 y(F)l(or)e(an)o(y)h(non{empt)o(y)g(subset)g Fs(W)23 b Fi(of)17 b Fs(E)s Fi(,)f(an)o(y)g Fs(y)j Fi(in)d Fs(W)24 b Fi(and)15 b Fs(x)j Fi(in)e Fs(E)s Fi(,)481 2379 y Fs(P)513 2386 y Fo(\014)540 2379 y Fv(\()p Fs(X)600 2388 y Fo(\034)t Fl(\()p Fo(W)680 2378 y Fh(c)700 2388 y Fl(\))732 2379 y Fv(=)e Fs(y)r(=X)877 2386 y Fl(0)913 2379 y Fv(=)g Fs(x)p Fv(\))31 b(=)1186 2236 y Fg(X)1119 2345 y Fo(g)q Fn(2)p Fo(G)1197 2350 y Fh(x;y)1250 2345 y Fl(\()p Fo(W)5 b Fl(\))1333 2283 y Fs(p)1358 2290 y Fo(\014)1385 2283 y Fv(\()p Fs(g)r Fv(\))p 1119 2368 330 2 v 1186 2378 a Fg(X)1146 2488 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(W)g Fl(\))1306 2426 y Fs(p)1331 2433 y Fo(\014)1358 2426 y Fv(\()p Fs(g)r Fv(\))1455 2379 y Fs(:)964 2670 y Ft(5)p eop %%Page: 6 6 6 5 bop 0 200 a Fx(Lemma)17 b(3.8.)28 b Fi(\(exit)17 b(time\))0 260 y(F)l(or)e(an)o(y)h(subset)g Fs(W)24 b Fi(of)17 b Fs(E)i Fi(and)d Fs(x)h Fi(in)f Fs(E)s Fi(,)180 477 y Fs(E)217 484 y Fo(\014)244 477 y Fv(\()p Fs(\034)6 b Fv(\()p Fs(W)364 457 y Fo(c)384 477 y Fv(\))p Fs(=X)469 484 y Fl(0)506 477 y Fv(=)14 b Fs(x)p Fv(\))31 b(=)731 334 y Fg(X)712 441 y Fo(y)q Fn(2)p Fo(W)802 431 y Fh(c)991 334 y Fg(X)880 443 y Fo(g)q Fn(2)p Fo(G)958 448 y Fh(x;y)1011 443 y Fl(\()p Fo(W)5 b Fn([f)p Fo(y)q Fn(g)p Fl(\))1183 381 y Fs(p)1208 388 y Fo(\014)1235 381 y Fv(\()p Fs(g)r Fv(\))p 712 466 587 2 v 907 477 a Fg(X)867 586 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(W)g Fl(\))1028 524 y Fs(p)1053 531 y Fo(\014)1080 524 y Fv(\()p Fs(g)r Fv(\))1335 477 y(=)1482 335 y Fg(X)1411 444 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(x)p Fn(6!)p Fo(W)g Fl(\))1634 382 y Fs(p)1659 389 y Fo(\014)1686 382 y Fv(\()p Fs(g)r Fv(\))p 1411 466 340 2 v 1482 477 a Fg(X)1442 586 y Fo(g)q Fn(2)p Fo(G)p Fl(\()p Fo(W)g Fl(\))1602 524 y Fs(p)1627 531 y Fo(\014)1654 524 y Fv(\()p Fs(g)r Fv(\))1756 477 y Fs(:)620 751 y Fw(4.)27 b(The)18 b(cycle)g(decomposition)60 841 y Fv(W)l(e)g(no)o(w)f(recall)h(brie\015y)f(some)h(basic)f(facts)i(and)e (de\014nitions)g(of)h(quan)o(tities)f(concerning)g(the)i(de-)0 901 y(comp)q(osition)13 b(of)h Fs(E)i Fv(in)o(to)e(cycles.)21 b(F)l(or)13 b(a)g(detailed)h(exp)q(osition)g(of)g(this)f(question,)h(w)o(e)g (refer)f(the)h(reader)0 960 y(to)j(Catoni)f([5,6])g(and)g(T)l(rouv)o(\023)-24 b(e)16 b([20,21].)0 1049 y Fx(De\014nition)21 b(4.1.)27 b Fv(\(cost)17 b(of)g(a)g(graph\))0 1109 y(W)l(e)g(de\014ne)f(the)g(cost)h(of)g(a)f(graph)g Fs(g)i Fv(o)o(v)o(er)e Fs(E)j Fv(b)o(y)724 1231 y Fs(V)11 b Fv(\()p Fs(g)r Fv(\))22 b(=)931 1183 y Fg(X)886 1292 y Fl(\()p Fo(x)p Fn(!)p Fo(y)q Fl(\))p Fn(2)p Fo(g)1057 1231 y Fs(V)11 b Fv(\()p Fs(x;)d(y)r Fv(\))p Fs(:)0 1403 y Fv(Clearly)l(,)16 b(w)o(e)g(ha)o(v)o(e)g(lim)440 1410 y Fo(\014)r Fn(!1)555 1403 y Fv(ln)8 b Fs(p)630 1410 y Fo(\014)656 1403 y Fv(\()p Fs(g)r Fv(\))p Fs(=\014)17 b Fv(=)d Fr(\000)p Fs(V)d Fv(\()p Fs(g)r Fv(\).)60 1492 y(W)l(e)17 b(next)g(de\014ne)f(a)g(delicate)h(but)f(extremely) h(useful)f(to)q(ol.)0 1580 y Fx(De\014nition)21 b(4.2.)27 b Fv(\(renormalized)15 b(comm)o(unication)f(cost\))0 1640 y(Lemma)i(3)p Fs(:)p Fv(7)p Fs(:)g Fv(implies)f(the)i(existence)g(of)f(the)h(limits)f(for)g (an)o(y)g(subset)g Fs(D)i Fv(of)f Fs(E)0 1775 y Fr(8)p Fs(x)c Fr(2)h Fs(D)51 b Fr(8)p Fs(y)15 b Fr(62)f Fs(D)127 b Fv(lim)475 1808 y Fo(\014)r Fn(!1)587 1775 y Fr(\000)635 1742 y Fv(1)p 632 1764 31 2 v 632 1810 a Fs(\014)677 1775 y Fv(ln)8 b Fs(P)759 1782 y Fo(\014)785 1775 y Fv(\()p Fs(X)845 1784 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))950 1775 y Fv(=)13 b Fs(y)r(=X)1094 1782 y Fl(0)1131 1775 y Fv(=)g Fs(x)p Fv(\))32 b(=)e Fs(C)1367 1782 y Fo(D)1402 1775 y Fv(\()p Fs(x;)8 b(y)r Fv(\))686 1877 y(=)30 b(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))k(:)e Fs(g)j Fr(2)e Fs(G)1143 1884 y Fo(x;y)1201 1877 y Fv(\()p Fs(D)1262 1857 y Fo(c)1283 1877 y Fv(\))8 b Fr(g)20 b(\000)g Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1862 1857 y Fo(c)1884 1877 y Fv(\))8 b Fr(g)p Fs(;)0 1977 y Fr(8)p Fs(x)13 b Fr(62)h Fs(D)51 b Fr(8)p Fs(y)15 b Fr(62)f Fs(D)127 b Fv(lim)475 2010 y Fo(\014)r Fn(!1)587 1977 y Fr(\000)635 1943 y Fv(1)p 632 1966 V 632 2011 a Fs(\014)677 1977 y Fv(ln)8 b Fs(P)759 1984 y Fo(\014)785 1977 y Fv(\()p Fs(X)845 1986 y Fo(\034)t Fl(\()p Fo(D)q(;)p Fl(1\))981 1977 y Fv(=)14 b Fs(y)r(=X)1126 1984 y Fl(0)1162 1977 y Fv(=)g Fs(x)p Fv(\))31 b(=)f Fs(C)1398 1984 y Fo(D)1434 1977 y Fv(\()p Fs(x;)8 b(y)r Fv(\))182 2079 y(=)30 b(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(V)g Fv(\()p Fs(g)r Fv(\))j(:)g Fs(z)i Fr(2)e Fs(D)f Fr([)e(f)p Fs(y)r Fr(g)p Fs(;)16 b(g)g Fr(2)e Fs(G)1145 2086 y Fo(z)q(;y)1201 2079 y Fv(\()p Fs(D)1262 2058 y Fo(c)1283 2079 y Fv(\))8 b Fr(g)20 b(\000)g Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1862 2058 y Fo(c)1884 2079 y Fv(\))8 b Fr(g)p Fs(:)0 2193 y Fv(F)l(or)16 b Fs(x)i Fv(in)e Fs(E)k Fv(and)d Fs(y)i Fv(in)e Fs(D)i Fv(w)o(e)e(set)g Fs(C)699 2200 y Fo(D)735 2193 y Fv(\()p Fs(x;)8 b(y)r Fv(\))16 b(=)f Fs(C)955 2202 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)1073 2193 y Fv(\()p Fs(x;)8 b(y)r Fv(\))19 b(so)e(that)g Fs(C)1412 2200 y Fo(D)1465 2193 y Fv(is)g(de\014ned)f(o)o(v)o(er)g Fs(E)e Fr(\002)e Fs(E)s Fv(.)0 2253 y(The)k(quan)o(tit)o(y)h Fs(C)338 2260 y Fo(D)373 2253 y Fv(\()p Fs(x;)8 b(y)r Fv(\))19 b(is)d(called)g(the)h (renormalized)d(comm)o(unication)g(cost)j(in)f Fs(D)q Fv(.)0 2313 y(F)l(or)f(a)i(set)g Fs(A)p Fv(,)f(w)o(e)h(de\014ne)f(also)g Fs(C)630 2320 y Fo(D)665 2313 y Fv(\()p Fs(x;)8 b(A)p Fv(\))16 b(=)d(min)o Fr(f)8 b Fs(C)1010 2320 y Fo(D)1046 2313 y Fv(\()p Fs(x;)g(y)r Fv(\))16 b(:)e Fs(y)i Fr(2)e Fs(A)8 b Fr(g)p Fv(.)0 2402 y Fj(R)m(emark.)27 b Fv(The)14 b(\014rst)f(form)o(ula)f(for)i(the)h (renormalized)d(comm)o(unication)g(cost)i(\(corresp)q(onding)e(to)j(the)0 2461 y(case)h Fs(x)f Fr(2)f Fs(D)q(;)j(y)f Fr(62)e Fs(D)q Fv(\))k(is)e(the)g (form)o(ula)f(giv)o(en)h(b)o(y)g(W)l(en)o(tzell)h(and)f(F)l(reidlin.)0 2550 y Fj(R)m(emark.)27 b Fv(F)l(or)15 b(an)o(y)h(set)h Fs(D)h Fv(and)e(an)o(y)g Fs(x)h Fv(in)f Fs(E)s Fv(,)g(w)o(e)h(ha)o(v)o(e)e Fs(C)1116 2557 y Fo(D)1152 2550 y Fv(\()p Fs(x;)8 b(x)p Fv(\))16 b(=)d(0.)964 2670 y Ft(6)p eop %%Page: 7 7 7 6 bop 0 200 a Fx(Lemma)17 b(4.3.)50 b Fi(F)l(or)15 b Fs(x)g Fr(62)f Fs(D)k Fi(and)e Fs(y)g Fr(62)e Fs(D)q Fi(,)407 303 y Fs(C)443 310 y Fo(D)479 303 y Fv(\()p Fs(x;)8 b(y)r Fv(\))24 b(=)e(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(C)1046 310 y Fo(D)1081 303 y Fv(\()p Fs(z)r(;)d(y)r Fv(\))16 b(:)d Fs(z)k Fr(2)d Fs(D)e Fr([)g(f)p Fs(y)r Fr(g)c(g)p Fs(:)0 407 y Fi(F)l(or)15 b Fs(x)g Fr(2)f Fs(D)q Fi(,)j Fs(y)f Fr(2)e Fs(D)k Fi(and)e Fs(z)g Fr(62)e Fs(D)q Fi(,)617 510 y Fs(C)653 517 y Fo(D)689 510 y Fv(\()p Fs(x;)8 b(z)r Fv(\))24 b Fr(\024)e Fs(C)923 517 y Fo(D)958 510 y Fv(\()p Fs(x;)8 b(y)r Fv(\))13 b(+)e Fs(C)1171 517 y Fo(D)1207 510 y Fv(\()p Fs(y)r(;)d(z)r Fv(\))p Fs(:)0 640 y Fj(Pr)m(o)m(of.)28 b Fv(The)12 b(\014rst)g(equation)g (is)g(a)g(direct)g(consequence)g(of)h(the)g(expression)e(of)h Fs(C)1509 647 y Fo(D)1545 640 y Fv(\()p Fs(x;)c(y)r Fv(\))14 b(and)e Fs(C)1802 647 y Fo(D)1838 640 y Fv(\()p Fs(z)r(;)c(y)r Fv(\))0 700 y(in)16 b(terms)g(of)h(graphs.)j(A)d(probabilistic)d(pro)q(of)j (is)f(also)g(p)q(ossible:)k(let)d Fs(x;)8 b(y)20 b Fv(b)q(elong)c(to)h Fs(D)1688 682 y Fo(c)1709 700 y Fv(;)f(w)o(e)g(ha)o(v)o(e)107 811 y Fs(P)139 818 y Fo(\014)166 811 y Fv(\()p Fs(X)226 820 y Fo(\034)t Fl(\()p Fo(D)q(;)p Fl(1\))362 811 y Fv(=)e Fs(y)r(=X)507 818 y Fl(0)543 811 y Fv(=)g Fs(x)p Fv(\))23 b(=)776 764 y Fg(X)727 873 y Fo(z)q Fn(2)p Fo(D)q Fn([f)p Fo(y)q Fn(g)905 811 y Fs(P)937 818 y Fo(\014)964 811 y Fv(\()p Fs(X)1024 820 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))1128 811 y Fv(=)13 b Fs(y)r(=X)1272 818 y Fl(0)1309 811 y Fv(=)h Fs(z)r Fv(\))p Fs(P)1438 818 y Fo(\014)1465 811 y Fv(\()p Fs(X)1525 818 y Fl(1)1562 811 y Fv(=)g Fs(z)r(=X)1706 818 y Fl(0)1742 811 y Fv(=)g Fs(x)p Fv(\))0 970 y(and)i(the)h(\014rst)e (equation)i(follo)o(ws)e(letting)i Fs(\014)j Fv(tend)c(to)h(in\014nit)o(y)l (.)0 1030 y(No)o(w)f(let)h Fs(x)e Fr(2)f Fs(D)q(;)j(y)f Fr(2)e Fs(D)k Fv(and)e Fs(z)g Fr(2)e Fs(D)719 1011 y Fo(c)740 1030 y Fv(.)22 b(W)l(e)17 b(can)f(decomp)q(ose)g Fs(P)1236 1037 y Fo(\014)1263 1030 y Fv(\()p Fs(X)1323 1039 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))1427 1030 y Fv(=)e Fs(z)r(=X)1571 1037 y Fl(0)1607 1030 y Fv(=)g Fs(x)p Fv(\))j(as)59 1133 y Fs(P)91 1140 y Fo(\014)118 1133 y Fv(\()p Fs(X)178 1142 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))365 1133 y Fv(=)d Fs(y)r(;)i(X)515 1142 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))619 1133 y Fv(=)e Fs(z)r(=X)763 1140 y Fl(0)799 1133 y Fv(=)g Fs(x)p Fv(\))20 b(+)f Fs(P)1009 1140 y Fo(\014)1036 1133 y Fv(\()p Fs(X)1096 1142 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))1283 1133 y Fv(=)13 b Fs(z)r(=X)1426 1140 y Fl(0)1463 1133 y Fv(=)h Fs(x)p Fv(\))711 1211 y Fr(\025)30 b Fs(P)812 1218 y Fo(\014)839 1211 y Fv(\()p Fs(X)899 1220 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))1086 1211 y Fv(=)13 b Fs(y)r(;)k(X)1236 1220 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))1340 1211 y Fv(=)c Fs(z)r(=X)1483 1218 y Fl(0)1520 1211 y Fv(=)g Fs(x)p Fv(\))711 1288 y(=)30 b Fs(P)812 1295 y Fo(\014)839 1288 y Fv(\()p Fs(X)899 1297 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))1086 1288 y Fv(=)13 b Fs(y)r(=X)1230 1295 y Fl(0)1267 1288 y Fv(=)g Fs(x)p Fv(\))8 b Fs(P)1406 1295 y Fo(\014)1434 1288 y Fv(\()p Fs(X)1494 1297 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))1598 1288 y Fv(=)14 b Fs(z)r(=X)1742 1295 y Fl(0)1779 1288 y Fv(=)f Fs(y)r Fv(\))p Fs(:)0 1391 y Fv(Letting)k Fs(\014)i Fv(tend)e(to)g(in\014nit)o(y)e(w)o(e)i(obtain)559 1495 y Fs(C)595 1502 y Fo(D)631 1495 y Fv(\()p Fs(x;)8 b(z)r Fv(\))32 b Fr(\024)e Fs(C)881 1504 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)1000 1495 y Fv(\()p Fs(x;)8 b(y)r Fv(\))21 b(+)e Fs(C)1229 1502 y Fo(D)1264 1495 y Fv(\()p Fs(y)r(;)8 b(z)r Fv(\))p Fs(:)0 1599 y Fv(Moreo)o(v)o(er)15 b(b)o(y)h(con)o(v)o(en)o(tion)f Fs(C)573 1608 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)692 1599 y Fv(\()p Fs(x;)8 b(y)r Fv(\))15 b(=)f Fs(C)910 1606 y Fo(D)945 1599 y Fv(\()p Fs(x;)8 b(y)r Fv(\).)51 b Ff(\003)60 1685 y Fv(F)l(or)15 b(another)h(construction)f(of)i(the)g(renormalized)d(comm)o(unication)h (cost,)h(see)h(T)l(rouv)o(\023)-24 b(e)16 b([20,21].)0 1771 y Fx(Prop)r(osition)j(4.4.)27 b Fi(\(prop)q(erties)16 b(of)h(the)g (renormalized)d(comm)o(unication)g(cost\))0 1831 y(F)l(or)h(an)o(y)h(subset)g Fs(D)i Fi(of)f Fs(E)i Fi(and)d(for)h(an)o(y)f(p)q(oin)o(t)g Fs(x)h Fi(in)f Fs(E)s Fi(,)g(the)h(cost)f Fs(C)1303 1838 y Fo(D)1339 1831 y Fv(\()p Fs(x;)8 b(D)1450 1813 y Fo(c)1472 1831 y Fv(\))17 b Fi(is)f(n)o(ull.)0 1891 y(F)l(or)f(an)o(y)h Fs(x;)8 b(y)r Fi(,)18 b(the)f(set)f(function)g Fs(A)f Fr(7!)e Fs(C)800 1898 y Fo(A)832 1891 y Fv(\()p Fs(x;)8 b(y)r Fv(\))18 b Fi(is)f(non{increasing)d(i.e.)516 1994 y Fs(A)h Fr(\032)e Fs(B)116 b Fv(=)-8 b Fr(\))113 b Fs(C)1004 2001 y Fo(A)1036 1994 y Fv(\()p Fs(x;)8 b(y)r Fv(\))23 b Fr(\025)f Fs(C)1270 2001 y Fo(B)1304 1994 y Fv(\()p Fs(x;)8 b(y)r Fv(\))p Fs(:)0 2124 y Fj(Pr)m(o)m(of.)28 b Fv(These)16 b(prop)q(erties)f(are)h(easy)h (consequences)e(of)i(the)g(probabilistic)d(de\014nition)h(of)i(the)g(renor-)0 2184 y(malized)22 b(cost.)40 b(Let)23 b(us)f(pro)o(v)o(e)f(the)i(last)f (assertion.)39 b(First)22 b(w)o(e)g(ha)o(v)o(e)g Fs(C)1436 2191 y Fo(A)1468 2184 y Fv(\()p Fs(x;)8 b(y)r Fv(\))25 b(=)f Fs(C)1706 2193 y Fo(A)p Fn(nf)p Fo(y)q Fn(g)1821 2184 y Fv(\()p Fs(x;)8 b(y)r Fv(\),)0 2244 y Fs(C)36 2251 y Fo(B)70 2244 y Fv(\()p Fs(x;)g(y)r Fv(\))23 b(=)e Fs(C)303 2253 y Fo(B)r Fn(nf)p Fo(y)q Fn(g)420 2244 y Fv(\()p Fs(x;)8 b(y)r Fv(\))23 b(and)d(w)o(e)h(ma)o(y) g(th)o(us)f(assume)g(that)h Fs(y)j Fv(is)d(in)f Fs(B)1457 2226 y Fo(c)1478 2244 y Fv(.)36 b(W)l(e)21 b(ha)o(v)o(e)g(the)g(inclu-)0 2304 y(sion)16 b Fs(B)143 2286 y Fo(c)177 2304 y Fr(\032)e Fs(A)267 2286 y Fo(c)304 2304 y Fv(so)i(that)h(for)f Fs(i)e Fv(=)g(0)i(or)g Fs(i)e Fv(=)g(1,)93 2407 y Fr(f)8 b Fs(X)167 2416 y Fo(\034)t Fl(\()p Fo(A;i)p Fl(\))294 2407 y Fv(=)13 b Fs(y)e Fr(g)30 b(\032)g(f)8 b Fs(X)579 2416 y Fo(\034)t Fl(\()p Fo(B)r(;i)p Fl(\))707 2407 y Fv(=)14 b Fs(y)c Fr(g)64 b Fv(=)-8 b Fr(\))63 b Fs(P)1059 2414 y Fo(\014)1086 2407 y Fv(\()8 b Fs(X)1154 2416 y Fo(\034)t Fl(\()p Fo(A;i)p Fl(\))1280 2407 y Fv(=)14 b Fs(y)c Fv(\))31 b Fr(\024)f Fs(P)1518 2414 y Fo(\014)1545 2407 y Fv(\()17 b Fs(X)1622 2416 y Fo(\034)t Fl(\()p Fo(B)r(;i)p Fl(\))1750 2407 y Fv(=)d Fs(y)c Fv(\))0 2511 y(and)16 b(\014nally)g Fs(C)284 2518 y Fo(A)316 2511 y Fv(\()p Fs(x;)8 b(y)r Fv(\))15 b Fr(\025)f Fs(C)534 2518 y Fo(B)568 2511 y Fv(\()p Fs(x;)8 b(y)r Fv(\).)51 b Ff(\003)964 2670 y Ft(7)p eop %%Page: 8 8 8 7 bop 0 200 a Fx(De\014nition)21 b(4.5.)27 b Fv(\(virtual)17 b(energy\))0 260 y(Lemma)f(3)p Fs(:)p Fv(6)p Fs(:)g Fv(implies)f(that)i(for)f (an)o(y)g Fs(x)h Fv(in)f Fs(E)s Fv(,)93 403 y(lim)76 435 y Fo(\014)r Fn(!1)188 403 y Fr(\000)236 369 y Fv(1)p 233 391 31 2 v 233 437 a Fs(\014)278 403 y Fv(ln)8 b Fs(\026)358 410 y Fo(\014)385 403 y Fv(\()p Fs(x)p Fv(\))31 b(=)g Fs(W)7 b Fv(\()p Fs(x)p Fv(\))482 504 y(=)31 b(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fr(f)p Fs(x)p Fr(g)p Fv(\))8 b Fr(g)21 