The paper appeared in the Functional Analysis and Its Application vol.32 No.2, 114-131 (1998) Below is a PostScript file. 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Fr(,)f Fq(k)d Fr(=)e(1)s Fq(;)9 b(:)e(:)g(:)h(;)h(n)c Fr(,)18 b(b)q(e)h(the)f(eigen)o(v)n(alues)g(of)k Fq(A)788 1063 y Fp(n)816 1057 y Fr(.)30 b(The)18 b(Wigner)g(limit)d(theorem)i ([1,)h(2])f(claims)f(that)i(the)-148 1107 y(empirical)12 b(distribution)i(function)f(for)h(the)g(n)o(um)o(b)q(ers)19 b Fq(\025)753 1113 y Fp(k)778 1107 y Fr(,)568 1200 y Fq(N)601 1206 y Fp(n)624 1200 y Fr(\()p Fq(\025)p Fr(\))12 b(=)743 1172 y(1)p 741 1191 V 741 1229 a Fq(n)771 1200 y Fr(#)p Ft(f)p Fq(k)g Fr(:)f Fq(\025)908 1206 y Fp(k)940 1200 y Fq(<)h(\025)p Ft(g)s Fq(;)-148 1288 y Fr(con)o(v)o(erges)j(in)f (probabilit)o(y)e(as)19 b Fq(n)11 b Ft(!)h(1)18 b Fr(to)c(a)f (distribution)h(corresp)q(onding)g(to)g(the)h(Wigner)e(semicircle)h(la) o(w)572 1393 y(lim)557 1418 y Fp(n)p Fn(!1)651 1393 y Fq(N)684 1399 y Fp(n)707 1393 y Fr(\()p Fq(\025)p Fr(\))e(=)819 1337 y Fm(Z)860 1347 y Fp(\025)842 1431 y Fn(\0001)910 1393 y Fq(\032)p Fr(\()p Fq(u)p Fr(\))7 b Fq(du)s(;)621 b Fr(\(1.2\))-148 1494 y(where)410 1577 y Fq(\032)p Fr(\()p Fq(u)p Fr(\))12 b(=)543 1519 y Fm(\032)588 1546 y Fr(0)206 b(for)14 b Fq(u)d(>)h Fr(1)h(or)h Fq(u)d(<)h Ft(\000)p Fr(1)s Fq(;)595 1592 y Fo(2)p 593 1599 21 2 v 593 1623 a Fp(\031)625 1574 y Ft(p)p 660 1574 114 2 v 34 x Fr(1)d Ft(\000)g Fq(u)755 1596 y Fo(2)815 1608 y Fr(for)14 b Ft(\000)p Fr(1)d Ft(\024)h Fq(u)f Ft(\024)h Fr(1)p Fq(:)-148 1671 y Fr(Later,)i(this)g(result)h(w)o(as)f(strengthened)i(b)o(y)d (Marc)o(henk)o(o,)h(P)o(astur,)g(L.)g(Arnold,)f(W)m(ac)o(h)o(ter,)g (Girk)o(o,)f(and)i(others)h([3{13].)-98 1722 y(Consider)g(the)20 b Fq(r)172 1728 y Fp(n)194 1722 y Fr(-neigh)o(b)q(orho)q(o)q(d)g Fq(O)507 1728 y Fp(n)548 1722 y Fr(of)14 b(the)h(righ)o(t)f(sp)q (ectrum)h(edge)20 b Fq(\025)13 b Fr(=)g(1)5 b(,)13 b(where)21 b Fq(r)1329 1728 y Fp(n)1351 1722 y Fq(n)1376 1707 y Fo(2)p Fp(=)p Fo(3)1441 1722 y Ft(!)12 b(1)19 b Fr(as)g Fq(n)13 b Ft(!)f(1)5 b Fr(.)-148 1772 y(F)m(or)12 b(example,)e(one)i (can)g(tak)o(e)17 b Fq(r)357 1778 y Fp(n)391 1772 y Ft(\030)11 b Fr(const)d Fq(=n)582 1757 y Fp(\015)608 1772 y Fr(,)17 b Fq(\015)d(<)e Fr(2)p Fq(=)p Fr(3)5 b(.)16 b(By)c(formally)d(applying) h(the)j(semicircle)e(la)o(w,)g(w)o(e)h(\014nd)g(that)-148 1830 y(the)j(n)o(um)o(b)q(er)e(of)g(eigen)o(v)n(alues)h(in)k Fq(O)422 1836 y Fp(n)463 1830 y Fr(b)q(eha)o(v)o(es)d(as)k(const)8 b Ft(\001)f Fq(r)815 1808 y Fo(3)p Fp(=)p Fo(2)814 