b(\000)e Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))k(:)e Fs(g)j Fr(2)e Fs(G)p Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fv(\))p Fs(;)j(y)f Fr(2)e Fs(E)d Fr(g)p Fs(:)0 629 y Fv(The)16 b(quan)o(tit)o(y)h Fs(W)7 b Fv(\()p Fs(x)p Fv(\))17 b(is)g(called)f(the)g(virtual)g(energy)h(of)f Fs(x)p Fv(.)0 689 y(F)l(or)f(an)i(arbitrary)e(set)h Fs(D)q Fv(,)i(w)o(e)e(de\014ne)g(its)g(virtual)g(energy)h Fs(W)7 b Fv(\()p Fs(D)q Fv(\))17 b(b)o(y)629 811 y Fs(W)7 b Fv(\()p Fs(D)q Fv(\))31 b(=)g(min)o Fr(f)8 b Fs(W)f Fv(\()p Fs(x)p Fv(\))15 b(:)e Fs(x)i Fr(2)f Fs(D)c Fr(g)p Fs(:)0 933 y Fv(The)16 b(b)q(ottom)h Fs(F)7 b Fv(\()p Fs(D)q Fv(\))18 b(of)f Fs(D)h Fv(is)e(the)h(set)g(of)f(p)q(oin)o(ts)g(of)h Fs(D)h Fv(with)f(virtual)f (energy)g Fs(W)7 b Fv(\()p Fs(D)q Fv(\))18 b(i.e.)577 1055 y Fs(F)7 b Fv(\()p Fs(D)q Fv(\))31 b(=)g Fr(f)8 b Fs(x)14 b Fr(2)g Fs(D)i Fv(:)d Fs(W)7 b Fv(\()p Fs(x)p Fv(\))15 b(=)f Fs(W)7 b Fv(\()p Fs(D)q Fv(\))h Fr(g)p Fs(:)0 1210 y Fx(De\014nition)21 b(4.6.)27 b Fv(\(lo)q(calized)17 b(virtual)f(energy\))0 1270 y(F)l(or)e(an)o(y)h(subset)g Fs(D)i Fv(of)e Fs(E)s Fv(,)g(an)o(y)g(p)q(oin)o (t)g Fs(x)h Fv(of)f Fs(E)j Fv(and)d(an)o(y)g Fs(y)i Fv(in)e Fs(D)q Fv(,)h(lemma)e(3)p Fs(:)p Fv(5)p Fs(:)h Fv(implies)f(the)i(existence)0 1330 y(of)h(the)g(limit)306 1479 y(lim)289 1512 y Fo(\014)r Fn(!1)401 1479 y Fr(\000)449 1446 y Fv(1)p 446 1468 V 446 1513 a Fs(\014)491 1479 y Fv(ln)558 1417 y Fn(1)542 1432 y Fg(X)541 1537 y Fo(n)p Fl(=0)624 1479 y Fs(P)656 1486 y Fo(\014)683 1479 y Fv(\()p Fs(X)743 1486 y Fo(n)785 1479 y Fv(=)c Fs(y)r(;)k(\034)6 b Fv(\()p Fs(D)q(;)i Fv(1\))14 b Fs(>)g(n=X)1212 1486 y Fl(0)1248 1479 y Fv(=)g Fs(x)p Fv(\))31 b(=)f Fs(W)1495 1486 y Fo(D)1532 1479 y Fv(\()p Fs(x;)8 b(y)r Fv(\))p Fs(:)0 1647 y Fv(The)13 b(quan)o(tit)o(y)g Fs(W)342 1654 y Fo(D)378 1647 y Fv(\()p Fs(x;)8 b(y)r Fv(\))15 b(is)e(the)g(logarithmic)f(rate)h(of)h(the)f(p)q(oten) o(tial)h(of)f(the)g(Mark)o(o)o(v)g(c)o(hain)f(starting)0 1707 y(from)k Fs(x)h Fv(and)f(killed)g(outside)g Fs(D)q Fv(.)0 1767 y(The)g(expression)f(of)i Fs(W)446 1774 y Fo(D)482 1767 y Fv(\()p Fs(x;)8 b(y)r Fv(\))19 b(in)d(terms)f(of)i(graphs)e(is)h(for)g(an)o(y)h Fs(x)g Fv(in)f Fs(D)i Fv(and)e Fs(y)j Fv(in)d Fs(D)q Fv(,)170 1889 y Fs(W)217 1896 y Fo(D)253 1889 y Fv(\()p Fs(x;)8 b(y)r Fv(\))32 b(=)e(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)g Fs(G)856 1896 y Fo(x;y)914 1889 y Fv(\()p Fs(D)975 1868 y Fo(c)1007 1889 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))c Fr(g)20 b(\000)f Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1706 1868 y Fo(c)1728 1889 y Fv(\))8 b Fr(g)0 2011 y Fv(and)16 b(for)g(an)o(y)g Fs(x)h Fv(in)f Fs(D)412 1993 y Fo(c)450 2011 y Fv(and)g Fs(y)j Fv(in)d Fs(D)q Fv(,)4 2133 y Fs(W)51 2140 y Fo(D)87 2133 y Fv(\()p Fs(x;)8 b(y)r Fv(\))24 b(=)e(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(V)g Fv(\()p Fs(g)r Fv(\))j(:)g Fs(z)i Fr(2)e Fs(D)q(;)j(g)e Fr(2)f Fs(G)1048 2140 y Fo(z)q(;y)1105 2133 y Fv(\()p Fs(D)1166 2113 y Fo(c)1198 2133 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))d Fr(g)k(\000)f Fv(min)o Fr(f)d Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1880 2113 y Fo(c)1902 2133 y Fv(\))8 b Fr(g)225 2211 y Fv(=)22 b(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(W)665 2218 y Fo(D)701 2211 y Fv(\()p Fs(z)r(;)d(y)r Fv(\))15 b(:)f Fs(z)i Fr(2)e Fs(D)c Fr(g)p Fs(:)0 2368 y Fx(Lemma)17 b(4.7.)28 b Fi(\(link)16 b(b)q(et)o(w)o(een)h(the)g(lo)q(calized)f(virtual)g(energy)g (and)g(the)h(renormalized)d(cost\))0 2428 y(Let)j Fs(D)h Fi(b)q(e)f(an)f (arbitrary)g(subset)f(of)i Fs(E)s Fi(,)f Fs(x)h Fi(in)f Fs(E)k Fi(and)15 b Fs(y)k Fi(in)d Fs(D)q Fi(.)23 b(Then)493 2550 y Fs(W)540 2557 y Fo(D)576 2550 y Fv(\()p Fs(x;)8 b(y)r Fv(\))21 b(+)e Fs(C)805 2559 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)924 2550 y Fv(\()p Fs(y)r(;)8 b(D)1033 2529 y Fo(c)1054 2550 y Fv(\))31 b(=)f Fs(C)1209 2559 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)1328 2550 y Fv(\()p Fs(x;)8 b(y)r Fv(\))p Fs(:)964 2670 y Ft(8)p eop %%Page: 9 9 9 8 bop 0 200 a Fj(Pr)m(o)m(of.)28 b Fv(Let)17 b(us)f(assume)f(\014rst)h (that)h Fs(x)g Fv(is)f(in)g Fs(D)q Fv(.)23 b(W)l(e)16 b(ha)o(v)o(e)170 325 y Fs(W)217 332 y Fo(D)253 325 y Fv(\()p Fs(x;)8 b(y)r Fv(\))32 b(=)e(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)g Fs(G)856 332 y Fo(x;y)914 325 y Fv(\()p Fs(D)975 304 y Fo(c)1007 325 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))c Fr(g)20 b(\000)f Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1706 304 y Fo(c)1728 325 y Fv(\))8 b Fr(g)0 450 y Fv(and)139 575 y Fs(C)175 584 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)293 575 y Fv(\()p Fs(y)r(;)g(D)402 555 y Fo(c)424 575 y Fv(\))31 b(=)f(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))k(:)e Fs(g)j Fr(2)e Fs(G)p Fv(\()p Fs(D)992 555 y Fo(c)1013 575 y Fv(\))8 b Fr(g)20 b(\000)f Fv(min)o Fr(f)8 b Fs(V)k Fv(\()p Fs(g)r Fv(\))i(:)f Fs(g)j Fr(2)e Fs(G)p Fv(\()p Fs(D)1592 555 y Fo(c)1624 575 y Fr([)e(f)p Fs(y)r Fr(g)p Fv(\))c Fr(g)p Fs(:)0 700 y Fv(Th)o(us)111 826 y Fs(W)158 833 y Fo(D)195 826 y Fv(\()p Fs(x;)g(y)r Fv(\))20 b(+)g Fs(C)424 835 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)542 826 y Fv(\()p Fs(y)r(;)8 b(D)651 805 y Fo(c)673 826 y Fv(\))30 b(=)h(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)1179 833 y Fo(x;y)1238 826 y Fv(\()p Fs(D)1299 805 y Fo(c)1331 826 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))d Fr(g)1127 903 y(\000)19 b Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1633 882 y Fo(c)1666 903 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))d Fr(g)722 980 y Fv(=)31 b Fs(C)828 989 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)946 980 y Fv(\()p Fs(x;)8 b(y)r Fv(\))p Fs(:)0 1105 y Fv(When)16 b Fs(x)h Fv(is)f(in)h Fs(D)341 1087 y Fo(c)378 1105 y Fv(w)o(e)f(can)h(write)78 1230 y Fs(W)125 1237 y Fo(D)161 1230 y Fv(\()p Fs(x;)8 b(y)r Fv(\))21 b(+)f Fs(C)391 1239 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)509 1230 y Fv(\()p Fs(y)r(;)8 b(D)618 1210 y Fo(c)640 1230 y Fv(\))30 b(=)h(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(W)1138 1237 y Fo(D)1174 1230 y Fv(\()p Fs(z)r(;)d(y)r Fv(\))15 b(:)f Fs(z)i Fr(2)e Fs(D)c Fr(g)19 b Fv(+)g Fs(C)1603 1239 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)1722 1230 y Fv(\()p Fs(y)r(;)8 b(D)1831 1210 y Fo(c)1852 1230 y Fv(\))689 1308 y(=)31 b(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(x;)d(z)r Fv(\))13 b(+)e Fs(C)1127 1317 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)1245 1308 y Fv(\()p Fs(z)r(;)d(y)r Fv(\))15 b(:)f Fs(z)i Fr(2)e Fs(D)c Fr(g)689 1385 y Fv(=)31 b Fs(C)795 1394 y Fo(D)q Fn(nf)p Fo(y)q Fn(g)913 1385 y Fv(\()p Fs(x;)8 b(y)r Fv(\))p Fs(:)51 b Ff(\003)0 1546 y Fj(R)m(emark.)27 b Fv(A)17 b(probabilistic)d(pro)q(of)i(is)g(also)g(p)q(ossible,)f(starting)h(from)g (the)h(iden)o(tit)o(y)0 1638 y Fg(\022)63 1646 y Fn(1)47 1661 y Fg(X)45 1766 y Fo(n)p Fl(=0)128 1708 y Fs(P)160 1715 y Fo(\014)187 1708 y Fv(\()p Fs(X)247 1715 y Fo(n)289 1708 y Fv(=)c Fs(y)r(;)k(\034)6 b Fv(\()p Fs(D)q Fv(\))15 b Fs(>)e(n=X)669 1715 y Fl(0)705 1708 y Fv(=)h Fs(x)p Fv(\))805 1638 y Fg(\023)851 1708 y Fs(P)883 1715 y Fo(\014)910 1708 y Fv(\()p Fs(X)970 1717 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))1157 1708 y Fr(62)g Fs(D)q(=X)1312 1715 y Fl(0)1349 1708 y Fv(=)f Fs(y)r Fv(\))31 b(=)1349 1828 y Fs(P)1381 1835 y Fo(\014)1408 1828 y Fv(\()p Fs(X)1468 1837 y Fo(\034)t Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))1655 1828 y Fv(=)13 b Fs(y)r(=X)1799 1835 y Fl(0)1836 1828 y Fv(=)g Fs(x)p Fv(\))p Fs(:)0 1956 y Fx(De\014nition)21 b(4.8.)27 b Fv(\(heigh)o(t)16 b(of)h(a)g(set\))0 2015 y(The)f(heigh)o(t)g Fs(H)t Fv(\()p Fs(D)q Fv(\))j(of)d(the)h(set)g Fs(D)h Fv(is)e(de\014ned)g(b)o (y)476 2164 y Fs(H)t Fv(\()p Fs(D)q Fv(\))32 b(=)e(max)707 2196 y Fo(x)p Fn(2)p Fo(D)870 2164 y Fv(lim)853 2197 y Fo(\014)r Fn(!1)974 2130 y Fv(1)p 971 2153 31 2 v 971 2198 a Fs(\014)1016 2164 y Fv(ln)8 b Fs(E)1103 2171 y Fo(\014)1130 2164 y Fv(\()p Fs(\034)e Fv(\()p Fs(D)q Fv(\))p Fs(=X)1323 2171 y Fl(0)1360 2164 y Fv(=)14 b Fs(x)p Fv(\))p Fs(:)0 2313 y Fv(By)j(lemma)f(3)p Fs(:)p Fv(8)p Fs(:)g Fv(this)g(limit)g(exists)g(and)g(is)g(equal)h(to)154 2439 y Fs(H)t Fv(\()p Fs(D)q Fv(\))32 b(=)e Fr(\000)8 b Fv(min)o Fr(f)g Fs(V)k Fv(\()p Fs(g)r Fv(\))i(:)f Fs(x)i Fr(2)f Fs(D)q(;)j(g)e Fr(2)f Fs(G)p Fv(\()p Fs(x)h Fr(6!)f Fs(D)1145 2418 y Fo(c)1174 2439 y Fr(g)d Fv(+)g(min)o Fr(f)d Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1708 2418 y Fo(c)1730 2439 y Fv(\))8 b Fr(g)p Fs(:)964 2670 y Ft(9)p eop %%Page: 10 10 10 9 bop 0 200 a Fx(De\014nition)21 b(4.9.)27 b Fv(\(b)q(oundary\))0 260 y(F)l(or)15 b(a)i(subset)f Fs(D)i Fv(of)f Fs(E)i Fv(w)o(e)d(de\014ne)g (its)h(b)q(oundary)e Fs(B)r Fv(\()p Fs(D)q Fv(\))481 365 y Fs(B)r Fv(\()p Fs(D)q Fv(\))23 b(=)f Fr(f)8 b Fs(y)17 b Fv(:)c Fs(y)j Fr(62)e Fs(D)q(;)j Fr(9)8 b Fs(x)15 b Fr(2)f Fs(D)51 b(V)11 b Fv(\()p Fs(x;)d(y)r Fv(\))16 b Fs(<)d Fr(1)8 b(g)0 477 y Fv(and)16 b(its)g(principal)f(b)q(oundary)609 464 y Fg(e)598 477 y Fs(B)s Fv(\()p Fs(D)q Fv(\))177 594 y Fg(e)166 606 y Fs(B)s Fv(\()p Fs(D)q Fv(\))23 b(=)f Fr(f)8 b Fs(y)16 b Fr(2)e Fs(B)r Fv(\()p Fs(D)q Fv(\))i(:)e Fr(9)p Fs(x)g Fr(2)g Fs(D)77 b Fv(lim)872 639 y Fo(\014)r Fn(!1)984 606 y Fr(\000)1032 572 y Fv(1)p 1029 595 31 2 v 1029 640 a Fs(\014)1074 606 y Fv(ln)8 b Fs(P)1156 613 y Fo(\014)1183 606 y Fv(\()p Fs(X)1243 615 y Fo(\034)t Fl(\()p Fo(D)q Fl(\))1347 606 y Fv(=)14 b Fs(y)r(=X)1492 613 y Fl(0)1528 606 y Fv(=)g Fs(x)p Fv(\))23 b(=)f(0)8 b Fr(g)p Fs(:)0 763 y Fj(R)m(emark.)27 b Fv(The)16 b(principal)f(b)q(oundary)g(of)i(a) f(set)h Fs(D)h Fv(ma)o(y)e(equiv)m(alen)o(tly)h(b)q(e)g(de\014ned)e(b)o(y)474 859 y Fg(e)463 872 y Fs(B)s Fv(\()p Fs(D)q Fv(\))23 b(=)f Fr(f)8 b Fs(y)16 b Fr(2)e Fs(B)r Fv(\()p Fs(D)q Fv(\))i(:)e Fr(9)p Fs(x)f Fr(2)i Fs(D)51 b(C)1197 879 y Fo(D)1233 872 y Fv(\()p Fs(x;)8 b(y)r Fv(\))15 b(=)f(0)8 b 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b(F)l(or)15 b(an)o(y)h(p)q(oin)o(ts)g Fs(x;)8 b(y)19 b Fi(in)e Fs(\031)r Fi(,)f(the)h(cost)g Fs(C)1158 1533 y Fo(\031)1184 1526 y Fv(\()p Fs(x;)8 b(y)r Fv(\))18 b Fi(is)f(n)o(ull.)0 1586 y(F)l(or)e(an)o(y)h(subset)g Fs(D)i Fi(of)f Fs(E)i Fi(con)o(taining)d Fs(\031)i Fi(and)e(for)g(an)o(y)h Fs(x;)8 b(y)19 b Fi(in)d Fs(\031)r Fi(,)h(w)o(e)f(ha)o(v)o(e)g(for)g(an)o(y)g Fs(z)j Fi(in)d Fs(E)556 1691 y(W)603 1698 y Fo(D)639 1691 y Fv(\()p Fs(z)r(;)8 b(x)p Fv(\))13 b Fr(\000)e Fs(W)862 1698 y Fo(D)898 1691 y Fv(\()p Fs(z)r(;)d(y)r Fv(\))23 b(=)f Fs(W)7 b Fv(\()p Fs(x)p Fv(\))12 b Fr(\000)f Fs(W)c Fv(\()p Fs(y)r Fv(\))485 1766 y Fs(C)521 1773 y Fo(D)557 1766 y Fv(\()p Fs(x;)h(z)r Fv(\))24 b(=)e Fs(C)791 1773 y Fo(D)827 1766 y Fv(\()p Fs(y)r(;)8 b(z)r Fv(\))p Fs(;)59 b(C)1047 1773 y Fo(D)1083 1766 y Fv(\()p Fs(z)r(;)8 b(x)p Fv(\))24 b(=)d Fs(C)1316 1773 y Fo(D)1352 1766 y Fv(\()p Fs(z)r(;)8 b(y)r Fv(\))0 1872 y Fi(and)16 b(for)g(an)o(y)g Fs(z)j Fi(in)d Fs(D)q Fi(,)h(w)o(e)g(ha)o(v)o(e)f 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Fv(in)o(to)d(its)h(maximal)g(sub)q(cycles) f(is)h(denoted)g(b)o(y)g Fr(M)p Fv(\()p Fs(D)q Fv(\).)0 2284 y(F)l(or)k Fs(x)h Fv(in)g Fs(D)i Fv(w)o(e)d(denote)h(b)o(y)g Fs(\031)r Fv(\()p Fs(x;)8 b(D)q Fv(\))19 b(the)f(unique)e(cycle)i(of)f Fr(M)p Fv(\()p Fs(D)q Fv(\))h(con)o(taining)e Fs(x)h Fv(\(for)g Fs(x)h Fv(in)e Fs(D)1855 2266 y Fo(c)1893 2284 y Fv(w)o(e)0 2344 y(mak)o(e)g(the)h(con)o(v)o(en)o(tion)e(that)i Fs(\031)r Fv(\()p Fs(x;)8 b(D)q Fv(\))16 b(=)d Fr(f)p Fs(x)p Fr(g)p Fv(\).)0 2430 y Fj(R)m(emark.)27 b Fv(In)16 b(case)g Fs(D)i Fv(is)f(a)f(cycle,)h(w)o (e)f(ha)o(v)o(e)g Fr(M)p Fv(\()p Fs(D)q Fv(\))f(=)f Fr(f)p Fs(D)q Fr(g)p Fv(.)60 2490 y(The)g(relev)m(ance)g(of)g(the)h(cycles)f(of)g Fr(M)p Fv(\()p Fs(D)q Fv(\))i(for)e(the)g(study)g(of)g(the)h(b)q(eha)o(viour) d(of)j(the)f(Mark)o(o)o(v)f(c)o(hain)0 2550 y(in)j Fs(D)i Fv(will)e(already)g (app)q(ear)g(in)g(the)h(next)g(lemma.)954 2670 y Ft(10)p eop %%Page: 11 11 11 10 bop 0 200 a Fx(Notation.)29 b Fv(Let)19 b Fs(g)h 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Fr(g)14 b Fv(:)f Fs(y)j Fr(2)e Fs(G)553 1600 y Fg(\011)594 1640 y Fr(\002)659 1578 y Fn(1)651 1593 y Fg([)644 1698 y Fo(r)q Fl(=0)722 1600 y Fg(\010)759 1640 y Fv(\()p Fs(\031)806 1647 y Fl(0)830 1640 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(\031)976 1647 y Fo(r)q Fn(\000)p Fl(1)1050 1640 y Fv(\))14 b Fr(2)g(M)p Fv(\()p Fs(D)f Fr(n)e Fs(G)p Fv(\))k(:)e Fs(\031)1427 1647 y Fo(k)1466 1640 y Fr(6)p Fv(=)h Fs(\031)1547 1647 y Fo(k)q Fn(\000)p Fl(1)1622 1640 y Fs(;)8 b Fv(1)14 b Fr(\024)g Fs(k)h(<)f(r)1862 1600 y Fg(\011)1571 1766 y Fr(\002)1610 1725 y Fg(\010)1647 1766 y Fr(f)p Fs(z)r Fr(g)g Fv(:)f Fs(z)k Fr(2)d Fs(D)1892 1745 y Fo(c)1921 1725 y Fg(\011)0 1849 y Fv(of)j(\014nite)f(sequences)f(of)i (cycles)g(starting)e(in)h Fs(G)p Fv(,)g(tra)o(v)o(eling)g(through)f Fr(M)p Fv(\()p Fs(D)d Fr(n)f Fs(G)p Fv(\))16 b(and)g(ending)f(in)h Fs(D)1915 1831 y Fo(c)1936 1849 y Fv(.)0 1909 y(W)l(e)h(de\014ne)f(a)g(cost)h (function)f Fs(V)598 1916 y Fo(x)623 1909 y Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))18 b(on)f(the)f(space)g(\011\()p Fs(D)q(;)8 b(G)p Fv(\))19 b(b)o(y)0 1993 y Fs(V)29 2000 y Fo(x)54 1993 y Fv(\()p Fs(D)q(;)8 b(G)p Fv(\)\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)g(\031)340 2000 y Fl(0)365 1993 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)512 2000 y Fo(r)q Fn(\000)p Fl(1)586 1993 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))31 b(=)602 2115 y Fs(W)649 2122 y Fo(D)685 2115 y Fv(\()p Fs(x;)8 b(y)r Fv(\))21 b(+)e Fs(V)12 b Fv(\()p Fs(y)r(;)c(\031)1014 2122 y Fl(0)1037 2115 y Fv(\))20 b(+)1135 2052 y Fo(r)q Fn(\000)p Fl(1)1134 2067 y Fg(X)1134 2175 y Fo(k)q Fl(=1)1215 2115 y Fs(C)1251 2122 y Fo(\031)1274 2127 y Fh(k)p Fd(\000)p Fe(1)1342 2115 y Fv(\()p Fs(\031)1389 2122 y Fo(k)q Fn(\000)p Fl(1)1465 2115 y Fs(;)8 b(\031)1515 2122 y Fo(k)1540 2115 y Fv(\))20 b(+)f Fs(C)1673 2122 y Fo(\031)1696 2127 y Fh(r)q Fd(\000)p Fe(1)1762 2115 y Fv(\()p Fs(\031)1809 2122 y Fo(r)q Fn(\000)p Fl(1)1883 2115 y Fs(;)8 b(z)r Fv(\))0 2243 y(where)16 b(w)o(e)g(recall)g (that)h(b)o(y)f(de\014nition,)233 2327 y Fs(V)11 b Fv(\()p Fs(y)r(;)d(\031)368 2334 y Fl(0)392 2327 y Fv(\))14 b(=)k(min)477 2357 y Fo(u)p Fn(2)p Fo(\031)551 2362 y Fe(0)587 2327 y Fs(V)11 b Fv(\()p Fs(y)r(;)d(u)p Fv(\))p Fs(;)109 b(C)901 2334 y Fo(\031)924 2339 y Fh(k)p Fd(\000)p Fe(1)992 2327 y Fv(\()p Fs(\031)1039 2334 y Fo(k)q Fn(\000)p Fl(1)1115 2327 y Fs(;)8 b(\031)1165 2334 y Fo(k)1190 2327 y Fv(\))14 b(=)19 b(min)1276 2357 y Fo(v)q Fn(2)p Fo(\031)1347 2362 y Fh(k)1385 2327 y Fs(C)1421 2334 y Fo(\031)1444 2339 y Fh(k)p Fd(\000)p Fe(1)1513 2327 y Fv(\()p Fs(\031)1560 2334 y Fo(k)q Fn(\000)p Fl(1)1636 2327 y Fs(;)8 b(v)r Fv(\))p Fs(:)0 2430 y Fv(Since)25 b(the)h(sets)g Fs(D)h Fv(and)e Fs(G)h Fv(will)g(b)q(e)g(\014xed)f(in)h(the)g(sequel,)h(w)o(e)f (will)f(drop)g(them)g(in)h(the)g(nota-)0 2490 y(tions)c(whenev)o(er)f(no)h (confusion)f(is)h(p)q(ossible:)32 b(for)22 b(instance)g(w)o(e)g(will)g(write) g Fs(V)1550 2497 y Fo(x)1575 2490 y Fs(;)8 b Fv(\011)p Fs(;)g(\031)r Fv(\()p Fs(X)t Fv(\))24 b(instead)0 2550 y(of)17 b Fs(V)86 2557 y Fo(x)111 2550 y Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))p Fs(;)g Fv(\011\()p Fs(D)q(;)g(G)p Fv(\))p Fs(;)g(\031)s Fv(\()p Fs(X)q(;)g(D)r(;)g (G)p Fv(\))q(.)954 2670 y Ft(13)p eop %%Page: 14 14 14 13 bop 0 200 a Fx(Theorem)17 b(5.2.)28 b Fi(\(estimation)16 b(of)h(the)g(probabilit)o(y)d(of)j(a)g(cycle)g(path\))0 260 y(There)d(exists)g(a)h(p)q(ositiv)o(e)f(constan)o(t)g Fs(K)740 267 y Fl(1)777 260 y Fi(\(dep)q(ending)g(only)g(on)g(the)h(cardinalit)o(y)e (of)i Fs(D)8 b Fr(n)f Fs(G)p Fi(\))15 b(suc)o(h)f(that)0 319 y(for)i(an)o(y)g(exit)i(cycle)f(path)f Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)8 b(\031)653 326 y Fl(0)676 319 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)823 326 y Fo(r)q Fn(\000)p Fl(1)897 319 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))17 b Fi(in)f Fv(\011\()p Fs(D)q(;)8 b(G)p Fv(\))18 b Fi(and)e(an)o(y)g Fs(\014)j Fi(w)o(e)e(ha)o(v)o(e)0 443 y Fv(\()p Fs(K)65 422 y Fn(\000)p Fl(1)61 457 y(1)119 443 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))214 423 y Fo(K)247 428 y Fe(1)270 443 y Fv(\))289 423 y Fo(r)q Fl(+1)370 443 y Fv(exp)9 b Fr(\000)p Fs(\014)s(V)554 450 y Fo(x)578 443 y Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)f(\031)723 450 y Fl(0)747 443 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(\031)893 450 y Fo(r)q Fn(\000)p Fl(1)967 443 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))381 519 y Fr(\024)30 b Fs(P)482 526 y Fo(\014)509 479 y Fg(\000)532 519 y Fs(\031)r Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))24 b(=)e(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)8 b(\031)933 526 y Fl(0)956 519 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)1103 526 y Fo(r)q Fn(\000)p Fl(1)1177 519 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))p Fs(=X)1359 526 y Fl(0)1396 519 y Fv(=)13 b Fs(x)1476 479 y Fg(\001)1530 519 y Fr(\024)856 601 y Fv(\()p Fs(K)917 608 y Fl(1)940 601 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1035 580 y Fn(\000)p Fo(K)1099 585 y Fe(1)1123 601 y Fv(\))1142 580 y Fo(r)q Fl(+1)1223 601 y Fv(exp)8 b Fr(\000)p Fs(\014)s(V)1406 608 y Fo(x)1431 601 y Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)g(\031)1576 608 y Fl(0)1599 601 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)1746 608 y Fo(r)q Fn(\000)p Fl(1)1819 601 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))p Fs(:)0 755 y Fj(Pr)m(o)m(of.)28 b Fv(Conditioning)17 b(b)o(y)h(the)h(last)f(visit)h(of)f(the)h(c)o(hain)f(to)h(the)f(set)h Fs(G)g Fv(and)f(applying)f(the)i(Mark)o(o)o(v)0 815 y(prop)q(ert)o(y)d(w)o(e) g(get)0 937 y Fs(P)32 944 y Fo(\014)59 897 y Fg(\000)82 937 y Fs(\031)r Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))24 b(=)e(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)8 b(\031)483 944 y Fl(0)506 937 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)653 944 y Fo(r)q Fn(\000)p Fl(1)726 937 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))p Fs(=X)908 944 y Fl(0)945 937 y Fv(=)14 b Fs(x)1026 897 y Fg(\001)1472 937 y Fv(\(2\))245 1057 y(=)324 995 y Fn(1)308 1009 y Fg(X)306 1115 y Fo(n)p Fl(=0)390 1057 y Fs(P)422 1064 y Fo(\014)448 1057 y Fv(\()p Fs(X)508 1064 y Fo(n)550 1057 y Fv(=)g Fs(y)r(;)8 b(\034)e Fv(\()p Fs(D)q Fv(\))15 b Fs(>)e(n=X)922 1064 y Fl(0)958 1057 y Fv(=)h Fs(x)p Fv(\))1119 995 y Fo(r)1096 1009 y Fg(Y)1092 1117 y Fo(k)q Fl(=0)1173 1057 y Fs(P)1205 1064 y Fo(\014)1232 1057 y Fv(\()p Fs(X)1292 1064 y Fo(\034)1310 1069 y Fh(k)1349 1057 y Fr(2)g Fs(\031)1424 1064 y Fo(k)1449 1057 y Fs(=X)1515 1064 y Fo(\034)1533 1069 y Fh(l)1564 1057 y Fr(2)g Fs(\031)1639 1064 y Fo(l)1655 1057 y Fs(;)8 b Fr(\000)p Fv(1)13 b Fr(\024)h Fs(l)h(<)e(k)r Fv(\))p Fs(:)17 1226 y Fv(By)k(lemma)e(3)p Fs(:)p Fv(5)p Fs(:)p Fv(,)h(there)h(exist) g(p)q(ositiv)o(e)f(constan)o(ts)f Fs(C)1049 1233 y Fl(1)1071 1226 y Fs(;)8 b(C)1133 1208 y Fn(0)1129 1239 y Fl(1)1168 1226 y Fv(suc)o(h)15 b(that)0 1388 y Fs(C)40 1368 y Fn(0)36 1401 y Fl(1)58 1388 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))153 1368 y Fo(C)182 1373 y Fe(1)213 1388 y Fv(exp)8 b Fr(\000)p Fs(\014)s(W)414 1395 y Fo(D)450 1388 y Fv(\()p Fs(x;)g(y)r Fv(\))15 b Fr(\024)649 1326 y Fn(1)633 1341 y Fg(X)632 1447 y Fo(n)p Fl(=0)715 1388 y Fs(P)747 1395 y Fo(\014)774 1388 y Fv(\()p Fs(X)834 1395 y Fo(n)876 1388 y Fv(=)e Fs(y)r(;)8 b(\034)e Fv(\()p Fs(D)q Fv(\))15 b Fs(>)f(n=X)1248 1395 y Fl(0)1284 1388 y Fv(=)g Fs(x)p Fv(\))1239 1522 y Fr(\024)g Fs(C)1332 1485 y Fd(0)1344 1500 y Fn(\000)p Fl(1)1328 1535 y(1)1398 1522 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1493 1501 y Fn(\000)p Fo(C)1553 1506 y Fe(1)1583 1522 y Fv(exp)9 b Fr(\000)p Fs(\014)s(W)1785 1529 y Fo(D)1821 1522 y Fv(\()p Fs(x;)f(y)r Fv(\))p Fs(:)0 1644 y Fv(Moreo)o(v)o(er)15 b(the)i(\014rst)f(term)g(of)g(the)h(pro)q(duct)f(in)g(form)o(ula)f(\(2\))i (satis\014es)115 1766 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))8 b(exp)h Fr(\000)p Fs(\014)s(V)i Fv(\()p Fs(y)r(;)d(\031)508 1773 y Fl(0)532 1766 y Fv(\))30 b Fr(\024)h Fs(P)683 1773 y Fo(\014)709 1766 y Fv(\()p Fs(X)769 1773 y Fo(\034)787 1778 y Fe(0)824 1766 y Fr(2)14 b Fs(\031)899 1773 y Fl(0)922 1766 y Fs(=X)988 1773 y Fo(\034)1006 1778 y Fd(\000)p Fe(1)1069 1766 y Fv(=)g Fs(y)r Fv(\))31 b Fr(\024)f Fs(a)p Fv(\()p Fs(\014)s Fv(\))1362 1746 y Fn(\000)p Fl(1)1416 1766 y Fr(j)p Fs(\031)1458 1773 y Fl(0)1481 1766 y Fr(j)8 b Fv(exp)h Fr(\000)p Fs(\014)s(V)h Fv(\()p Fs(y)r(;)e(\031)1792 1773 y Fl(0)1816 1766 y Fv(\))0 1889 y(and)16 b(for)g Fs(k)f Fr(\025)f Fv(1)j(w)o(e)f(ha)o(v)o(e)0 2011 y Fs(P)32 2018 y Fo(\014)59 2011 y Fv(\()p Fs(X)119 2018 y Fo(\034)137 2023 y Fh(k)176 2011 y Fr(2)e Fs(\031)251 2018 y Fo(k)276 2011 y Fs(=X)342 2018 y Fo(\034)360 2023 y Fh(l)391 2011 y Fr(2)g Fs(\031)466 2018 y Fo(l)482 2011 y Fs(;)8 b Fr(\000)p Fv(1)14 b Fr(\024)f Fs(l)i(<)e(k)r Fv(\))31 b(=)533 b(\(3\))483 2050 y Fg(X)449 2154 y Fo(u)p Fn(2)p Fo(\031)523 2159 y Fh(k)p Fd(\000)p Fe(1)596 2097 y Fs(P)628 2104 y Fo(\014)655 2097 y Fv(\()p Fs(X)715 2104 y Fo(\034)733 2109 y Fh(k)772 2097 y Fr(2)15 b Fs(\031)848 2104 y Fo(k)872 2097 y Fs(=X)938 2104 y Fo(\034)956 2109 y Fh(k)p Fd(\000)p Fe(1)1039 2097 y Fv(=)f Fs(u)p Fv(\))8 b Fs(P)1180 2104 y Fo(\014)1207 2097 y Fv(\()p Fs(X)1267 2104 y Fo(\034)1285 2109 y Fh(k)p Fd(\000)p Fe(1)1368 2097 y Fv(=)13 b Fs(u=X)1515 2104 y Fo(\034)1533 2109 y Fh(l)1564 2097 y Fr(2)h Fs(\031)1639 2104 y Fo(l)1655 2097 y Fs(;)8 b Fr(\000)p Fv(1)13 b Fr(\024)h Fs(l)h(<)e(k)r Fv(\))p Fs(:)17 2271 y Fv(Y)l(et)k(there)f(exists)h(a)f(p)q(ositiv)o(e)h(constan)o(t)f Fs(C)836 2278 y Fl(2)874 2271 y Fv(suc)o(h)g(that)g(for)h(an)o(y)f Fs(u)g Fv(in)g Fs(\031)1396 2278 y Fo(k)q Fn(\000)p Fl(1)1472 2271 y Fv(,)0 2394 y Fs(C)40 2373 y Fn(\000)p Fl(1)36 2407 y(2)93 2394 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))188 2373 y Fo(C)217 2378 y Fe(2)247 2394 y Fv(exp)9 b Fr(\000)p Fs(\014)s(C)438 2401 y Fo(\031)461 2406 y Fh(k)p Fd(\000)p Fe(1)528 2394 y Fv(\()p Fs(\031)575 2401 y Fo(k)q Fn(\000)p Fl(1)652 2394 y Fs(;)f(\031)702 2401 y Fo(k)727 2394 y Fv(\))634 2469 y Fr(\024)31 b Fs(P)736 2476 y Fo(\014)762 2469 y Fv(\()p Fs(X)822 2476 y Fo(\034)840 2481 y Fh(k)879 2469 y Fr(2)15 b Fs(\031)955 2476 y Fo(k)979 2469 y Fs(=X)1045 2476 y Fo(\034)1063 2481 y Fh(k)p Fd(\000)p Fe(1)1146 2469 y Fv(=)f Fs(u)p Fv(\))30 b Fr(\024)1207 2550 y Fs(C)1243 2557 y Fl(2)1265 2550 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1360 2529 y Fn(\000)p Fo(C)1420 2534 y Fe(2)1451 2550 y Fv(exp)9 b Fr(\000)p Fs(\014)s(C)1642 2557 y Fo(\031)1665 2562 y Fh(k)p Fd(\000)p Fe(1)1732 2550 y Fv(\()p Fs(\031)1779 2557 y Fo(k)q Fn(\000)p Fl(1)1856 2550 y Fs(;)f(\031)1906 2557 y Fo(k)1931 2550 y Fv(\))954 2670 y Ft(14)p eop %%Page: 15 15 15 14 bop 0 200 a Fv(whence,)15 b(substituting)f(this)h(inequalit)o(y)g(in)g (the)g(previous)f(equation)i(and)e(summing)f(o)o(v)o(er)i Fs(u)e Fr(2)h Fs(\031)1860 207 y Fo(k)q Fn(\000)p Fl(1)1936 200 y Fv(,)0 260 y(w)o(e)i(get)0 335 y Fs(C)40 314 y Fn(\000)p Fl(1)36 349 y(2)93 335 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))188 315 y Fo(C)217 320 y Fe(2)247 335 y Fv(exp)9 b Fr(\000)p Fs(\014)s(C)438 342 y Fo(\031)461 347 y Fh(k)p Fd(\000)p Fe(1)528 335 y Fv(\()p Fs(\031)575 342 y Fo(k)q Fn(\000)p Fl(1)652 335 y Fs(;)f(\031)702 342 y Fo(k)727 335 y Fv(\))524 410 y Fr(\024)30 b Fs(P)625 417 y Fo(\014)652 410 y Fv(\()p Fs(X)712 417 y Fo(\034)730 422 y Fh(k)769 410 y Fr(2)14 b Fs(\031)844 417 y Fo(k)869 410 y Fs(=X)935 417 y Fo(\034)953 422 y Fh(l)985 410 y Fr(2)g Fs(\031)1060 417 y Fo(l)1075 410 y Fs(;)8 b Fr(\000)p Fv(1)14 b Fr(\024)g Fs(l)g(<)g(k)r Fv(\))30 b Fr(\024)1185 489 y Fs(C)1221 496 y Fl(2)1243 489 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1338 468 y Fn(\000)p Fo(C)1398 473 y Fe(2)1429 489 y Fv(exp)9 b Fr(\000)p Fs(\014)s(C)1620 496 y Fo(\031)1643 501 y Fh(k)p Fd(\000)p Fe(1)1710 489 y Fv(\()p Fs(\031)1757 496 y Fo(k)q Fn(\000)p Fl(1)1833 489 y Fs(;)f(\031)1883 496 y Fo(k)1908 489 y Fv(\))g Fs(:)0 564 y Fv(and)16 b(form)o(ula)f(\(2\))i(yields)f(the)h(result.)49 b Ff(\003)60 644 y Fv(The)14 b(cost)g(of)g(a)g(cycle)h(path)f(includes)f(the) h(sum)f(of)h(the)h(costs)f(of)g(eac)o(h)g(of)g(its)g(arro)o(ws)e(\()p Fs(\031)1710 651 y Fo(k)q Fn(\000)p Fl(1)1800 644 y Fr(!)i Fs(\031)1892 651 y Fo(k)1917 644 y Fv(\).)0 704 y(Th)o(us)f(an)o(y)h(path)h (of)f(b)q(ounded)g(cost)h(has)e(a)i(b)q(ounded)e(n)o(um)o(b)q(er)g(of)i(arro) o(ws)d(of)j(p)q(ositiv)o(e)f(cost.)22 b(Ho)o(w)o(ev)o(er)0 764 y(it)e(ma)o(y)g(ha)o(v)o(e)f(an)h(arbitrary)f(large)g(n)o(um)o(b)q(er)f (of)i(arro)o(ws)f(of)h(n)o(ull)f(cost)h(if)g(there)g(exist)h(lo)q(ops)e(of)h (n)o(ull)0 824 y(cost.)0 904 y Fx(De\014nition)h(5.3.)28 b Fv(The)17 b(cycle)g(path)g(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)8 b(\031)874 911 y Fl(0)897 904 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)1044 911 y Fo(r)q Fn(\000)p Fl(1)1118 904 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))17 b(is)g(optimal)f(kno)o(wing)g(that)h Fs(X)1831 911 y Fl(0)1868 904 y Fv(=)e Fs(x)0 963 y Fv(if)i(its)f(cost)h Fs(V)247 970 y Fo(x)272 963 y Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)8 b(\031)417 970 y Fl(0)440 963 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)587 970 y Fo(r)q Fn(\000)p Fl(1)661 963 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))17 b(is)f(equal)g(to)h(the)g(in\014m)o(um)219 1039 y(inf)s Fr(f)8 b Fs(V)341 1046 y Fo(x)367 1039 y Fv(\()p Fs( )418 1046 y Fn(\000)p Fl(1)472 1039 y Fs(;)g(:)g(:)g(:)17 b(;)8 b( )623 1046 y Fo(s)p Fn(\000)p Fl(1)696 1039 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))14 b(:)g Fs(s)g Fr(2)g Fk(N)p Fs(;)8 b Fv(\()p Fs( )1046 1046 y Fn(\000)p Fl(1)1102 1039 y Fs(;)g(:)g(:)g(:)16 b(;)8 b( )1252 1046 y Fo(s)p Fn(\000)p Fl(1)1325 1039 y Fs(;)g Fr(f)p Fs(z)r Fr(g)p Fv(\))15 b Fr(2)f Fv(\011\()p Fs(D)q(;)8 b(G)p Fv(\))g Fr(g)p Fs(:)60 1135 y Fv(Let)17 b Fr(G)179 1142 y Fo(x)220 1135 y Fv(b)q(e)g(the)g(graph)e(con)o(taining)g(the)i(arro)o(ws)e (of)i(all)f(optimal)f(paths)h(kno)o(wing)g(that)h Fs(X)1793 1142 y Fl(0)1829 1135 y Fv(=)d Fs(x)p Fv(.)0 1215 y Fx(Prop)r(osition)22 b(5.4.)27 b Fi(The)19 b(cycle)h(path)f Fv(\()p Fs(\031)827 1222 y Fn(\000)p Fl(1)881 1215 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)1028 1222 y Fo(r)1050 1215 y Fv(\))20 b Fi(is)e(optimal)g(kno)o(wing)g (that)i Fs(X)1677 1222 y Fl(0)1717 1215 y Fv(=)e Fs(x)i Fi(if)f(and)0 1275 y(only)i(if)g(it)g(is)f(con)o(tained)g(in)h(the)g(optimal)f(graph)g Fr(G)1018 1282 y Fo(x)1063 1275 y Fi(\(that)i(is,)f(all)g(its)g(arro)o(ws)e (do)h(app)q(ear)g(in)h(the)0 1335 y(optimal)16 b(graph\).)0 1415 y Fj(R)m(emark.)27 b Fv(That)19 b(an)o(y)g(optimal)g(path)g(b)q(elongs)g (to)h Fr(G)1013 1422 y Fo(x)1057 1415 y Fv(is)f(ob)o(vious.)30 b(The)19 b(in)o(teresting)f(p)q(oin)o(t)i(is)f(that)0 1474 y(eac)o(h)c(path)f(con)o(tained)h(in)f(the)i(optimal)e(graph)g Fr(G)942 1481 y Fo(x)982 1474 y Fv(yields)g(also)h(an)g(optimal)f(path)h(kno) o(wing)f Fs(X)1819 1481 y Fl(0)1855 1474 y Fv(=)g Fs(x)p Fv(.)0 1554 y Fj(Pr)m(o)m(of.)28 b Fv(Let)17 b(us)g(de\014ne)f(the)i(cost)f(of)g(a)h (b)q(eginning)d(path)i(\()p Fs(\031)1137 1561 y Fn(\000)p Fl(1)1192 1554 y Fs(;)8 b(\031)1242 1561 y Fl(0)1264 1554 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(\031)1411 1561 y Fo(k)1436 1554 y Fv(\))18 b(where)f(the)g Fs(\031)1732 1561 y Fo(k)1757 1554 y Fv('s)f(b)q(elong)0 1614 y(to)h Fr(M)p Fv(\()p Fs(D)c Fr(n)e Fs(G)p Fv(\))17 b(b)o(y)187 1739 y Fs(V)216 1746 y Fo(x)241 1739 y Fv(\()p Fs(\031)288 1746 y Fn(\000)p Fl(1)343 1739 y Fs(;)8 b(\031)393 1746 y Fl(0)416 1739 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(\031)562 1746 y Fo(k)588 1739 y Fv(\))30 b(=)h Fs(W)754 1746 y Fo(D)790 1739 y Fv(\()p Fs(x;)8 b(\031)887 1746 y Fn(\000)p Fl(1)942 1739 y Fv(\))19 b(+)g Fs(V)12 b Fv(\()p Fs(\031)1126 1746 y Fn(\000)p Fl(1)1180 1739 y Fs(;)c(\031)1230 1746 y Fl(0)1253 1739 y Fv(\))19 b(+)1374 1677 y Fo(k)1350 1692 y Fg(X)1354 1799 y Fo(l)p Fl(=1)1430 1739 y Fs(C)1466 1746 y Fo(\031)1489 1751 y Fh(l)p Fd(\000)p Fe(1)1550 1739 y Fv(\()p Fs(\031)1597 1746 y Fo(l)p Fn(\000)p Fl(1)1664 1739 y Fs(;)8 b(\031)1714 1746 y Fo(l)1729 1739 y Fv(\))p Fs(:)0 1862 y Fv(Then)17 b(for)h(an)o(y)f(optimal)g(path)g(\()p Fs(\031)652 1869 y Fn(\000)p Fl(1)707 1862 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)854 1869 y Fo(r)876 1862 y Fv(\))18 b(and)f(an)o(y)h Fs(k)f(<)f(r)q Fv(,)j(\()p Fs(\031)1308 1869 y Fn(\000)p Fl(1)1362 1862 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)1509 1869 y Fo(k)1534 1862 y Fv(\))18 b(is)f(optimal)g(among)0 1922 y(the)h(paths)e(ending)h(in)g Fs(\031)472 1929 y Fo(k)496 1922 y Fv(.)25 b(Con)o(v)o(ersely)l(,)16 b(assume)g(that)i(\()p Fs(\031)1127 1929 y Fn(\000)p Fl(1)1181 1922 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)1328 1929 y Fo(r)1350 1922 y Fv(\))18 b(b)q(elongs)f(to)g Fr(G)1658 1929 y Fo(x)1700 1922 y Fv(i.e.)25 b(that)17 b(all)0 1982 y(the)12 b(arro)o(ws)e(of)h(the)h (path)g(do)f(app)q(ear)g(in)g(the)h(graph)e Fr(G)1004 1989 y Fo(x)1029 1982 y Fv(.)20 b(W)l(e)12 b(pro)o(v)o(e)f(b)o(y)g(induction)f (that)i(\()p Fs(\031)1704 1989 y Fn(\000)p Fl(1)1759 1982 y Fs(;)c(:)g(:)g(:)17 b(;)8 b(\031)1906 1989 y Fo(k)1931 1982 y Fv(\))0 2042 y(is)19 b(optimal)g(among)f(the)i(paths)f(ending)g(in)g Fs(\031)883 2049 y Fo(k)908 2042 y Fv(.)31 b(Supp)q(ose)19 b(the)g(result)g(is)g(true)h(at)g(rank)f Fs(k)14 b Fr(\000)f Fv(1.)31 b(By)0 2101 y(the)23 b(v)o(ery)g(de\014nition)f(of)i(the)f(optimal)f (graph)g Fr(G)959 2108 y Fo(x)984 2101 y Fv(,)i(there)f(exists)g(an)g (optimal)f(path)h Fs(\015)j Fv(suc)o(h)c(that)0 2161 y(the)f(arro)o(w)e(\()p Fs(\031)281 2168 y Fo(k)q Fn(\000)p Fl(1)358 2161 y Fs(;)8 b(\031)408 2168 y Fo(k)433 2161 y Fv(\))21 b(is)f(in)h Fs(\015)s Fv(.)34 b(Let)22 b Fs(\015)787 2168 y Fo(k)q Fn(\000)p Fl(1)883 2161 y Fv(b)q(e)f(the)g(path)g Fs(\015)i Fv(truncated)d(at)i Fs(\031)1540 2168 y Fo(k)q Fn(\000)p Fl(1)1636 2161 y Fv(and)f Fs(\015)1764 2168 y Fo(k)1809 2161 y Fv(b)q(e)g(the)0 2221 y(path)f Fs(\015)j Fv(truncated)e(at)f Fs(\031)493 2228 y Fo(k)518 2221 y Fv(.)34 b(By)22 b(the)e(induction)g(h)o(yp)q(othesis)f Fs(V)1242 2228 y Fo(x)1267 2221 y Fv(\()p Fs(\015)1312 2228 y Fo(k)q Fn(\000)p Fl(1)1388 2221 y Fv(\))i(=)f Fs(V)1516 2228 y Fo(x)1541 2221 y Fv(\()p Fs(\031)1588 2228 y Fn(\000)p Fl(1)1643 2221 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(\031)1789 2228 y Fo(k)q Fn(\000)p Fl(1)1865 2221 y Fv(\))21 b(so)0 2281 y(that)c Fs(V)137 2288 y Fo(x)162 2281 y Fv(\()p Fs(\015)207 2288 y Fo(k)232 2281 y Fv(\))d(=)g Fs(V)347 2288 y Fo(x)372 2281 y Fv(\()p Fs(\031)419 2288 y Fn(\000)p Fl(1)473 2281 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)620 2288 y Fo(k)645 2281 y Fv(\).)22 b(Since)16 b Fs(\015)856 2288 y Fo(k)897 2281 y Fv(is)g(optimal,)g(\()p Fs(\031)1191 2288 y Fn(\000)p Fl(1)1245 2281 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)1392 2288 y Fo(k)1417 2281 y Fv(\))17 b(is)f(also)g(optimal.)49 b Ff(\003)0 2361 y Fj(R)m(emark.)27 b Fv(The)20 b(same)g(kind)h(of)g(pro)q (of)f(app)q(ears)g(in)g(the)h(study)g(of)g(the)g(w)o(ell)f(kno)o(wn)g (dynamic)g(pro-)0 2420 y(gramming)14 b(algorithm.)60 2490 y(W)l(e)f(study)f (no)o(w)g(the)h(link)f(b)q(et)o(w)o(een)h(the)g(cycle)g(path)f(cost)h(and)f (the)h(renormalized)d(comm)o(unication)0 2550 y(cost.)954 2670 y Ft(15)p eop %%Page: 16 16 16 15 bop 0 200 a Fx(Theorem)17 b(5.5.)50 b Fi(Let)17 b Fs(D)h Fi(b)q(e)e(a)h(domain,)e(let)h Fs(x)h Fi(b)q(elong)f(to)h Fs(D)h Fi(and)d(let)i Fs(y)i Fi(b)q(e)d(a)h(p)q(oin)o(t)e(of)i Fs(E)s Fi(.)k(Let)d Fs(\031)1925 207 y Fo(x)0 260 y Fi(and)g Fs(\031)127 267 y Fo(y)171 260 y Fi(b)q(e)h(the)h(cycles)f(of)g Fr(M)p Fv(\()p Fs(D)q Fv(\))i Fi(con)o(taining)c Fs(x)j Fi(and)e Fs(y)k Fi(resp)q(ectiv)o(ely)d(\(if)g Fs(y)j Fi(is)c(not)h(in)g Fs(D)q Fi(,)h(w)o(e)f(put)0 319 y Fs(\031)28 326 y Fo(y)66 319 y Fv(=)14 b Fr(f)p Fs(y)r Fr(g)p Fi(\).)22 b(The)17 b(cost)f Fs(C)491 326 y Fo(D)527 319 y Fv(\()p Fs(x;)8 b(y)r Fv(\))18 b Fi(is)e(equal)h(to)g (the)f(in\014m)o(um)82 461 y Fv(inf)151 391 y Fg(\032)231 399 y Fo(r)205 414 y Fg(X)205 521 y Fo(k)q Fl(=1)285 461 y Fs(C)321 468 y Fo(\031)344 473 y Fh(k)p Fd(\000)p Fe(1)413 461 y Fv(\()p Fs(\031)460 468 y Fo(k)q Fn(\000)p Fl(1)536 461 y Fs(;)8 b(\031)586 468 y Fo(k)611 461 y Fv(\))14 b(:)g Fs(r)h Fr(2)f Fk(N)p Fs(;)k(\031)851 468 y Fl(0)887 461 y Fv(=)c Fs(\031)968 468 y Fo(x)994 461 y Fs(;)i Fv(\()p Fs(\031)1071 468 y Fl(1)1094 461 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)1241 468 y Fo(r)q Fn(\000)p Fl(1)1315 461 y Fv(\))14 b Fr(2)g(M)p Fv(\()p Fs(D)q Fv(\))1535 441 y Fo(r)q Fn(\000)p Fl(1)1609 461 y Fs(;)i(\031)1667 468 y Fo(r)1704 461 y Fv(=)d Fs(\031)1784 468 y Fo(y)1817 391 y Fg(\033)1854 461 y Fs(:)0 635 y Fj(R)m(emark.)27 b Fv(Prop)q(osition)18 b(4)p Fs(:)p Fv(11)p Fs(:)g Fv(sho)o(ws)g(that)i Fs(C)893 642 y Fo(D)929 635 y Fv(\()p Fs(x;)8 b(y)r Fv(\))20 b(=)e Fs(C)1156 642 y Fo(D)1192 635 y Fv(\()p Fs(\031)1239 642 y Fo(x)1265 635 y Fs(;)8 b(\031)1315 642 y Fo(y)1339 635 y Fv(\))20 b(is)f(indep)q(enden) o(t)f(of)h(the)h(pair)0 695 y(\()p Fs(x;)8 b(y)r Fv(\))18 b(c)o(hosen)e(in)g Fs(\031)378 702 y Fo(x)414 695 y Fr(\002)11 b Fs(\031)492 702 y Fo(y)516 695 y Fv(.)0 780 y Fj(R)m(emark.)27 b Fv(If)12 b(w)o(e)h(had)e (not)i(used)f(maximal)f(cycles)i(in)f(theorem)g(5)p Fs(:)p Fv(5)p Fs(:)g Fv(as)g(w)o(ell)g(as)g(in)g(the)g(construction)g(of)0 840 y(the)j(exit)g(cycle)g(path,)f(the)h(renormalized)d(comm)o(unication)h (cost)h(w)o(ould)f(ha)o(v)o(e)h(b)q(een)h(strictly)f(inferior)0 900 y(to)j(the)h(in\014m)o(um)d(of)i(the)g(cycle)h(paths)f(costs)g(and)f(all) h(the)g(exit)h(cycle)g(paths)e(of)i(\014nite)e(length)h(w)o(ould)0 959 y(ha)o(v)o(e)k(had)h(an)g(exp)q(onen)o(tially)f(v)m(anishing)g (probabilit)o(y)g(to)h(b)q(e)g(tak)o(en)h(during)d(the)i(last)g(excursion.)0 1019 y(This)15 b(sho)o(ws)f(that)i(the)g(maximal)f(cycles)h(of)g Fr(M)p Fv(\()p 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Fl(0)1345 1686 y Fr(!)e(\001)8 b(\001)g(\001)14 b(!)f Fs(\031)1572 1693 y Fo(r)1610 1686 y Fv(b)q(e)i(the)g(sequence)0 1746 y(of)h(the)g (successiv)o(e)f(cycles)i(of)f Fr(M)p Fv(\()p Fs(D)q Fv(\))h(visited)f(b)o(y) g(this)f(path.)22 b(Since)15 b(the)i(path)e(of)h(p)q(oin)o(ts)g(starts)f(at)i Fs(x)0 1806 y Fv(and)e(ends)f(at)i Fs(y)h Fv(w)o(e)e(ha)o(v)o(e)g Fs(\031)523 1813 y Fl(0)559 1806 y Fv(=)f Fs(\031)640 1813 y Fo(x)681 1806 y Fv(and)g Fs(\031)804 1813 y Fo(r)841 1806 y Fv(=)f Fs(\031)921 1813 y Fo(y)946 1806 y Fv(.)21 b(In)15 b(addition)f(w)o(e)h(remo)o(v)o(e)g(the)g(lo)q(ops)g(in)g(this)f(cycle)0 1865 y(path,)i(in)g(order)g(to)g(obtain)g(a)h(path)f(of)h(distinct)f(cycles.) 22 b(No)o(w)407 1966 y Fs(C)443 1973 y Fo(D)478 1966 y Fv(\()p Fs(x;)8 b(y)r Fv(\))32 b(=)f Fs(V)11 b Fv(\()p 753 1938 V Fs(g)r Fv(\))19 b Fr(\000)g Fv(min)p Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(D)1324 1945 y Fo(c)1357 1966 y Fr([)d(f)p Fs(y)r Fr(g)p Fv(\))d Fr(g)p Fs(:)0 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Fo(c)1333 2344 y Fv(\))8 b Fr(g)1385 2274 y Fg(\023)1442 2344 y Fv(+)19 b Fs(V)11 b Fv(\()p 1559 2317 V Fs(g)r Fr(j)1599 2351 y Fo(\031)1622 2356 y Fh(y)1646 2344 y Fv(\))d Fr(\000)1215 2446 y Fg(X)1072 2555 y Fo(\031)q Fn(2M)p Fl(\()p Fo(D)q Fn(nf)p Fo(y)q Fn(g)p Fl(\))p Fo(;\031)q Fn(\032)p Fo(\031)1407 2560 y Fh(y)1363 2493 y Fv(min)o Fr(f)g Fs(V)k Fv(\()p Fs(g)r Fv(\))i(:)g Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(\031)1800 2473 y Fo(c)1821 2493 y Fv(\))8 b Fr(g)p Fs(:)954 2670 y Ft(16)p eop %%Page: 17 17 17 16 bop 0 200 a Fv(Ho)o(w)o(ev)o(er)15 b(the)i(set)f(of)h(cycles)f Fr(f)8 b Fs(\031)16 b Fv(:)e Fs(\031)i Fr(2)e(M)p Fv(\()p Fs(D)e Fr(n)e(f)p Fs(y)r Fr(g)p Fv(\))p Fs(;)17 b(\031)f Fr(\032)e Fs(\031)1176 207 y Fo(y)1208 200 y Fr(g)j Fv(is)e(exactly)i(the)g(maximal)e (partition)0 260 y Fr(M)p Fv(\()p Fs(\031)107 267 y Fo(y)143 260 y Fr(n)c(f)p Fs(y)r Fr(g)p Fv(\))17 b(whence)f(b)o(y)g(applying)g(once)g (more)g(lemma)f(4)p Fs(:)p Fv(13)p Fs(:)p Fv(,)h(the)h(cost)f Fs(C)1493 267 y Fo(D)1529 260 y Fv(\()p Fs(x;)8 b(y)r Fv(\))18 b(is)e(equal)h(to)101 354 y Fg(X)0 463 y Fo(\031)q Fn(2M)p Fl(\()p Fo(D)q Fl(\))p Fo(;\031)q Fn(6)p Fl(=)p Fo(\031)254 468 y Fh(y)283 331 y Fg(\022)320 401 y Fs(V)11 b Fv(\()p 379 374 26 2 v Fs(g)r Fr(j)419 408 y Fo(\031)446 401 y Fv(\))p Fr(\000)p Fv(min)o Fr(f)d Fs(V)k Fv(\()p Fs(g)r Fv(\))i(:)f Fs(g)j Fr(2)e Fs(G)p Fv(\()p Fs(\031)941 381 y Fo(c)962 401 y Fv(\))8 b Fr(g)1014 331 y Fg(\023)1060 401 y Fv(+)g Fs(V)j Fv(\()p 1166 374 V Fs(g)r Fr(j)1206 408 y Fo(\031)1229 413 y Fh(y)1253 401 y Fv(\))e Fr(\000)f Fv(min)o Fr(f)g Fs(V)k Fv(\()p Fs(g)r Fv(\))i(:)f Fs(g)j Fr(2)e Fs(G)p Fv(\()p Fs(\031)1765 381 y Fo(c)1763 413 y(y)1788 401 y Fr([f)p Fs(y)r Fr(g)p Fv(\))8 b Fr(g)0 570 y Fv(Since)p 135 543 V 21 w Fs(g)23 b Fv(b)q(elongs)d(to)i Fs(G)470 577 y Fo(x;y)529 570 y Fv(\()p Fs(D)590 552 y Fo(c)625 570 y Fr([)15 b(f)p Fs(y)r Fr(g)p Fv(\),)23 b(for)e(an)o(y)f Fs(\031)k Fv(in)d Fr(M)p Fv(\()p Fs(D)q Fv(\))15 b Fr(n)g Fs(\031)1322 577 y Fo(y)1346 570 y Fv(,)22 b(its)g(restriction)p 1700 543 V 20 w Fs(g)r Fr(j)1740 577 y Fo(\031)1788 570 y Fv(b)q(elongs)0 630 y(to)17 b Fs(G)p Fv(\()p Fs(\031)149 612 y Fo(c)170 630 y Fv(\).)22 b(Moreo)o(v)o(er,)p 458 603 V 15 w Fs(g)r Fr(j)498 637 y Fo(\031)521 642 y Fh(y)561 630 y Fv(b)q(elongs)16 b(to)h Fs(G)p Fv(\()p Fs(\031)889 612 y Fo(c)887 642 y(y)923 630 y Fr([)11 b(f)p Fs(y)r Fr(g)p Fv(\).)23 b(As)16 b(a)h(consequence,)378 794 y Fs(C)414 801 y Fo(D)450 794 y Fv(\()p Fs(x;)8 b(y)r Fv(\))32 b Fr(\025)666 732 y Fo(r)q Fn(\000)p Fl(1)665 747 y Fg(X)665 854 y Fo(k)q Fl(=0)746 724 y Fg(\022)782 794 y Fs(V)11 b Fv(\()p 841 766 V Fs(g)r Fr(j)881 801 y Fo(\031)904 806 y Fh(k)929 794 y Fv(\))g Fr(\000)g Fv(min)o Fr(f)d Fs(V)j Fv(\()p Fs(g)r Fv(\))j(:)g Fs(g)i Fr(2)e Fs(G)p Fv(\()p Fs(\031)1446 773 y Fo(c)1444 806 y(k)1469 794 y Fv(\))8 b Fr(g)1521 724 y Fg(\023)1558 794 y Fs(:)0 951 y Fv(\(where)16 b Fs(\031)191 958 y Fl(0)214 951 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(\031)361 958 y Fo(r)400 951 y Fv(is)16 b(the)h(sequence)g(of)f(cycles)h(previously)e(describ)q(ed\).) 0 1011 y(By)d(construction)e(of)i(the)f Fs(\031)513 1018 y Fo(k)538 1011 y Fv('s,)h(there)f(are)g Fs(x)826 1018 y Fo(k)862 1011 y Fv(in)g Fs(\031)943 1018 y Fo(k)979 1011 y Fv(and)g Fs(y)1095 1018 y Fo(k)1131 1011 y Fv(in)g Fs(\031)1212 1018 y Fo(k)q Fl(+1)1299 1011 y Fv(suc)o(h)f(that)p 1509 984 V 12 w Fs(g)q Fr(j)1548 1018 y Fo(\031)1571 1023 y Fh(k)1607 1011 y Fv(is)h(in)g Fs(G)1744 1018 y Fo(x)1767 1023 y Fh(k)1788 1018 y Fo(;y)1820 1023 y Fh(k)1845 1011 y Fv(\()p Fs(\031)1894 993 y Fo(c)1892 1025 y(k)1917 1011 y Fv(\),)0 1071 y(th)o(us)21 b Fs(V)11 b Fv(\()p 174 1043 V Fs(g)r Fr(j)214 1078 y Fo(\031)237 1083 y Fh(k)262 1071 y Fv(\))k Fr(\000)f Fv(min)o Fr(f)8 b Fs(V)j Fv(\()p Fs(g)r Fv(\))24 b(:)f Fs(g)h Fr(2)f Fs(G)p Fv(\()p Fs(\031)822 1053 y Fo(c)820 1085 y(k)846 1071 y Fv(\))8 b Fr(g)32 b(\025)f Fs(C)1036 1078 y Fo(\031)1059 1083 y Fh(k)1083 1071 y Fv(\()p Fs(x)1130 1078 y Fo(k)1155 1071 y Fs(;)8 b(y)1201 1078 y Fo(k)1226 1071 y Fv(\).)39 b(W)l(e)22 b(obtain)g(\014nally)f(the)h (desired)0 1130 y(inequalit)o(y)634 1235 y Fs(C)670 1242 y Fo(D)706 1235 y Fv(\()p Fs(x;)8 b(y)r Fv(\))32 b Fr(\025)922 1173 y Fo(r)q Fn(\000)p Fl(1)921 1188 y Fg(X)921 1295 y Fo(k)q Fl(=0)1002 1235 y Fs(C)1038 1242 y Fo(\031)1061 1247 y Fh(k)1085 1235 y Fv(\()p Fs(\031)1132 1242 y Fo(k)1157 1235 y Fs(;)8 b(\031)1207 1242 y Fo(k)q Fl(+1)1283 1235 y Fv(\))p Fs(:)0 1373 y Fv(Con)o(v)o(ersely)l(,)22 b(let)h Fs(\031)375 1380 y Fl(0)421 1373 y Fr(!)g(\001)8 b(\001)g(\001)24 b(!)f Fs(\031)677 1380 y Fo(r)722 1373 y Fv(b)q(e)f(a)g(cycle)h(path)f(in)g Fr(M)p Fv(\()p Fs(D)q Fv(\))i(starting)d(at)i Fs(\031)1607 1380 y Fl(0)1653 1373 y Fv(=)g Fs(\031)1743 1380 y Fo(x)1768 1373 y Fv(,)h(ending)0 1432 y(at)16 b Fs(\031)88 1439 y Fo(r)125 1432 y Fv(=)d Fs(\031)205 1439 y Fo(y)245 1432 y Fv(and)j(realizing)e(the)i (in\014m)o(um)e(of)i(the)g(theorem.)21 b(Let)c(us)e(in)o(tro)q(duce)g(the)h (stopping)e(times)247 1542 y 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Fv(\()p Fs(y)r(;)p 962 1703 V 8 w(\031)993 1737 y Fl(0)1015 1730 y Fv(\))20 b(+)1113 1668 y Fo(r)q Fn(\000)p Fl(1)1112 1683 y Fg(X)1112 1790 y Fo(k)q Fl(=1)1193 1730 y Fs(C)p 1229 1718 25 2 v 8 x Fo(\031)1253 1743 y Fh(k)p Fd(\000)p Fe(1)1321 1730 y Fv(\()p 1340 1703 31 2 v Fs(\031)1371 1737 y Fo(k)q Fn(\000)p Fl(1)1446 1730 y Fs(;)p 1468 1703 V 8 w(\031)1499 1737 y Fo(k)1523 1730 y Fv(\))g(+)f Fs(C)p 1656 1718 25 2 v 8 x Fo(\031)1680 1743 y Fh(r)q Fd(\000)p Fe(1)1747 1730 y Fv(\()p 1766 1703 31 2 v Fs(\031)1796 1737 y Fo(r)q Fn(\000)p Fl(1)1869 1730 y Fs(;)8 b(z)r Fv(\))p Fs(:)0 1889 y Fv(Notice)20 b(that)f(the)g(pruned) f(cycle)h(path)g(of)g(\()p Fs(X)892 1896 y Fo(n)920 1889 y Fv(\))h(is)e(a)h(partition)f(of)h(the)h(cycle)f(path)g(of)g(\()p Fs(X)1776 1896 y Fo(n)1804 1889 y Fv(\):)27 b(it)20 b(is)0 1949 y(obtained)c(b)o(y)h(regrouping)d(equiv)m(alen)o(t)j(cycles)g(in)g Fr(M)p Fv(\()p Fs(D)c Fr(n)e Fs(G)p Fv(\).)24 b(Theorem)16 b(5)p Fs(:)p Fv(5)p Fs(:)g Fv(sho)o(ws)g(that)h(the)g Fs(V)1900 1956 y Fo(x)1925 1949 y Fv({)0 2010 y(cost)f(of)g(a)f(cycle)h(path)g(and)f (the)p 616 1970 41 2 v 15 w Fs(V)656 2017 y Fo(x)682 2010 y Fv({cost)g(of)h(the)g(corresp)q(onding)d(pruned)h(cycle)i(path)g(are)f(equal) g(i.e.)0 2070 y(w)o(e)h(ha)o(v)o(e)g Fs(V)217 2077 y Fo(x)242 2070 y Fv(\()p Fs(\031)r Fv(\()p Fs(X)q(;)8 b(D)q(;)g(G)p Fv(\)\))26 b(=)p 601 2030 V 21 w Fs(V)642 2077 y Fo(x)667 2070 y Fv(\()p 686 2043 31 2 v Fs(\031)r Fv(\()p Fs(X)q(;)8 b(D)q(;)g(G)p Fv(\)\).)0 2158 y Fx(Theorem)17 b(6.2.)28 b Fi(\(estimation)16 b(of)h(the)g(probabilit)o(y)d(of)j(a)g(pruned)e(cycle)i(path\))0 2218 y(There)d(exists)g(a)h(p)q(ositiv)o(e)f(constan)o(t)g Fs(K)740 2225 y Fl(2)777 2218 y Fi(\(dep)q(ending)g(only)g(on)g(the)h (cardinalit)o(y)e(of)i Fs(D)8 b Fr(n)f Fs(G)p Fi(\))15 b(suc)o(h)f(that)0 2279 y(for)i(an)o(y)g(pruned)f(cycle)i(path)g Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)p 696 2252 V 8 w(\031)726 2286 y Fl(0)749 2279 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 868 2252 V 8 w(\031)898 2286 y Fo(r)q 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Fr(g)p Fv(\))p Fs(=X)1360 2475 y Fl(0)1398 2468 y Fv(=)13 b Fs(x)1478 2428 y Fg(\001)1532 2468 y Fr(\024)841 2550 y Fv(\()p Fs(K)902 2557 y Fl(2)925 2550 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1020 2529 y Fn(\000)p Fo(K)1084 2534 y Fe(2)1108 2550 y Fv(\))1127 2529 y Fo(r)q Fl(+1)1208 2550 y Fv(exp)8 b Fr(\000)p Fs(\014)p 1362 2510 41 2 v 3 w(V)1402 2557 y Fo(x)1427 2550 y Fv(\()p Fr(f)p Fs(y)r Fr(g)p Fs(;)p 1544 2523 31 2 v 8 w(\031)1575 2557 y Fl(0)1597 2550 y Fs(;)g(:)g(:)g(:)17 b(;)p 1716 2523 V 8 w(\031)1746 2557 y Fo(r)q Fn(\000)p Fl(1)1819 2550 y Fs(;)8 b Fr(f)p Fs(z)r Fr(g)p Fv(\))p Fs(:)954 2670 y Ft(19)p eop %%Page: 20 20 20 19 bop 0 200 a Fj(Pr)m(o)m(of.)28 b Fv(This)17 b(is)g(the)i(same)e(kind)g (of)h(pro)q(of)g(as)g(for)f(theorem)h(5)p Fs(:)p Fv(2)p Fs(:)p Fv(.)25 b(The)18 b(only)g(di\013erence)f(is)h(that)g(w)o(e)0 260 y(no)o(w)e(use)g(the)h(estimation)e(\(for)i(some)f(constan)o(t)g Fs(C)977 267 y Fl(4)999 260 y Fv(\),)h(for)f(all)g Fs(u)g Fv(in)p 1297 232 31 2 v 16 w Fs(\031)1327 267 y Fo(k)q Fn(\000)p Fl(1)1403 260 y Fv(,)0 389 y Fs(C)40 368 y Fn(\000)p Fl(1)36 402 y(4)93 389 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))188 369 y Fo(C)217 374 y Fe(4)247 389 y Fv(exp)9 b Fr(\000)p Fs(\014)s(C)p 438 377 25 2 v 8 x Fo(\031)461 402 y Fh(k)p Fd(\000)p Fe(1)530 389 y Fv(\()p 549 362 31 2 v Fs(\031)579 396 y Fo(k)q Fn(\000)p Fl(1)655 389 y Fs(;)p 677 362 V 8 w(\031)707 396 y Fo(k)732 389 y Fv(\))629 464 y Fr(\024)31 b Fs(P)731 471 y Fo(\014)757 464 y Fv(\()p Fs(X)p 817 451 23 2 v 7 x Fo(\034)840 477 y Fh(k)879 464 y Fr(2)p 926 436 31 2 v 14 w Fs(\031)956 471 y Fo(k)980 464 y Fs(=X)p 1046 451 23 2 v 7 x Fo(\034)1069 477 y Fh(k)p Fd(\000)p Fe(1)1151 464 y Fv(=)14 b Fs(u)p Fv(\))30 b Fr(\024)1100 545 y Fs(C)1136 552 y Fl(4)1158 545 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1253 525 y Fn(\000)p Fo(C)1313 530 y Fe(4)1344 545 y Fv(exp)8 b Fr(\000)p Fs(\014)s(C)p 1534 532 25 2 v 8 x Fo(\031)1558 558 y Fh(k)p Fd(\000)p Fe(1)1626 545 y Fv(\()p 1645 518 31 2 v Fs(\031)1676 552 y Fo(k)q Fn(\000)p Fl(1)1751 545 y Fs(;)p 1773 518 V 8 w(\031)1804 552 y Fo(k)1828 545 y Fv(\))p Fs(:)50 b Ff(\003)60 717 y Fv(The)16 b(k)o(ey)h(fact)g(that)g(mak)o(es)p 608 677 41 2 v 16 w Fs(V)648 724 y Fo(x)690 717 y Fv(a)f(go)q(o)q(d)h(cost)g (function)f(to)h(w)o(ork)f(with)g(is)g(the)h(follo)o(wing)0 814 y Fx(Lemma)22 b(6.3.)63 b Fi(Let)22 b Fv(\()p 474 787 31 2 v Fs(\031)504 821 y Fl(0)527 814 y Fs(;)8 b(:)g(:)g(:)16 b(;)p 645 787 V 8 w(\031)676 821 y Fo(r)698 814 y Fv(\))21 b Fi(b)q(e)g(a)g(sequence)g(suc)o(h)f(that)p 1296 787 V 21 w Fs(\031)1326 821 y Fo(k)1372 814 y Fr(2)p 1426 774 60 2 v 21 w(M)p Fv(\()p Fs(D)c Fr(n)e Fs(G)p Fv(\))p Fi(,)p 1695 787 31 2 v 22 w Fs(\031)1725 821 y Fo(k)1771 814 y Fr(6)p Fv(=)p 1830 787 V 20 w Fs(\031)1861 821 y Fo(k)q Fn(\000)p Fl(1)1936 814 y Fi(,)0 874 y Fv(1)g Fr(\024)f Fs(k)j Fr(\024)d Fs(r)q Fi(.)23 b(There)15 b(exists)g(a)g(p)q(ositiv)o(e)g(constan)o(t)g Fs(\016)i Fi(indep)q(enden)o(t)d(of)i Fv(\()p 1342 846 V Fs(\031)1373 881 y Fl(0)1395 874 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 1514 846 V 8 w(\031)1544 881 y Fo(r)1566 874 y Fv(\))f Fi(suc)o(h)e(that,)i(if)f Fs(r)j Fi(is)0 935 y(greater)e(than)g(the)h(cardinalit)o(y)e(of)p 677 895 60 2 v 17 w Fr(M)p Fv(\()p Fs(D)e Fr(n)e Fs(G)p Fv(\))p Fi(,)17 b(then)651 1040 y Fo(r)625 1055 y Fg(X)625 1163 y Fo(k)q Fl(=1)714 1103 y Fs(C)p 750 1090 25 2 v 7 x Fo(\031)774 1116 y Fh(k)p Fd(\000)p Fe(1)842 1103 y Fv(\()p 861 1075 31 2 v Fs(\031)892 1110 y Fo(k)q Fn(\000)p Fl(1)967 1103 y Fs(;)p 989 1075 V 8 w(\031)1019 1110 y Fo(k)1044 1103 y Fv(\))31 b Fr(\025)f Fs(\016)i(>)f Fv(0)p Fs(:)0 1322 y Fj(R)m(emark.)c Fv(The)16 b(graph)f(of)h(the)h(jumps)e(of)i(n)o(ull)e(cost)h(o)o(v)o(er)p 1104 1282 60 2 v 15 w Fr(M)p Fv(\()p Fs(D)d Fr(n)d Fs(G)p Fv(\))17 b(has)f(no)g(lo)q(op.)22 b(W)l(e)16 b(could)g(ha)o(v)o(e)0 1384 y(replaced)f(the)g(cardinalit)o(y)g(of)p 584 1344 V 15 w Fr(M)p Fv(\()p Fs(D)c Fr(n)e Fs(G)p Fv(\))16 b(b)o(y)f(the)h(diameter)f(of) g(this)g(graph,)g(that)h(is)f(the)g(n)o(um)o(b)q(er)f(of)0 1443 y(v)o(ertices)i(of)h(its)f(longest)h(paths.)0 1541 y Fj(Pr)m(o)m(of.)28 b Fv(Since)18 b Fs(r)j Fv(is)d(greater)h(than)f(or)g(equal)h(to)g Fr(j)p 948 1501 V(M)p Fv(\()p Fs(D)c Fr(n)d Fs(G)p Fv(\))p Fr(j)p Fv(,)20 b(t)o(w)o(o)e(elemen)o(ts)p 1528 1514 31 2 v 18 w Fs(\031)1558 1548 y Fo(i)1594 1541 y Fv(and)p 1693 1514 V 18 w Fs(\031)1723 1548 y Fo(j)1744 1541 y Fv(,)h(0)f Fr(\024)f Fs(i)h(<)0 1601 y(j)i Fr(\024)e Fs(r)i Fv(of)f(the)h(sequence)p 496 1573 V 18 w Fs(\031)527 1608 y Fl(0)549 1601 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 668 1573 V 8 w(\031)698 1608 y Fo(r)739 1601 y Fv(ha)o(v)o(e)h(to)h(b)q (e)h(equal.)28 b(Since)p 1280 1573 V 19 w Fs(\031)1310 1608 y Fo(i)1344 1601 y Fr(6)p Fv(=)p 1401 1573 V 18 w Fs(\031)1431 1608 y Fo(i)p Fl(+1)1498 1601 y Fv(,)19 b(then)g Fs(j)h(>)e(i)13 b Fv(+)f(1)19 b(and)p 0 1633 V 0 1661 a Fs(\031)30 1668 y Fo(i)47 1661 y Fs(;)p 69 1633 V 8 w(\031)99 1668 y Fo(i)p Fl(+1)166 1661 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 285 1633 V 8 w(\031)315 1668 y Fo(j)350 1661 y Fv(=)p 403 1633 V 14 w Fs(\031)433 1668 y Fo(i)466 1661 y Fv(is)f(a)g(lo)q(op.)22 b(Therefore)15 b(one)h(of)h(its)f (jumps)f(at)i(least)f(has)g(a)g(p)q(ositiv)o(e)g(cost)h(and)0 1720 y(the)g(sum)e(of)i(the)g(lemma)e(is)h(not)h(smaller)e(than)257 1848 y Fs(\016)24 b Fv(=)e(min)o Fr(f)p Fs(C)p 508 1835 28 2 v 7 x Fo(\031)535 1848 y Fv(\()p 554 1820 31 2 v Fs(\031)r(;)8 b(y)r Fv(\))15 b(:)p 694 1820 V 14 w Fs(\031)h Fr(2)p 785 1808 60 2 v 14 w(M)p Fv(\()p Fs(D)d Fr(n)e Fs(G)p Fv(\))p Fs(;)25 b(y)16 b Fr(2)e Fs(B)r Fv(\()p 1197 1820 31 2 v Fs(\031)s Fv(\))p Fs(;)25 b(C)p 1322 1835 28 2 v 7 x Fo(\031)1349 1848 y Fv(\()p 1368 1820 31 2 v Fs(\031)r(;)8 b(y)r Fv(\))15 b Fs(>)e Fv(0)8 b Fr(g)p Fs(:)50 b Ff(\003)0 2015 y Fx(Corollary)20 b(6.4.)28 b Fi(The)16 b(cost)p 578 1975 41 2 v 17 w Fs(V)618 2022 y Fo(x)659 2015 y Fi(giv)o(es)h(a)f(con)o(trol)g(on)g(the)h(length)f(i.e.)0 2075 y Fr(8)p Fs(\025)d(>)g Fv(0)50 b Fr(9)8 b Fs(R)p Fv(\()p Fs(\025)p Fv(\))51 b Fr(8)p Fs(r)14 b Fr(\025)g Fs(R)p Fv(\()p Fs(\025)p Fv(\))50 b Fr(8)p Fv(\()p 711 2047 31 2 v Fs(\031)741 2082 y Fl(0)763 2075 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 882 2047 V 8 w(\031)912 2082 y Fo(r)934 2075 y Fv(\))d Fr(2)p 1015 2035 39 2 v 15 w Fv(\011)o(\()p Fs(D)f Fr(n)e Fs(G)p Fv(\))p 1320 2035 41 2 v 100 w Fs(V)1361 2082 y Fo(x)1386 2075 y Fv(\()p 1405 2047 31 2 v Fs(\031)1435 2082 y Fl(0)1458 2075 y Fs(;)d(:)g(:)g(:)16 b(;)p 1576 2047 V 8 w(\031)1607 2082 y Fo(r)1629 2075 y Fv(\))e Fr(\025)g Fs(\025:)0 2174 y Fx(Corollary)20 b(6.5.)51 b Fi(The)16 b(cost)h(function)p 797 2134 41 2 v 17 w Fs(V)837 2181 y Fo(x)879 2174 y Fi(is)f(a)h(go)q(o)q(d)g(rate)g(function)f(on)p 1460 2134 39 2 v 16 w Fv(\011)h Fi(i.e.)22 b(its)17 b(lev)o(el)g(sets)f(are)0 2234 y(compact)g(\(\014nite)h(in)f(our)g(situation\).)0 2333 y Fj(Pr)m(o)m(of.)28 b Fv(Corollary)20 b(6)p Fs(:)p Fv(4)p Fs(:)i Fv(sho)o(ws)f(that)h(the)g(sets)g Fr(f)8 b Fv(\()p 990 2305 31 2 v Fs(\031)1021 2340 y Fl(0)1043 2333 y Fs(;)g(:)g(:)g(:)17 b(;)p 1162 2305 V 8 w(\031)1193 2340 y Fo(r)1215 2333 y Fv(\))23 b(:)p 1294 2293 41 2 v 23 w Fs(V)1334 2340 y Fo(x)1359 2333 y Fv(\()p 1378 2305 31 2 v Fs(\031)1409 2340 y Fl(0)1431 2333 y Fs(;)8 b(:)g(:)g(:)17 b(;)p 1550 2305 V 8 w(\031)1580 2340 y Fo(r)1602 2333 y Fv(\))24 b Fr(\024)e Fs(\025)8 b Fr(g)23 b Fv(con)o(tain)0 2393 y(only)16 b(sequences)g(of)h(b)q(ounded)f(length)g (and)g(are)g(th)o(us)f(\014nite.)50 b Ff(\003)60 2490 y Fv(The)13 b(length)h(of)f(the)h(cycle)h(path)e Fs(\031)r Fv(\()p Fs(X)t Fv(\))i(\(i.e.)21 b(the)14 b(n)o(um)o(b)q(er)d(of)j(jumps)f(it)h(in)o(v)o (olv)o(es\))e(will)h(b)q(e)h(denoted)0 2550 y(b)o(y)i Fr(j)p Fs(\031)r Fv(\()p Fs(X)t Fv(\))p Fr(j)i Fv(in)e(the)g(sequel.)22 b(F)l(or)16 b(instance)g Fr(j)p Fv(\()p Fs(\031)884 2557 y Fl(0)907 2550 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(\031)1053 2557 y Fo(r)1076 2550 y Fv(\))p Fr(j)14 b Fv(=)g Fs(r)q Fv(.)954 2670 y Ft(20)p eop %%Page: 21 21 21 20 bop 0 200 a Fx(Prop)r(osition)20 b(6.6.)55 b Fi(There)18 b(exist)g(a)g(p)q(ositiv)o(e)g(constan)o(t)f Fs(K)1194 207 y Fl(3)1235 200 y Fi(and)g(a)i(function)e Fs(h)f Fv(:)g Fk(N)i Fr(!)e Fk(R)1806 207 y Fl(+)1854 200 y Fi(suc)o(h)0 260 y(that)h Fv(lim)177 267 y Fl(+)p Fn(1)258 260 y Fs(h)d Fv(=)f(+)p Fr(1)j Fi(and)361 364 y Fs(P)393 371 y Fo(\014)420 364 y Fv(\()p Fr(j)p 453 336 31 2 v Fs(\031)s Fv(\()p Fs(X)t Fv(\))p Fr(j)e(\025)g Fs(R=X)752 371 y Fl(0)788 364 y Fv(=)g Fs(x)p Fv(\))31 b Fr(\024)f Fv(\()p Fs(K)1049 371 y Fl(3)1072 364 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1167 343 y Fn(\000)p Fo(K)1231 348 y Fe(3)1255 364 y Fv(\))1274 343 y Fo(R)1315 364 y Fv(exp)9 b Fr(\000)p Fs(\014)s(h)p Fv(\()p Fs(R)p Fv(\))p Fs(:)0 493 y Fj(Pr)m(o)m(of.)34 596 y Fs(P)66 603 y Fo(\014)93 596 y Fv(\()p Fr(j)p 126 568 V Fs(\031)s Fv(\()p Fs(X)t Fv(\))p Fr(j)14 b(\025)g Fs(R=X)425 603 y Fl(0)461 596 y Fv(=)g Fs(x)p Fv(\))140 688 y(=)271 640 y Fg(X)209 747 y Fo(y)q(;)p 242 727 25 2 v(\031)267 752 y Fe(0)287 747 y Fo(;:::)6 b(;)p 353 727 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1776 y Fv(\))h(+)g Fs(V)i Fv(\()p Fs(s)357 1783 y Fl(0)380 1776 y Fs(;)d(s)425 1783 y Fl(1)448 1776 y Fv(\))i(+)543 1713 y Fo(r)517 1728 y Fg(X)517 1835 y Fo(k)q Fl(=1)597 1720 y Fg(\020)627 1776 y Fs(W)d Fv(\()p Fs(s)723 1783 y Fl(2)p Fo(k)768 1776 y Fv(\))q(+)q Fs(V)k Fv(\()p Fs(s)910 1783 y Fl(2)p Fo(k)955 1776 y Fs(;)d(s)1000 1783 y Fl(2)p Fo(k)q Fl(+1)1095 1776 y Fv(\))q Fr(\000)q Fs(H)t Fv(\()p Fs(\031)r Fv(\()p Fs(s)1291 1783 y Fl(2)p Fo(k)1337 1776 y Fs(;)g(D)r Fr(n)q Fs(G)p Fv(\)\))q Fr(\000)q Fs(W)f Fv(\()p Fs(\031)r Fv(\()p Fs(s)1691 1783 y Fl(2)p Fo(k)1737 1776 y Fs(;)h(D)r Fr(n)q Fs(G)p Fv(\)\))1905 1720 y Fg(\021)1936 1776 y Fs(:)60 1956 y Fv(Since)22 b(the)g(sets)h Fs(D)h Fv(and)e Fs(G)h Fv(will)f(b)q(e)g(\014xed)h(in)f(the)g(sequel,)i(w)o(e)e(will)g(drop)g (them)g(in)g(the)h(nota-)0 2016 y(tions)f(whenev)o(er)f(no)h(confusion)g(is)g (p)q(ossible:)32 b(for)22 b(instance)g(w)o(e)g(will)g(write)g Fs(V)1551 2023 y Fo(x)1576 2016 y Fs(;)8 b Fr(S)t Fs(;)g Fr(S)t Fv(\()p Fs(X)t Fv(\))24 b(instead)0 2076 y(of)17 b Fs(V)86 2083 y Fo(x)111 2076 y Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))p Fs(;)g Fr(S)t Fv(\()p Fs(D)q(;)g(G)p Fv(\))p Fs(;)g Fr(S)d Fv(\()p Fs(X)q(;)j(D)r(;)g(G)p Fv(\))q(.)0 2136 y(The)16 b(next)h(lemma)f(giv)o(es)g (the)h(probabilit)o(y)e(of)h(one)h(\014xed)f(jump)g(of)g(a)h(saddle)e(path.)0 2223 y Fx(Lemma)i(7.2.)28 b Fi(\(last)17 b(p)q(oin)o(t)f(and)g(exit)h(p)q (oin)o(t)f(of)h(a)f(cycle\))0 2283 y(There)g(exists)h(a)f(p)q(ositiv)o(e)g (constan)o(t)g Fs(K)750 2290 y Fl(5)790 2283 y Fi(suc)o(h)f(that)i(for)f(an)o (y)g(cycle)h Fs(\031)r Fi(,)g Fs(x;)8 b(y)17 b Fr(2)d Fs(\031)k Fi(and)e Fs(z)h Fr(2)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))p Fi(,)0 2393 y Fs(K)46 2372 y Fn(\000)p Fl(1)42 2406 y(5)99 2393 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))194 2373 y Fo(K)227 2378 y Fe(5)259 2393 y Fv(exp)8 b Fr(\000)p Fs(\014)413 2353 y Fg(\000)436 2393 y Fs(W)f Fv(\()p Fs(y)r Fv(\))12 b(+)f Fs(V)g Fv(\()p Fs(y)r(;)d(z)r Fv(\))k Fr(\000)f Fs(H)t Fv(\()p Fs(\031)r Fv(\))h Fr(\000)f Fs(W)c Fv(\()p Fs(\031)r Fv(\))1126 2353 y Fg(\001)515 2468 y Fr(\024)30 b Fs(P)616 2475 y Fo(\014)643 2468 y Fv(\()p Fs(X)703 2477 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))798 2468 y Fv(=)13 b Fs(z)r(;)8 b(X)938 2477 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))p Fn(\000)p Fl(1)1085 2468 y Fv(=)13 b Fs(y)r(=X)1229 2475 y Fl(0)1266 2468 y Fv(=)g Fs(x)p Fv(\))32 b Fr(\024)790 2550 y Fs(K)832 2557 y Fl(5)854 2550 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))949 2529 y Fn(\000)p Fo(K)1013 2534 y Fe(5)1045 2550 y Fv(exp)9 b Fr(\000)p Fs(\014)1200 2510 y Fg(\000)1222 2550 y Fs(W)e Fv(\()p Fs(y)r Fv(\))12 b(+)f Fs(V)g Fv(\()p Fs(y)r(;)d(z)r Fv(\))k Fr(\000)f Fs(H)t Fv(\()p Fs(\031)r Fv(\))i Fr(\000)d Fs(W)d Fv(\()p Fs(\031)r Fv(\))1912 2510 y Fg(\001)1936 2550 y Fs(:)954 2670 y Ft(22)p eop %%Page: 23 23 23 22 bop 0 200 a Fj(Pr)m(o)m(of)20 b(of)e(lemma.)26 b Fv(By)17 b(summing)d(o)o(v)o(er)i(the)h(p)q(ossible)e(v)m(alues)h(of)h(the)g(last)f (instan)o(t)g(in)g Fs(\031)r Fv(,)h(w)o(e)f(obtain)57 372 y Fs(P)89 379 y Fo(\014)116 372 y Fv(\()p Fs(X)176 380 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))271 372 y Fv(=)d Fs(z)r(;)8 b(X)411 380 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))p Fn(\000)p Fl(1)558 372 y Fv(=)13 b Fs(y)r(=X)702 379 y Fl(0)739 372 y Fv(=)g Fs(x)p Fv(\))23 b(=)938 309 y Fn(1)922 324 y Fg(X)922 431 y Fo(k)q Fl(=0)1003 372 y Fs(P)1035 379 y Fo(\014)1062 372 y Fv(\()p Fs(\034)6 b Fv(\()p Fs(\031)r Fv(\))14 b Fs(>)g(k)r(;)i(X)1343 379 y Fo(k)1381 372 y Fv(=)e Fs(y)r(;)8 b(X)1523 379 y Fo(k)q Fl(+1)1612 372 y Fv(=)14 b Fs(z)r(=X)1756 379 y Fl(0)1793 372 y Fv(=)f Fs(x)p Fv(\))619 539 y(=)680 468 y Fg(\022)741 476 y Fn(1)725 491 y Fg(X)724 599 y Fo(k)q Fl(=0)805 539 y Fs(P)837 546 y Fo(\014)864 539 y Fv(\()p Fs(\034)6 b Fv(\()p Fs(\031)r Fv(\))15 b Fs(>)e(k)r(;)j(X)1145 546 y Fo(k)1184 539 y Fv(=)d Fs(y)r(=X)1328 546 y Fl(0)1365 539 y Fv(=)h Fs(x)p Fv(\))1465 468 y Fg(\023)1502 539 y Fs(P)1534 546 y Fo(\014)1561 539 y Fv(\()p Fs(X)1621 546 y Fl(1)1658 539 y Fv(=)f Fs(z)r(=X)1801 546 y Fl(0)1838 539 y Fv(=)h Fs(y)r Fv(\))p Fs(:)0 718 y Fv(Lemma)i(3)p Fs(:)p Fv(5)p Fs(:)g Fv(yields)g(the)h(existence)g(of)f(a)h(p)q(ositiv)o(e)f (constan)o(t)g Fs(C)1229 725 y Fl(5)1268 718 y Fv(suc)o(h)f(that)0 890 y Fs(C)40 868 y Fn(\000)p Fl(1)36 903 y(5)93 890 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))188 869 y Fo(C)217 874 y Fe(5)247 890 y Fv(exp)9 b Fr(\000)p Fs(\014)s(W)449 897 y Fo(\031)475 890 y Fv(\()p Fs(x;)f(y)r Fv(\))16 b Fr(\024)674 827 y Fn(1)658 842 y Fg(X)657 949 y Fo(k)q Fl(=0)738 890 y Fs(P)770 897 y Fo(\014)797 890 y Fv(\()p Fs(\034)6 b Fv(\()p Fs(\031)r Fv(\))15 b Fs(>)e(k)r(;)j(X)1078 897 y Fo(k)1117 890 y Fv(=)d Fs(y)r(=X)1261 897 y Fl(0)1298 890 y Fv(=)g Fs(x)p Fv(\))1310 1019 y Fr(\024)h Fs(C)1399 1026 y Fl(5)1421 1019 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1516 998 y Fn(\000)p Fo(C)1576 1003 y Fe(5)1606 1019 y Fv(exp)9 b Fr(\000)p Fs(\014)s(W)1808 1026 y Fo(\031)1834 1019 y Fv(\()p Fs(x;)f(y)r Fv(\))0 1153 y(By)18 b(lemma)d(4)p Fs(:)p Fv(7)p Fs(:)p Fv(,)i(w)o(e)f(kno)o(w)h(that)g Fs(W)707 1160 y Fo(\031)734 1153 y Fv(\()p Fs(x;)8 b(y)r Fv(\))16 b(=)e Fs(C)953 1162 y Fo(\031)q Fn(nf)p Fo(y)q Fn(g)1063 1153 y Fv(\()p Fs(x;)8 b(y)r Fv(\))13 b Fr(\000)e Fs(C)1276 1162 y Fo(\031)q Fn(nf)p Fo(y)q Fn(g)1385 1153 y Fv(\()p Fs(y)r(;)d(\031)1482 1135 y Fo(c)1503 1153 y Fv(\).)24 b(Since)16 b Fs(\031)j Fv(is)e(a)g(cycle,)0 1213 y(the)g(cost)f Fs(C)224 1222 y Fo(\031)q Fn(nf)p Fo(y)q Fn(g)334 1213 y Fv(\()p Fs(x;)8 b(y)r Fv(\))18 b(is)e(n)o(ull.)21 b(Moreo)o(v)o(er)15 b(prop)q(osition)f(4)p Fs(:)p Fv(14)p Fs(:)i Fv(giv)o(es)390 1348 y Fr(\000)p Fs(W)476 1355 y Fo(\031)503 1348 y Fv(\()p Fs(x;)8 b(y)r Fv(\))24 b(=)e Fs(C)738 1357 y Fo(\031)q Fn(nf)p Fo(y)q Fn(g)847 1348 y Fv(\()p Fs(y)r(;)8 b(\031)944 1328 y Fo(c)965 1348 y Fv(\))23 b(=)13 b Fs(H)g Fv(\()p Fs(\031)r Fv(\))f(+)f Fs(W)c Fv(\()p Fs(\031)r Fv(\))12 b Fr(\000)f Fs(W)c Fv(\()p Fs(y)r Fv(\))p Fs(:)0 1483 y Fv(The)16 b(required)g(estimation)g(follo)o(ws)f(easily)l(.)49 b Ff(\003)0 1587 y Fx(Corollary)20 b(7.3.)50 b Fi(F)l(or)15 b(an)o(y)h(cycle)h Fs(\031)r Fi(,)g(an)o(y)f(p)q(oin)o(t)g Fs(x)h Fi(in)f Fs(\031)j Fi(and)d(an)o(y)g(p)q(oin)o(t)g Fs(y)j Fi(in)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))p Fi(,)i(w)o(e)e(ha)o(v)o(e)343 1722 y Fs(C)379 1729 y Fo(\031)405 1722 y Fv(\()p Fs(x;)8 b(y)r Fv(\))24 b(=)e(min)o Fr(f)8 b Fs(W)f Fv(\()p Fs(z)r Fv(\))12 b(+)f Fs(V)g Fv(\()p Fs(z)r(;)d(y)r Fv(\))15 b(:)f Fs(z)i Fr(2)e Fs(\031)r Fr(g)e(\000)e Fs(H)t Fv(\()p Fs(\031)r Fv(\))j Fr(\000)e Fs(W)c Fv(\()p Fs(\031)r Fv(\))p Fs(:)0 1901 y Fx(Theorem)17 b(7.4.)28 b Fi(\(estimation)16 b(of)h(the)g(probabilit)o(y)d(of)j(a)g(saddle)e(path\))0 1961 y(There)f(exists)g(a)h(p)q(ositiv)o(e)f(constan)o(t)g Fs(K)740 1968 y Fl(6)777 1961 y Fi(\(dep)q(ending)g(only)g(on)g(the)h(cardinalit)o(y)e (of)i Fs(D)8 b Fr(n)f Fs(G)p Fi(\))15 b(suc)o(h)f(that)0 2021 y(for)i(an)o(y)g(saddle)g(path)g Fv(\()p Fs(s)481 2028 y Fl(0)504 2021 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(s)646 2028 y Fl(2)p Fo(r)q Fl(+1)739 2021 y Fv(\))17 b Fi(in)f Fr(S)t Fv(\()p Fs(D)d Fr(n)e Fs(G)p Fv(\))17 b Fi(and)f(an)o(y)g Fs(\014)j Fi(w)o(e)e(ha)o(v)o(e)0 2157 y Fv(\()p Fs(K)65 2136 y Fn(\000)p Fl(1)61 2171 y(6)119 2157 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))214 2137 y Fo(K)247 2142 y Fe(6)270 2157 y Fv(\))289 2137 y Fo(r)q Fl(+1)370 2157 y Fv(exp)9 b Fr(\000)p Fs(\014)s(V)554 2164 y Fo(x)578 2157 y Fv(\()p Fs(s)620 2164 y Fl(0)643 2157 y Fs(;)f(:)g(:)g(:)17 b(;)8 b(s)785 2164 y Fl(2)p Fo(r)q Fl(+1)878 2157 y Fv(\))514 2232 y Fr(\024)30 b Fs(P)615 2239 y Fo(\014)642 2232 y Fv(\()p Fr(S)t Fv(\()p Fs(X)t Fv(\))15 b(=)e(\()p Fs(s)887 2239 y Fl(0)910 2232 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(s)1052 2239 y Fl(2)p Fo(r)q Fl(+1)1145 2232 y Fv(\))p Fs(=X)1230 2239 y Fl(0)1267 2232 y Fv(=)13 b Fs(x)p Fv(\))32 b Fr(\024)1042 2311 y Fv(\()p Fs(K)1103 2318 y Fl(6)1126 2311 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1221 2290 y Fn(\000)p Fo(K)1285 2295 y Fe(6)1309 2311 y Fv(\))1328 2290 y Fo(r)q Fl(+1)1409 2311 y Fv(exp)8 b Fr(\000)p Fs(\014)s(V)1592 2318 y Fo(x)1617 2311 y Fv(\()p Fs(s)1659 2318 y Fl(0)1682 2311 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(s)1824 2318 y Fl(2)p Fo(r)q Fl(+1)1917 2311 y Fv(\))p Fs(:)0 2490 y Fj(Pr)m(o)m(of.)28 b Fv(If)22 b Fs(S)s Fv(\()p Fs(X)t Fv(\))g(=)g(\()p Fs(s)454 2497 y Fl(0)477 2490 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(s)619 2497 y Fl(2)p Fo(r)q Fl(+1)712 2490 y Fv(\),)23 b(the)f(last)g(visit)f(of)h (the)g(c)o(hain)f(\()p Fs(X)1423 2497 y Fo(n)1450 2490 y Fv(\))i(in)e Fs(G)h Fv(o)q(ccurred)e(at)i Fs(s)1913 2497 y Fl(0)1936 2490 y Fv(.)0 2550 y(Conditioning)10 b(according)g(to)i(the)g(time)g(of)g(this)f (last)g(visit)h(and)f(then)g(applying)g(the)h(Mark)o(o)o(v)e(prop)q(ert)o(y)l (,)954 2670 y Ft(23)p eop %%Page: 24 24 24 23 bop 0 200 a Fv(w)o(e)16 b(obtain)0 315 y Fs(P)32 322 y Fo(\014)59 315 y Fv(\()p Fr(S)t Fv(\()p Fs(X)t Fv(\))f(=)e(\()p Fs(s)304 322 y Fl(0)328 315 y Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(s)469 322 y Fl(2)p Fo(r)q Fl(+1)562 315 y Fv(\))p Fs(=X)647 322 y Fl(0)684 315 y Fv(=)14 b Fs(x)p Fv(\))0 431 y(=)86 369 y Fn(1)69 384 y 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Fv({cost.)954 2670 y Ft(27)p eop %%Page: 28 28 28 27 bop 0 200 a Fx(Prop)r(osition)19 b(8.8.)27 b Fi(\(concen)o(tration)16 b(of)h(the)g(la)o(w)f(of)h(the)f(saddle)g(path\))567 341 y Fv(lim)550 374 y Fo(\014)r Fn(!1)679 341 y Fs(P)711 348 y Fo(\014)738 341 y Fv(\()p Fs(V)786 348 y Fo(x)811 341 y Fv(\()p Fr(S)t Fv(\()p Fs(X)t Fv(\)\))f(=)f(0)p Fs(=X)1125 348 y Fl(0)1161 341 y Fv(=)g Fs(x)p Fv(\))31 b(=)f(1)p Fs(:)60 514 y Fv(W)l(e)20 b(\014nally)e(giv)o(e)i(a)f(criterion)g(to)g(decide)h(whether)f(a)g(saddle)g (path)g(has)g(a)g(p)q(ositiv)o(e)g(asymptotic)0 574 y(probabilit)o(y)c(to)i (o)q(ccur.)0 685 y Fx(Prop)r(osition)i(8.9.)27 b Fi(Assume)16 b(that)h Fv(lim)8 b(inf)737 718 y Fo(\014)r Fn(!1)866 685 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))15 b Fs(>)f Fv(0)p Fi(.)22 b(Then)95 854 y Fv(lim)8 b(inf)112 886 y Fo(\014)r Fn(!1)249 854 y Fs(P)281 861 y Fo(\014)308 854 y Fv(\()p Fr(S)t Fv(\()p Fs(X)t Fv(\))15 b(=)f(\()p Fs(s)554 861 y Fl(0)577 854 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(s)719 861 y Fl(2)p 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Fs(;)8 b(:)g(:)g(:)16 b(;)8 b(s)846 1114 y Fl(2)p Fo(r)q Fl(+1)939 1107 y Fv(\))19 b Fi(b)q(e)f(a)f (saddle)g(path)h(in)f Fr(S)t Fv(\()p Fs(D)q(;)8 b(G)p Fv(\))20 b Fi(of)e(n)o(ull)e Fs(V)1800 1114 y Fo(x)1825 1107 y Fi({cost.)0 1166 y(This)e(saddle)g(path)h(visits)g(successiv)o(ely)f(the)i(three)f(sets)g Fr(V)t Fv(\()p Fs(G;)8 b(D)1239 1148 y Fo(c)1260 1166 y Fv(\))p Fs(;)17 b Fr(R)p Fv(\()p Fs(G;)8 b(D)1474 1148 y Fo(c)1496 1166 y Fv(\))p Fs(;)17 b Fr(V)t Fv(\()p Fs(D)1642 1148 y Fo(c)1662 1166 y Fs(;)8 b(G)p Fv(\))p Fi(:)22 b(if)16 b(w)o(e)f(let)0 1226 y Fs(r)22 1233 y Fn(\003)59 1226 y Fv(=)f(min)o Fr(f)p Fs(k)h Fv(:)f Fs(s)312 1233 y Fl(2)p Fo(k)q Fl(+1)421 1226 y Fr(62)g(V)t Fv(\()p Fs(G;)8 b(D)625 1208 y Fo(c)646 1226 y Fv(\))g Fr(g)17 b Fi(and)f Fs(r)835 1208 y Fn(\003)873 1226 y Fv(=)d(min)o Fr(f)p Fs(k)i Fr(\025)f Fs(r)1149 1233 y Fn(\003)1186 1226 y Fv(:)g Fs(s)1237 1233 y Fl(2)p Fo(k)q Fl(+1)1346 1226 y Fr(62)g(R)p Fv(\()p Fs(G;)8 b(D)1557 1208 y Fo(c)1579 1226 y Fv(\))g 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Fs(;)8 b(D)q Fv(\))g Fr(g)p Fi(.)24 b(Th)o(us)15 b(a)i(saddle)e(path)i (of)0 1689 y(n)o(ull)f Fs(V)128 1696 y Fo(x)153 1689 y Fi({cost)h(ma)o(y)f(b) q(e)h(decomp)q(osed)f(in)o(to)h(three)f(parts:)22 b(an)17 b(ascending)e(part) i Fv(\()p Fs(s)1561 1696 y Fo(k)1586 1689 y Fs(;)g Fv(0)d Fr(\024)g Fs(k)i Fr(\024)e Fv(2)p Fs(r)1851 1696 y Fn(\003)1875 1689 y Fv(\))p Fi(,)j(a)0 1749 y(ridge)f(part)g Fv(\()p Fs(s)275 1756 y Fo(k)300 1749 y Fs(;)g Fv(2)p Fs(r)377 1756 y Fn(\003)415 1749 y Fs(<)d(k)i Fr(\024)f Fv(2)p Fs(r)609 1731 y Fn(\003)633 1749 y Fv(\))j Fi(and)f(a)g(descending)f(part)h Fv(\()p Fs(s)1209 1756 y Fo(k)1234 1749 y Fs(;)h Fv(2)p Fs(r)1313 1731 y Fn(\003)1350 1749 y Fs(<)d(k)h Fr(\024)f Fv(2)p Fs(r)f Fv(+)e(1\))p Fi(.)0 1842 y Fj(Pr)m(o)m(of.)28 b Fv(W)l(e)23 b(\014rst)e(pro)o(v)o(e)h(that)h(the) g(t)o(w)o(o)f(indices)f Fs(r)996 1824 y Fl(1)995 1854 y Fn(\003)1044 1842 y Fv(=)i(min)o Fr(f)p Fs(k)j Fv(:)e Fs(s)1327 1849 y Fl(2)p Fo(k)q Fl(+1)1446 1842 y Fr(62)g Fs(\031)r Fv(\()p Fs(s)1575 1849 y Fl(0)1598 1842 y Fs(;)8 b(D)q Fv(\))g Fr(g)24 b Fv(and)e Fs(r)1864 1824 y Fl(2)1863 1854 y Fn(\003)1911 1842 y Fv(=)0 1902 y(min)o Fr(f)p Fs(k)h Fv(:)f Fs(s)216 1909 y Fl(2)p Fo(k)q Fl(+1)333 1902 y Fr(62)f(V)t Fv(\()p Fs(G;)8 b(D)544 1884 y Fo(c)566 1902 y Fv(\))g Fr(g)22 b Fv(are)e(equal.)36 b(W)l(e)22 b(ha)o(v)o(e)f Fs(\031)r Fv(\()p Fs(s)1175 1909 y Fl(0)1198 1902 y Fs(;)8 b(D)q Fv(\))23 b Fr(\032)e(V)t Fv(\()p Fs(G;)8 b(D)1521 1884 y Fo(c)1542 1902 y Fv(\))22 b(so)f(that)h Fs(r)1785 1884 y Fl(1)1784 1914 y Fn(\003)1829 1902 y Fr(\024)g Fs(r)1913 1884 y Fl(2)1912 1914 y Fn(\003)1936 1902 y Fv(.)0 1962 y(Theorem)f(9)p Fs(:)p Fv(1)p Fs(:)i Fv(sho)o(ws)e(that)h Fs(C)615 1971 y Fo(D)q Fn(n)p Fo(G)703 1962 y Fv(\()p Fs(s)745 1970 y Fl(2)p Fo(r)784 1961 y Fe(1)783 1980 y Fd(\003)805 1970 y Fl(+1)857 1962 y Fs(;)8 b(D)921 1944 y Fo(c)943 1962 y Fv(\))24 b(=)f(0)g(so)f(that)h Fs(s)1300 1970 y Fl(2)p Fo(r)1339 1961 y Fe(1)1338 1980 y Fd(\003)1359 1970 y Fl(+1)1435 1962 y Fv(is)f(not)g(in)h Fr(V)t Fv(\()p Fs(G;)8 b(D)1807 1944 y Fo(c)1828 1962 y Fv(\))23 b(and)0 2026 y Fs(r)23 2008 y Fl(1)22 2038 y Fn(\003)60 2026 y Fr(\025)14 b Fs(r)136 2008 y Fl(2)135 2038 y Fn(\003)159 2026 y Fv(.)22 b(Th)o(us)15 b Fs(r)343 2033 y Fn(\003)380 2026 y Fv(=)f Fs(r)456 2008 y Fl(1)455 2038 y Fn(\003)493 2026 y Fv(=)f Fs(r)568 2008 y Fl(2)567 2038 y Fn(\003)592 2026 y Fv(.)0 2086 y(That)24 b(the)h(p)q(oin)o(ts)e Fs(s)406 2093 y Fo(k)431 2086 y Fs(;)17 b Fv(0)26 b Fr(\024)h Fs(k)h Fr(\024)e Fv(2)p Fs(r)745 2093 y Fn(\003)793 2086 y Fv(are)e(in)g Fr(V)t Fv(\()p Fs(G;)8 b(D)1107 2068 y Fo(c)1128 2086 y Fv(\))25 b(is)f(straigh)o(tforw)o(ard;)h(in)f(fact)h (they)f(all)0 2146 y(b)q(elong)19 b(to)g Fs(\031)r Fv(\()p Fs(s)297 2153 y Fl(0)320 2146 y Fs(;)8 b(D)q Fv(\))21 b(\(remem)o(b)q(er)c (that)j Fs(s)812 2153 y Fl(2)p Fo(k)875 2146 y Fr(2)e Fs(\031)r Fv(\()p Fs(s)998 2153 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1095 2146 y Fs(;)8 b(D)14 b Fr(n)f Fs(G)p Fv(\)\).)30 b(Theorem)19 b(9)p 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Fv(W)l(e)d(no)o(w)f(study)g(the)h(times)f Fs(\034)618 1023 y Fl(1)648 1016 y Fr(\000)7 b Fs(\034)716 1023 y Fl(0)738 1016 y Fs(;)h(:)g(:)g(:)17 b(;)8 b(\034)879 1023 y Fo(r)q Fl(+1)959 1016 y Fr(\000)f Fs(\034)1027 1023 y Fo(r)1064 1016 y Fv(b)q(et)o(w)o(een)14 b(the)h(jumps)f(along)g(the)h(saddle)e(path)0 1076 y(as)k(w)o(ell)h(as)f(the) h(times)g(the)g(pro)q(cess)f(sp)q(ends)g(in)g(the)h(p)q(oin)o(ts)f(of)h(the)h (cycle)f(path,)g(kno)o(wing)f(that)h(the)0 1136 y(Mark)o(o)o(v)e(c)o(hain)h (follo)o(ws)f(a)h(\014xed)g(saddle)g(path.)24 b(The)17 b(k)o(ey)h(lies)f(in)g (the)g(follo)o(wing)f(W)l(en)o(tzell{F)l(reidlin)0 1195 y(t)o(yp)q(e)h(form)o (ulas.)0 1290 y Fx(Lemma)g(10.1.)28 b Fi(\(conditional)15 b(exp)q(ected)j(n)o (um)o(b)q(er)c(of)j(visits)f(b)q(efore)h(exit\))0 1350 y(F)l(or)e(an)i (arbitrary)e(set)h Fs(W)7 b Fi(,)17 b(t)o(w)o(o)f(p)q(oin)o(ts)g Fs(x;)8 b(z)19 b Fi(in)d Fs(W)981 1331 y Fo(c)1018 1350 y Fi(and)g Fs(y)j Fi(in)d Fs(W)7 b Fi(,)16 b(w)o(e)h(ha)o(v)o(e)23 1515 y Fn(1)7 1530 y Fg(X)5 1635 y Fo(n)p Fl(=0)88 1577 y Fs(P)120 1584 y Fo(\014)147 1537 y Fg(\000)170 1577 y Fs(X)211 1584 y Fo(n)252 1577 y Fv(=)d Fs(z)r(;)8 b(\034)e Fv(\()p Fs(W)453 1556 y Fo(c)473 1577 y Fv(\))15 b Fs(>)e(n=X)655 1584 y Fl(0)692 1577 y Fv(=)g Fs(x;)8 b(X)835 1586 y Fo(\034)t Fl(\()p Fo(W)915 1576 y Fh(c)935 1586 y Fl(\))967 1577 y Fv(=)14 b Fs(y)1046 1537 y Fg(\001)1083 1577 y Fv(=)1261 1434 y Fg(X)1141 1543 y Fo(g)1160 1548 y Fe(1)1180 1543 y Fn(2)p Fo(G)1238 1548 y Fh(x;z)1290 1543 y Fl(\()p Fo(W)5 b Fn([f)p Fo(z)q Fn(g)p Fl(\))1444 1481 y Fs(p)1469 1488 y Fo(\014)1496 1481 y Fv(\()p Fs(g)1539 1488 y Fl(1)1561 1481 y Fv(\))1647 1434 y Fg(X)1572 1543 y Fo(g)1591 1548 y Fe(2)1611 1543 y Fn(2)p Fo(G)1669 1548 y Fh(z)q(;y)1720 1543 y Fl(\()p Fo(W)g Fl(\))1802 1481 y Fs(p)1827 1488 y Fo(\014)1854 1481 y Fv(\()p Fs(g)1897 1488 y Fl(2)1920 1481 y Fv(\))p 1141 1566 798 2 v 1227 1577 a Fg(X)1177 1686 y Fo(g)1196 1691 y Fe(1)1216 1686 y Fn(2)p Fo(G)p Fl(\()p Fo(W)g Fl(\))1373 1624 y Fs(p)1398 1631 y Fo(\014)1425 1624 y Fv(\()p Fs(g)1468 1631 y Fl(1)1490 1624 y Fv(\))1594 1577 y Fg(X)1517 1686 y Fo(g)1536 1691 y Fe(2)1557 1686 y Fn(2)p Fo(G)1615 1691 y Fh(x;y)1667 1686 y Fl(\()p Fo(W)g Fl(\))1766 1624 y Fs(p)1791 1631 y Fo(\014)1818 1624 y Fv(\()p Fs(g)1861 1631 y Fl(2)1884 1624 y Fv(\))0 1842 y Fj(Pr)m(o)m(of.)28 b Fv(W)l(e)16 b(just)h(write)f(\(where)h(1\()p Fs(A)p Fv(\))g(denotes)g(the)f(c)o(haracteristic)g(function)g(of)h(the)f(ev)o (en)o(t)h Fs(A)p Fv(\))108 1944 y Fn(1)92 1959 y Fg(X)91 2064 y Fo(n)p Fl(=0)174 2006 y Fs(P)206 2013 y Fo(\014)233 2006 y Fv(\()p Fs(X)293 2013 y Fo(n)334 2006 y Fv(=)d Fs(z)r(;)8 b(\034)e Fv(\()p Fs(W)535 1985 y Fo(c)555 2006 y Fv(\))15 b Fs(>)e(n=X)737 2013 y Fl(0)774 2006 y Fv(=)g Fs(x;)8 b(X)917 2015 y Fo(\034)t Fl(\()p Fo(W)997 2005 y Fh(c)1017 2015 y Fl(\))1049 2006 y Fv(=)14 b Fs(y)r Fv(\))31 b(=)448 2171 y(=)534 2109 y Fn(1)518 2124 y Fg(X)517 2231 y Fo(k)q Fl(=0)621 2134 y Fs(P)653 2141 y Fo(\014)680 2134 y Fv(\()p 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Fv(\))p 621 2327 1233 2 v 971 2373 a Fs(P)1003 2380 y Fo(\014)1030 2373 y Fv(\()p Fs(X)1090 2382 y Fo(\034)t Fl(\()p Fo(W)1170 2372 y Fh(c)1190 2382 y Fl(\))1222 2373 y Fv(=)f Fs(y)r(=X)1366 2380 y Fl(0)1403 2373 y Fv(=)g Fs(x)p Fv(\))0 2507 y(and)j(the)h(result)e(follo)o(ws)h(b)o(y)g (applying)g(lemmas)f(3)p Fs(:)p Fv(5)p Fs(:)h Fv(and)g(3)p Fs(:)p Fv(7)p Fs(:)p Fv(.)49 b Ff(\003)954 2670 y Ft(32)p eop %%Page: 33 33 33 32 bop 0 200 a Fx(Lemma)17 b(10.2.)28 b Fi(\(conditional)15 b(exp)q(ected)j(exit)f(time\))0 260 y(F)l(or)e(an)i(arbitrary)e(set)h Fs(W)7 b Fi(,)17 b Fs(x)g Fi(in)f Fs(W)690 242 y Fo(c)727 260 y Fi(and)g Fs(y)j Fi(in)d Fs(W)7 b Fi(,)16 b(w)o(e)g(ha)o(v)o(e)70 489 y Fs(E)107 496 y Fo(\014)134 489 y Fv(\()p Fs(\034)6 b Fv(\()p Fs(W)254 469 y Fo(c)275 489 y Fv(\))p Fs(=X)360 496 y Fl(0)396 489 y Fv(=)14 b Fs(x;)8 b(X)540 498 y Fo(\034)t Fl(\()p Fo(W)620 488 y Fh(c)640 498 y Fl(\))672 489 y Fv(=)14 b Fs(y)r Fv(\))31 b(=)894 346 y Fg(X)876 453 y Fo(z)q Fn(2)p Fo(W)965 443 y 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Fv(In)g(general)g(these)h(quan)o(tities)f(do)g(really)g(dep)q(end)h(on)f (the)h(conditioning)e(ev)o(en)o(t.)21 b(Ho)o(w)o(ev)o(er)14 b(w)o(e)h(will)0 2550 y(only)h(b)q(e)h(concerned)f(with)g(the)h(case)f(of)h (a)g(cycle.)954 2670 y Ft(33)p eop %%Page: 34 34 34 33 bop 0 200 a Fx(Prop)r(osition)19 b(10.5.)49 b Fi(F)l(or)16 b(a)g(cycle)h Fs(\031)r Fi(,)g(t)o(w)o(o)f(p)q(oin)o(ts)g Fs(x;)8 b(z)19 b Fi(in)d Fs(\031)j Fi(and)d Fs(y)j Fi(in)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))i Fi(w)o(e)e(ha)o(v)o(e)91 344 y Fv(lim)74 377 y Fo(\014)r Fn(!1)211 310 y Fv(1)p 208 333 31 2 v 208 378 a Fs(\014)253 344 y Fv(ln)321 282 y Fn(1)305 297 y Fg(X)303 402 y Fo(n)p Fl(=0)386 344 y Fs(P)418 351 y Fo(\014)445 304 y Fg(\000)468 344 y Fs(X)509 351 y Fo(n)550 344 y Fv(=)e Fs(z)r(;)8 b(\034)e Fv(\()p Fs(\031)r Fv(\))15 b Fs(>)e(n=X)909 351 y Fl(0)945 344 y Fv(=)h Fs(x;)8 b(X)1089 353 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1184 344 y Fv(=)14 b Fs(y)1263 304 y Fg(\001)1316 344 y Fv(=)31 b Fs(W)7 b Fv(\()p Fs(\031)r 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Fv(whic)o(h)13 b(is)h(exactly)i Fs(W)7 b Fv(\()p Fs(\031)r Fv(\))g(+)g Fs(H)t Fv(\()p Fs(\031)r Fv(\))g Fr(\000)g Fs(W)g Fv(\()p Fs(z)r Fv(\))17 b(\(b)o(y)d(prop)q(osition)f(4)p Fs(:)p Fv(14)p Fs(:)p Fv(\).)21 b(This)14 b(pro)o(v)o(es)e(the)j(\014rst)f(form)o(ula)0 1156 y(stated)24 b(in)e(the)i(prop)q(osition.)41 b(T)l(aking)23 b(the)g(in\014m)o(um)e(o)o(v)o(er)i Fs(z)j Fv(in)d Fs(\031)i Fv(\(whic)o(h)e(corresp)q(onds)e(to)j(the)0 1215 y(quan)o(tit)o(y)16 b(app)q(earing)f(in)i(corollary)e(10)p Fs(:)p Fv(4)p Fs(:)p Fv(\),)h(w)o(e)h(obtain)f Fs(H)t Fv(\()p Fs(\031)r Fv(\),)h(pro)o(ving)e(the) i(second)f(form)o(ula.)48 b Ff(\003)0 1302 y Fx(Prop)r(osition)17 b(10.6.)27 b Fi(\(estimation)14 b(of)h(the)g(conditioned)f(n)o(um)o(b)q(er)e (of)j(visits)f(to)h(a)g(state)g(b)q(efore)g(exit\))0 1421 y(F)l(or)g(an)o(y)h (cycle)i Fs(\031)r Fi(,)e(an)o(y)g Fs(x;)8 b(u)17 b Fi(in)f Fs(\031)r Fi(,)h(an)o(y)f Fs(y)j Fi(in)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))p 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Fv(=)30 b(inf)12 b Fr(f)c Fs(n)14 b(>)g(\022)854 2557 y Fo(k)q Fn(\000)p Fl(1)944 2550 y Fv(:)f Fs(X)1012 2557 y Fo(n)1053 2550 y Fv(=)h Fs(u)8 b Fr(g)50 b Fs(;)24 b Fv(for)17 b Fs(k)e Fr(\025)e Fv(1)p Fs(:)954 2670 y Ft(34)p eop %%Page: 35 35 35 34 bop 0 200 a Fv(F)l(or)15 b(an)o(y)h Fs(N)k Fr(\025)13 b Fs(M)20 b(>)14 b Fv(0)i(w)o(e)g(ha)o(v)o(e)6 325 y Fs(P)38 332 y Fo(\014)65 325 y Fv(\()p Fs(\036)d(>)h(N)5 b(;)17 b(X)297 334 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))392 325 y Fv(=)c Fs(y)r(=X)536 332 y Fl(0)573 325 y Fv(=)g Fs(x)p Fv(\))23 b(=)f Fs(P)788 332 y Fo(\014)815 325 y Fv(\()p Fs(\022)857 332 y Fo(N)909 325 y Fs(<)14 b(\034)6 b Fv(\()p Fs(\031)r Fv(\))p Fs(;)17 b(X)1130 334 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1224 325 y Fv(=)d Fs(y)r(=X)1369 332 y Fl(0)1406 325 y Fv(=)f Fs(x)p Fv(\))239 402 y(=)22 b Fs(P)332 409 y Fo(\014)359 402 y Fv(\()p Fs(\022)401 409 y Fl(0)438 402 y Fs(<)14 b(\034)6 b Fv(\()p Fs(\031)r Fv(\))p Fs(=X)653 409 y Fl(0)689 402 y Fv(=)14 b Fs(x)p Fv(\))8 b 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Fv(\()p Fs(X)419 623 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))514 614 y Fv(=)14 b Fs(y)r(=X)659 621 y Fl(0)695 614 y Fv(=)g Fs(x)p Fv(\))803 544 y Fg(\022)847 581 y Fs(E)884 588 y Fo(\014)910 581 y Fv(\()p Fs(\036=X)1025 588 y Fl(0)1062 581 y Fv(=)f Fs(u)p Fv(\))p 847 603 316 2 v 977 649 a Fs(M)1168 544 y Fg(\023)1205 555 y Fn(b)p Fo(N)q(=)n(M)t Fn(c)0 769 y Fv(Let)k(us)f(tak)o(e)h Fs(M)28 b Fv(=)22 b Fr(d)p Fs(e)8 b(E)488 776 y Fo(\014)515 769 y Fv(\()p Fs(\036=X)630 776 y Fl(0)667 769 y Fv(=)13 b Fs(u)p Fv(\))p Fr(e)p Fv(.)22 b(W)l(e)17 b(obtain)28 926 y Fs(P)60 933 y Fo(\014)87 926 y Fv(\()p Fs(\036)d(>)f(N)s(;)j(X)316 935 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))411 926 y Fv(=)e Fs(y)r(=X)556 933 y Fl(0)592 926 y Fv(=)g Fs(x)p Fv(\))23 b Fr(\024)f Fs(P)808 933 y Fo(\014)835 926 y Fv(\()p Fs(X)895 935 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))990 926 y Fv(=)13 b Fs(y)r(=X)1134 933 y Fl(0)1171 926 y Fv(=)g Fs(x)p Fv(\))18 b(exp)8 b Fr(\000)1419 856 y Fg(\026)1648 892 y Fs(N)p 1454 914 433 2 v 1454 960 a(e)g(E)1522 967 y Fo(\014)1549 960 y Fv(\()p Fs(\036=X)1664 967 y Fl(0)1701 960 y Fv(=)13 b Fs(u)p Fv(\))e(+)g(1)1893 856 y Fg(\027)0 1079 y Fv(so)16 b(that)289 1162 y Fs(P)321 1169 y Fo(\014)348 1162 y Fv(\()p Fs(\036)e(>)f(N)q(=X)570 1171 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))665 1162 y Fv(=)h Fs(y)r(;)8 b(X)807 1169 y Fl(0)843 1162 y Fv(=)14 b Fs(x)p Fv(\))23 b Fr(\024)f Fv(exp)8 b Fr(\000)1158 1091 y Fg(\026)1387 1128 y Fs(N)p 1193 1150 V 1193 1196 a(e)g(E)1261 1203 y Fo(\014)1288 1196 y Fv(\()p Fs(\036=X)1403 1203 y Fl(0)1440 1196 y Fv(=)13 b Fs(u)p Fv(\))e(+)g(1)1632 1091 y Fg(\027)0 1290 y Fv(from)16 b(whic)o(h)f(it)i(is)f(easy)h(to)g(deduce)f(the)g(\014rst)g (equation)h(of)f(prop)q(osition)f(10)p Fs(:)p Fv(6)p Fs(:)p Fv(.)0 1350 y(W)l(e)i(no)o(w)f(pro)o(v)o(e)f(the)i(second)f(inequalit)o(y)l (.)21 b(W)l(e)c(ha)o(v)o(e)0 1520 y Fs(P)32 1527 y Fo(\014)59 1520 y Fv(\()p Fs(\036)d(<)f(N)s(;)j(X)288 1529 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))383 1520 y Fv(=)e Fs(y)r(=X)528 1527 y 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2670 y Ft(35)p eop %%Page: 36 36 36 35 bop 0 200 a Fv(and)20 b(the)h(second)g(inequalit)o(y)f(stated)h(in)g (prop)q(osition)e(10)p Fs(:)p Fv(6)p Fs(:)i Fv(will)f(follo)o(w)g(from)h(the) g(t)o(w)o(o)f(follo)o(wing)0 260 y(equations:)498 407 y(\(6\))126 b(lim)669 439 y Fo(\014)r Fn(!1)804 370 y Fs(P)836 377 y Fo(\014)863 370 y Fv(\()p Fs(X)923 379 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1018 370 y Fv(=)14 b Fs(y)r(=X)1163 377 y Fl(0)1199 370 y Fv(=)g Fs(u)p Fv(\))p 804 395 496 2 v 804 441 a Fs(P)836 448 y Fo(\014)863 441 y Fv(\()p Fs(X)923 450 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1018 441 y Fv(=)g Fs(y)r(=X)1163 448 y Fl(0)1199 441 y Fv(=)g Fs(x)p Fv(\))1336 407 y(=)30 b(1)8 b Fs(;)368 553 y Fv(\(7\))109 b(lim)8 b(inf)557 586 y Fo(\014)r Fn(!1)703 553 y Fr(\000)751 519 y Fv(1)p 748 542 31 2 v 748 587 a Fs(\014)793 553 y Fv(ln)849 516 y Fs(P)881 523 y Fo(\014)907 516 y Fv(\()p Fs(X)967 525 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(u)p Fn(g)p Fl(\))1147 516 y Fv(=)14 b Fs(y)r(=X)1292 523 y Fl(0)1328 516 y Fv(=)g Fs(x)p Fv(\))p 849 542 581 2 v 891 587 a Fs(P)923 594 y Fo(\014)950 587 y Fv(\()p Fs(X)1010 596 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1105 587 y Fv(=)g Fs(y)r(=X)1250 594 y Fl(0)1286 587 y Fv(=)g Fs(x)p Fv(\))1465 553 y Fs(>)30 b Fv(0)8 b Fs(:)66 705 y Fv(F)l(orm)o(ula)14 b(\(6\))k(follo)o(ws)d(from)h(the)h(three)f(equations:)131 823 y Fs(P)163 830 y Fo(\014)190 823 y Fv(\()p Fs(X)250 832 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))345 823 y Fv(=)e Fs(y)r(=X)490 830 y Fl(0)527 823 y Fv(=)f Fs(u)p Fv(\))22 b Fr(\025)g Fs(P)742 830 y Fo(\014)769 823 y Fv(\()p Fs(X)829 832 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(x)p Fn(g)p Fl(\))1008 823 y Fv(=)13 b Fs(x=X)1154 830 y Fl(0)1191 823 y Fv(=)h Fs(u)p Fv(\))8 b Fs(P)1332 830 y Fo(\014)1359 823 y Fv(\()p Fs(X)1419 832 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1514 823 y Fv(=)13 b Fs(y)r(=X)1658 830 y Fl(0)1695 823 y Fv(=)h Fs(x)p Fv(\))8 b Fs(;)132 900 y(P)164 907 y Fo(\014)190 900 y Fv(\()p Fs(X)250 909 y Fo(\034)t Fl(\()p Fo(\031)q 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b(us)f(\014nally)g(pro)o(v)o(e)f(form)o(ula)g(\(7\).)23 b(W)l(e)17 b(ha)o(v)o(e)51 1235 y Fs(P)83 1242 y Fo(\014)110 1235 y Fv(\()p Fs(X)170 1244 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(u)p Fn(g)p Fl(\))350 1235 y Fv(=)d Fs(y)r(=X)495 1242 y Fl(0)531 1235 y Fv(=)g Fs(x)p Fv(\))p 51 1260 V 94 1306 a Fs(P)126 1313 y Fo(\014)153 1306 y Fv(\()p Fs(X)213 1315 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))308 1306 y Fv(=)f Fs(y)r(=X)452 1313 y Fl(0)489 1306 y Fv(=)g Fs(x)p Fv(\))660 1272 y(=)22 b(1)11 b Fr(\000)f Fs(P)838 1279 y Fo(\014)865 1272 y Fv(\()p Fs(X)925 1281 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(u)p Fn(g)p Fl(\))1105 1272 y Fv(=)k Fs(u=X)1253 1279 y Fl(0)1288 1272 y Fv(=)g Fs(x)p Fv(\))1403 1235 y Fs(P)1435 1242 y Fo(\014)1462 1235 y Fv(\()p Fs(X)1522 1244 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1617 1235 y Fv(=)g Fs(y)r(=X)1762 1242 y Fl(0)1798 1235 y Fv(=)g Fs(u)p Fv(\))p 1403 1260 496 2 v 1403 1306 a Fs(P)1435 1313 y Fo(\014)1462 1306 y Fv(\()p Fs(X)1522 1315 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1617 1306 y Fv(=)g Fs(y)r(=X)1762 1313 y Fl(0)1798 1306 y Fv(=)g Fs(x)p Fv(\))185 1386 y(\(8\))114 b Fr(\024)22 b Fv(1)11 b Fr(\000)509 1346 y Fg(\000)532 1386 y Fv(1)g Fr(\000)g Fs(P)650 1393 y Fo(\014)677 1386 y Fv(\()p Fs(X)737 1395 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(u)p Fn(g)p Fl(\))916 1386 y Fr(6)p Fv(=)j Fs(u=X)1064 1393 y Fl(0)1100 1386 y Fv(=)f Fs(x)p Fv(\))1199 1346 y Fg(\001)q(\000)1246 1386 y Fv(1)e Fr(\000)g Fs(P)1364 1393 y Fo(\014)1391 1386 y Fv(\()p Fs(X)1451 1395 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(x)p Fn(g)p Fl(\))1629 1386 y Fr(6)p Fv(=)j Fs(x=X)1776 1393 y Fl(0)1813 1386 y Fv(=)f Fs(u)p Fv(\))1913 1346 y Fg(\001)1936 1386 y Fs(:)17 1504 y Fv(Moreo)o(v)o(er,)h(w)o(e)j(kno)o(w)f(that)467 1642 y(lim)8 b(inf)484 1674 y Fo(\014)r Fn(!1)614 1642 y Fr(\000)661 1608 y Fv(1)p 659 1630 31 2 v 659 1676 a Fs(\014)704 1642 y Fv(ln)f Fs(P)785 1649 y Fo(\014)812 1642 y Fv(\()p Fs(X)872 1651 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(u)p Fn(g)p Fl(\))1052 1642 y Fr(6)p Fv(=)14 b Fs(u)o(=X)1199 1649 y Fl(0)1236 1642 y Fv(=)f Fs(x)p Fv(\))31 b Fs(>)g Fv(0)8 b Fs(;)468 1772 y Fv(lim)g(inf)486 1804 y Fo(\014)r Fn(!1)615 1772 y Fr(\000)662 1738 y Fv(1)p 660 1761 V 660 1806 a Fs(\014)705 1772 y Fv(ln)f Fs(P)786 1779 y Fo(\014)813 1772 y Fv(\()p Fs(X)873 1781 y Fo(\034)t Fl(\()p Fo(\031)q Fn(nf)p Fo(x)p Fn(g)p Fl(\))1052 1772 y Fr(6)p Fv(=)14 b Fs(x=X)1199 1779 y Fl(0)1236 1772 y Fv(=)f Fs(u)p Fv(\))30 b Fs(>)h Fv(0)8 b Fs(:)0 1917 y Fv(W)l(e)20 b(can)g(th)o(us)f(mak)o(e)g(an)h(asymptotic)f(expansion)g(in)g(form)o(ula)g (\(8\),)i(from)e(whic)o(h)g(form)o(ula)f(\(7\))i(fol-)0 1976 y(lo)o(ws.)49 b Ff(\003)0 2066 y Fx(Corollary)20 b(10.7.)27 b Fi(F)l(or)16 b(an)o(y)g(cycle)h Fs(\031)r Fi(,)f(an)o(y)h Fs(x;)8 b(u)16 b Fi(in)g Fs(\031)r Fi(,)h(an)o(y)f Fs(y)j Fi(in)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))p Fi(,)i(an)o(y)e(p)q(ositiv)o(e)g Fs(\017)p Fi(,)h(w)o(e)f(ha)o(v)o(e)19 2239 y Fv(lim)2 2271 y Fo(\014)r Fn(!1)122 2239 y Fs(P)154 2246 y Fo(\014)181 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Ft(36)p eop %%Page: 37 37 37 36 bop 0 200 a Fx(Corollary)20 b(10.8.)27 b Fi(\(estimation)16 b(of)h(the)g(conditioned)e(exit)j(time\))0 260 y(F)l(or)d(eac)o(h)h Fs(\017)f(>)e Fv(0)p Fi(,)j(eac)o(h)h(cycle)g Fs(\031)r Fi(,)f Fs(x)h Fi(in)f Fs(\031)r Fi(,)h Fs(y)i Fi(in)d Fs(B)r Fv(\()p Fs(\031)r Fv(\))p Fi(,)i(w)o(e)e(ha)o(v)o(e)98 366 y Fv(lim)81 398 y Fo(\014)r Fn(!1)201 366 y Fs(P)233 373 y Fo(\014)260 296 y Fg(\022)305 366 y Fv(exp)8 b Fs(\014)s Fv(\()p Fs(H)t Fv(\()p Fs(\031)r Fv(\))13 b Fr(\000)e Fs(\017)p Fv(\))22 b Fr(\024)g Fs(\034)6 b Fv(\()p Fs(\031)r Fv(\))23 b Fr(\024)f Fv(exp)8 b Fs(\014)s Fv(\()p Fs(H)t Fv(\()p Fs(\031)r Fv(\))13 b(+)d Fs(\017)p Fv(\))p Fs(=X)1331 373 y Fl(0)1368 366 y Fv(=)k Fs(x;)j(X)1521 375 y Fo(\034)t Fl(\()p Fo(\031)q Fl(\))1616 366 y Fv(=)c Fs(y)1694 296 y Fg(\023)1761 366 y Fv(=)31 b(1)p Fs(:)0 492 y Fx(Prop)r(osition)19 b(10.9.)27 b Fi(F)l(or)15 b(an)o(y)i(saddle)e(path)h Fv(\()p Fs(s)950 499 y Fl(0)973 492 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(s)1115 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2505 y Fs(;)j(S)1650 2512 y Fo(\027)1670 2517 y Fh(k)p Fd(\000)p Fe(1)1751 2505 y Fv(=)d Fs(A)1841 2512 y Fo(k)q Fn(\000)p Fl(1)1917 2505 y Fv(\))p Fs(:)954 2670 y Ft(38)p eop %%Page: 39 39 39 38 bop 17 200 a Fv(W)l(e)16 b(ha)o(v)o(e)g(to)h(estimate)g(the)f(term)0 319 y(\(10\))95 b Fs(P)215 326 y Fo(\014)242 319 y Fv(\()p Fs(X)302 326 y Fo(\027)322 331 y Fh(k)344 326 y Fn(\000)p Fl(1)411 319 y Fv(=)14 b Fs(s)487 326 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)583 319 y Fs(;)i(X)654 326 y Fo(\027)674 331 y Fh(k)712 319 y Fv(=)e Fs(s)788 326 y Fl(2)p Fo(k)833 319 y Fs(;)i(S)894 326 y Fo(\027)914 331 y Fh(k)952 319 y Fv(=)d Fs(A)1041 326 y Fo(k)1066 319 y Fs(=X)1132 326 y Fo(\027)1152 331 y Fh(k)p Fd(\000)p Fe(1)1234 319 y Fv(=)h Fs(s)1310 326 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)1406 319 y Fs(;)i(S)1467 326 y Fo(\027)1487 331 y Fh(k)p Fd(\000)p Fe(1)1569 319 y Fv(=)d Fs(A)1658 326 y Fo(k)q Fn(\000)p Fl(1)1734 319 y Fv(\))p Fs(:)17 441 y Fv(Let)k Fr(I)133 448 y Fo(k)174 441 y Fv(b)q(e)g(the)g(collection)f(of)h(all)f(the)h(sequences)f(\()p Fs(u)1035 448 y Fl(0)1057 441 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(u)1205 448 y Fo(m)1242 441 y Fv(\))17 b(in)g Fs(E)1377 423 y Fo(m)p Fl(+1)1481 441 y Fv(suc)o(h)e(that)60 500 y Fr(\017)49 b Fs(u)163 507 y Fl(0)199 500 y Fv(=)14 b Fs(s)275 507 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)370 500 y Fv(,)60 560 y Fr(\017)49 b Fs(u)163 567 y Fl(0)185 560 y Fs(;)8 b(:)g(:)g(:)17 b(;)8 b(u)333 567 y Fo(m)387 560 y Fv(are)16 b(distinct)h(p)q(oin)o(ts,)60 620 y Fr(\017)49 b Fs(A)171 627 y Fo(k)q Fn(\000)p Fl(1)258 620 y Fr([)12 b(f)c Fs(u)365 627 y Fl(0)387 620 y Fs(;)g(:)g(:)g(:)17 b(;)8 b(u)535 627 y Fo(m)580 620 y Fr(g)14 b Fv(=)g Fs(A)709 627 y Fo(k)734 620 y Fv(,)60 680 y Fr(\017)49 b Fs(C)170 689 y Fo(A)200 694 y Fh(k)p Fd(\000)p Fe(1)266 689 y Fn([f)7 b Fo(u)344 694 y Fe(0)363 689 y Fo(;:::)f(;u)453 694 y Fh(l)p Fd(\000)p Fe(1)518 689 y Fn(g)540 680 y Fv(\()p Fs(s)582 687 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)678 680 y Fs(;)i(u)729 687 y Fo(l)744 680 y Fv(\))15 b(=)e(0.)0 740 y(If)k Fr(I)77 747 y Fo(k)118 740 y Fv(is)f(empt)o(y)l(,)g(then)g(the)h(term)f(in)h(form)o(ula)d (\(10\))j(is)g(zero,)f(otherwise)g(it)g(is)g(equal)h(to)98 852 y Fg(X)0 961 y Fl(\()p Fo(u)40 966 y Fe(0)59 961 y Fo(;:::)6 b(;u)149 966 y Fh(m)181 961 y Fl(\))p Fn(2I)247 966 y Fh(k)290 837 y Fo(m)276 852 y Fg(Y)276 959 y Fo(l)p Fl(=1)348 899 y Fs(P)380 906 y Fo(\014)407 859 y Fg(\000)430 899 y Fs(X)471 906 y Fo(\013)497 911 y Fh(l)528 899 y Fv(=)13 b Fs(u)609 906 y Fo(l)624 899 y Fs(;)k(\027)680 906 y Fo(k)718 899 y Fs(>)c(\013)802 906 y Fo(l)817 899 y Fs(=X)883 906 y Fo(\013)909 911 y Fh(l)p Fd(\000)p Fe(1)984 899 y Fv(=)h Fs(u)1066 906 y Fo(l)p Fn(\000)p Fl(1)1131 899 y Fs(;)j(S)1193 906 y Fo(u)1217 911 y Fh(l)p Fd(\000)p Fe(1)1291 899 y Fv(=)c Fs(A)1380 906 y Fo(k)q Fn(\000)p Fl(1)1467 899 y Fr([)e(f)d Fs(u)1573 906 y Fl(0)1596 899 y Fs(;)g(:)g(:)g(:)16 b(;)8 b(u)1743 906 y Fo(l)p Fn(\000)p Fl(2)1818 899 y Fr(g)1843 859 y Fg(\001)50 1037 y Fv(\(11\))291 b Fr(\002)11 b Fs(P)511 1044 y Fo(\014)538 997 y Fg(\000)560 1037 y Fs(X)601 1044 y Fo(\027)621 1049 y Fh(k)643 1044 y Fn(\000)p Fl(1)711 1037 y Fv(=)i Fs(s)786 1044 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)882 1037 y Fs(;)j(X)953 1044 y Fo(\027)973 1049 y Fh(k)1012 1037 y Fv(=)d Fs(s)1087 1044 y Fl(2)p Fo(k)1132 1037 y Fs(;)k(S)1194 1044 y Fo(\027)1214 1049 y Fh(k)1251 1037 y Fv(=)d Fs(A)1341 1044 y Fo(k)1366 1037 y Fs(=X)1432 1044 y Fo(\013)1458 1049 y Fh(m)1507 1037 y Fv(=)g Fs(u)1589 1044 y Fo(m)1626 1037 y Fs(;)i(S)1687 1044 y Fo(\013)1713 1049 y Fh(m)1746 1044 y Fl(+1)1813 1037 y Fv(=)d Fs(A)1902 1044 y Fo(k)1927 997 y Fg(\001)17 1156 y Fv(where)j Fs(\013)193 1163 y Fl(0)229 1156 y Fv(=)d Fs(\027)306 1163 y Fo(k)q Fn(\000)p Fl(1)398 1156 y Fv(and)j Fs(\013)527 1163 y Fo(l)556 1156 y Fv(=)d(inf)t Fr(f)8 b Fs(n)14 b(>)g(\013)831 1163 y Fo(l)p Fn(\000)p Fl(1)897 1156 y Fs(=X)963 1163 y Fo(n)1004 1156 y Fr(62)g Fs(S)1082 1163 y Fo(n)1117 1156 y Fr(g)p Fv(.)22 b(But)107 1275 y Fs(P)139 1282 y 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Fs(u)438 1521 y Fo(l)453 1514 y Fs(;)i(\027)508 1521 y Fo(k)546 1514 y Fs(<)e(\013)631 1521 y Fo(l)646 1514 y Fs(=)8 b Fr(\001)g(\001)g(\001)h Fv(\))22 b Fr(\024)g Fs(P)880 1521 y Fo(\014)907 1514 y Fv(\()p Fs(\027)951 1521 y Fo(k)989 1514 y Fs(<)14 b(\013)1074 1521 y Fo(l)1089 1514 y Fs(=)8 b Fr(\001)g(\001)g(\001)g Fv(\))23 b Fr(\024)f Fs(C)1327 1521 y Fl(7)1349 1514 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1444 1493 y Fn(\000)p Fo(C)1504 1498 y Fe(8)1535 1514 y Fv(exp\()p Fr(\000)p Fs(\015)s(\014)s Fv(\))8 b Fs(;)456 1592 y(C)496 1571 y Fn(\000)p Fl(1)492 1605 y(7)549 1592 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))644 1572 y Fo(C)673 1577 y Fe(8)717 1592 y Fr(\024)14 b Fs(P)802 1599 y Fo(\014)829 1592 y Fv(\()p Fs(X)889 1599 y Fo(\013)915 1604 y Fh(l)946 1592 y Fv(=)g Fs(u)1028 1599 y Fo(l)1042 1592 y Fs(=)8 b Fr(\001)g(\001)g(\001)h Fv(\))23 b Fr(\024)f Fs(C)1281 1599 y Fl(7)1302 1592 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1397 1572 y Fn(\000)p Fo(C)1457 1577 y Fe(8)1480 1592 y Fs(:)0 1711 y Fv(Therefore)16 b(for)g Fs(\014)j Fv(large)d(enough)g(and)g(for)g(some)g(p) q(ositiv)o(e)g(constan)o(ts)g Fs(C)1382 1718 y Fl(9)1404 1711 y Fs(;)8 b(C)1462 1718 y Fl(10)1504 1711 y Fv(,)0 1830 y(\(12\))246 b Fs(C)374 1809 y Fn(\000)p Fl(1)370 1844 y(9)427 1830 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))522 1810 y Fo(C)551 1815 y Fe(10)612 1830 y Fr(\024)22 b Fs(P)705 1837 y Fo(\014)732 1830 y Fv(\()p Fs(X)792 1837 y Fo(\013)818 1842 y Fh(l)850 1830 y Fv(=)13 b Fs(u)931 1837 y Fo(l)946 1830 y Fs(;)j(\027)1001 1837 y Fo(k)1039 1830 y Fr(\025)e Fs(\013)1124 1837 y Fo(l)1139 1830 y Fs(=)8 b Fr(\001)g(\001)g(\001)h Fv(\))22 b Fr(\024)g Fs(C)1377 1837 y Fl(9)1399 1830 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))1494 1810 y Fn(\000)p Fo(C)1554 1815 y Fe(10)1602 1830 y Fs(:)17 1949 y Fv(W)l(e)16 b(also)g(ha)o(v)o(e)g(for)g(some)g(p)q(ositiv)o(e)h(constan)o (ts)e Fs(C)959 1956 y Fl(11)1001 1949 y Fs(;)8 b(C)1059 1956 y Fl(12)0 2068 y Fs(C)40 2047 y Fn(\000)p Fl(1)36 2082 y(11)93 2068 y Fs(a)p Fv(\()p Fs(\014)s 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Fl(+1)1676 2143 y Fv(=)d Fs(A)1765 2150 y Fo(k)1790 2143 y Fv(\))675 2222 y Fr(\024)30 b Fs(C)780 2229 y Fl(11)822 2222 y Fs(a)p Fv(\()p Fs(\014)s Fv(\))917 2201 y Fn(\000)p Fo(C)977 2206 y Fe(12)1033 2222 y Fv(exp)8 b Fr(\000)p Fs(\014)1187 2181 y Fg(\000)1210 2222 y Fs(W)1257 2229 y Fo(A)1287 2234 y Fh(k)1311 2222 y Fv(\()p Fs(u)1359 2229 y Fo(m)1396 2222 y Fs(;)g(s)1441 2229 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1537 2222 y Fv(\))k(+)f Fs(V)g Fv(\()p Fs(s)1700 2229 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1796 2222 y Fs(;)d(s)1841 2229 y Fl(2)p Fo(k)1886 2222 y Fv(\))1905 2181 y Fg(\001)1936 2222 y Fs(:)15 2341 y Fv(Let)15 b(us)e(remark)h(ev)o(en)o(tually)g(that)h Fs(W)721 2348 y Fo(A)751 2353 y Fh(k)775 2341 y Fv(\()p Fs(u)823 2348 y Fo(m)861 2341 y Fs(;)8 b(s)906 2348 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1001 2341 y Fv(\))15 b(=)e Fs(W)1134 2348 y Fo(A)1164 2353 y Fh(k)1189 2341 y Fv(\()p Fs(s)1231 2348 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1327 2341 y Fs(;)8 b(s)1372 2348 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1467 2341 y Fv(\))15 b(b)q(ecause)g(it)f(is)g(p)q(ossible)0 2401 y(to)f(go)g(from)e Fs(u)262 2408 y Fo(m)313 2401 y Fv(to)i Fs(s)393 2408 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)501 2401 y Fv(in)f Fs(A)592 2408 y Fo(k)630 2401 y Fv(at)h(n)o(ull)e(cost.)21 b(Putting)12 b(together)h(\(9\))p Fs(;)8 b Fv(\(11\))p Fs(;)g Fv(\(12\))p Fs(;)g Fv(\(13\))16 b(and)c(noticing)0 2460 y(that)20 b Fs(P)143 2467 y Fo(\014)169 2460 y Fv(\()p Fs(X)229 2469 y Fo(\034)t Fl(\()p Fo(D)q Fn(n)p Fo(G)p Fl(\))389 2460 y Fv(=)e Fs(z)r(;)f(\027)527 2467 y Fo(r)q Fl(+1)617 2460 y Fs(>)h(\034)6 b Fv(\()p Fs(D)q Fv(\))p Fs(=X)848 2467 y Fo(\027)868 2472 y Fh(r)910 2460 y Fv(=)18 b Fs(s)990 2467 y Fl(2)p Fo(r)1032 2460 y Fv(\))i(is)f(logarithmically)e(equiv)m(alen)o(t)i(to)h(zero,)f(w)o(e)0 2520 y(obtain)d(the)h(estimates)f(stated)h(in)f(the)h(theorem.)48 b Ff(\003)954 2670 y Ft(39)p eop %%Page: 40 40 40 39 bop 0 200 a Fx(Theorem)17 b(11.2.)27 b Fi(F)l(or)15 b(an)o(y)h(p)q (ositiv)o(e)f Fs(\025)h 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Fl(2)p Fo(k)1054 2550 y Fv(\))p Fs(;)j(U)d Fv(\()p Fs(s)1176 2557 y Fl(2)p Fo(k)q Fl(+1)1273 2550 y Fv(\)\))20 b Fr(\000)f Fv(max)o(\()p Fs(U)5 b Fv(\()p Fs(s)1581 2557 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)1678 2550 y Fv(\))p Fs(;)j(U)d Fv(\()p Fs(s)1800 2557 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)1897 2550 y Fv(\)\))p Fs(:)954 2670 y Ft(41)p eop %%Page: 42 42 42 41 bop 0 200 a Fj(Pr)m(o)m(of.)28 b Fv(Since)13 b Fs(s)307 207 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)417 200 y Fv(do)q(es)i(not)f(b)q(elong)g (to)g Fs(\031)r Fv(\()p Fs(s)901 207 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)997 200 y Fs(;)8 b(D)g Fr(n)f Fs(G)p Fv(\))15 b(\(whic)o(h)f(coincides)f (with)h Fs(\031)r Fv(\()p Fs(s)1724 207 y Fl(2)p Fo(k)1769 200 y Fs(;)8 b(D)g Fr(n)f Fs(G)p Fv(\)\))0 260 y(and)21 b Fs(V)11 b Fv(\()p Fs(s)184 267 y Fl(2)p Fo(k)q Fn(\000)p Fl(2)280 260 y Fs(;)d(s)325 267 y Fl(2)p Fo(k)q Fn(\000)p Fl(1)421 260 y Fv(\))21 b(is)g(\014nite)g(\(whic)o(h)g(implies)f(b)o(y)h(rev)o(ersibilit)o (y)e(that)j Fs(V)11 b Fv(\()p Fs(s)1534 267 y 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Ft(43)p eop %%Page: 44 44 44 43 bop 830 200 a Fw(References)33 280 y Ft(1.)21 b(L.)14 b(Alonso{R.)g(Cerf,)e Fq(The)j(thr)n(e)n(e)f(dimensional)h(p)n(olyomino)n(es) g(of)g(minimal)g(ar)n(e)n(a)p Ft(,)e(preprin)o(t)f(\(1995\).)33 329 y(2.)21 b(G.)d(Ben)g(Arous{R.)f(Cerf,)h Fq(Metastability)h(of)g(the)g (thr)n(e)n(e)f(dimensional)h(Ising)h(mo)n(del)f(on)g(a)g(torus)g(at)g(very)g (low)87 379 y(temp)n(er)n(atur)n(es)p Ft(,)13 b(preprin)o(t)f(\(1995\).)33 429 y(3.)21 b(M.)e(Cassandro{A.)f(Galv)o(es{E.)h(Olivieri{M.E.)e(V)l(ares,)k Fq(Metastable)f(b)n(ehaviour)f(of)h(sto)n(chastic)h(dynamics:)30 b(a)87 479 y(p)n(athwise)16 b(appr)n(o)n(ach)p Ft(,)c(Jour.)h(Stat.)h(Ph)o (ys.)f Fc(35)18 b(nos.)e(5/6)f Ft(\(1984\),)h(603{634.)33 529 y(4.)21 b(O.)14 b(Catoni,)f Fq(L)n(ar)n(ge)h(deviations)i(for)f(anne)n(aling) p Ft(,)e(PhD)h(Thesis,)e(Univ)o(ersit)o(y)g(P)o(aris)h(XI)h(\(1990\).)33 578 y(5.)21 b(O.)14 b(Catoni,)f Fq(Notes)j(de)f(c)n(ours)g(de)g(DEA)f Ft(\(1995\),)h(Univ)o(ersit)o(\023)-20 b(e)12 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