1835 y Fp(n)866 1830 y Fq(n)e Fr(.)18 b(Let)d(us)f(renormalize)f(the)h (eigen)o(v)n(alues)g(b)o(y)g(setting)681 1906 y Fq(\025)705 1912 y Fp(k)737 1906 y Fr(=)e(1)d Ft(\000)g Fq(\022)871 1912 y Fp(k)892 1906 y Fq(r)911 1912 y Fp(n)-148 1982 y Fr(and)14 b(place)g(the)h(mass)666 2066 y Fq(\026)691 2072 y Fp(n)714 2066 y Fr(\()p Fq(\022)749 2072 y Fp(k)770 2066 y Fr(\))c(=)884 2038 y(1)p 846 2056 97 2 v 846 2105 a Fq(nr)891 2083 y Fo(3)p Fp(=)p Fo(2)890 2109 y Fp(n)-148 2174 y Fr(at)16 b(eac)o(h)g(p)q(oin)o(t)21 b Fq(\022)135 2180 y Fp(k)160 2174 y Fr(.)j(W)m(e)16 b(th)o(us)g(obtain)f(a)g (measure)21 b Fq(\026)724 2180 y Fp(n)767 2174 y Fr(on)16 b(the)g(real)g(line)f(suc)o(h)i(that)k Fq(\026)1281 2180 y Fp(n)1303 2174 y Fr(\()p Fl(R)1352 2159 y Fo(1)1368 2174 y Fr(\))15 b(=)g Fq(r)1466 2152 y Fn(\000)p Fo(3)p Fp(=)p Fo(2)1465 2179 y Fp(n)1549 2174 y Fr(.)24 b(The)16 b(main)-148 2223 y(result)f(of)e(the)i(presen)o(t)g(pap)q(er)g(can)f(b) q(e)g(stated)h(as)f(follo)o(ws.)p -148 2286 600 2 v -93 2314 a Fk(\003)-70 2326 y Fj(This)d(w)o(ork)f(w)o(as)h(supp)q(orted)d (b)o(y)i(RFBR)h(gran)o(t)e(No.)i(96-01-0037)c(and)j(NSF)h(gran)o(t)e (No.)i(DMS-9706794)d(\(Y)m(a.)15 b(G.)10 b(Sinai\))f(and)h(b)o(y)g(NSF) -148 2367 y(gran)o(t)h(No.)g(DMS-9304580)e(\(A.)j(B.)f(Soshnik)o(o)o (v\).)p -148 2435 1910 2 v -98 2471 a(Princeton)k(Univ)o(ersit)o(y)m(,) h(Dept.)30 b(of)16 b(Mathematics,)f(Princeton,)h(New)h(Jersey)m(,)g (USA;)h(Institute)c(for)i(Adv)n(anced)f(Study)m(,)h(Sc)o(ho)q(ol)f(of) -148 2510 y(Mathematics,)g(Princeton,)h(New)h(Jersey)m(,)g(USA.)g(T)m (ranslated)e(from)g(F)m(unktsional)1013 2495 y Fn(0)1022 2510 y Fj(n)o(yi)h(Analiz)g(i)g(Ego)g(Prilozheniy)o(a,)f(V)m(ol.)h(32,) i(No.)e(2,)-148 2550 y(pp.)11 b(56{79,)f(April{June,)f(1998.)15 b(Original)10 b(article)g(submitted)e(Septem)o(b)q(er)h(5,)i(1997.)-148 2610 y Fo(114)358 b Fj(0016{2663)o(/98)o(/32)o(02{)o(01)o(14)s($12.50) 801 2609 y(c)788 2610 y Ft(\015)o Fj(1998)11 b(Plen)o(um)f(Publishing)f (Corp)q(oration)p eop %%Page: 115 2 115 1 bop -98 -71 a Fs(Theorem)14 b(1)i(\(Main)e(Theorem\).)22 b Fi(As)e Fq(n)11 b Ft(!)g(1)5 b Fi(,)14 b(the)g(me)n(asur)n(es)19 b Fq(\026)1034 -65 y Fp(n)1076 -71 y Fi(we)n(akly)c(c)n(onver)n(ge)f (in)h(pr)n(ob)n(ability)e(on)i(e)n(ach)-148 -21 y(\014nite)h(interval)e (to)h(a)h(me)n(asur)n(e)j Fq(\026)h Fi(c)n(onc)n(entr)n(ate)n(d)c(on)f (the)h(half-line)j Fl(R)976 -36 y Fo(+)1021 -21 y Fi(and)d(absolutely)f (c)n(ontinuous)h(with)e(r)n(esp)n(e)n(ct)h(to)-148 29 y(the)g(L)n(eb)n(esgue)g(me)n(asur)n(e.)k(The)c(density)20 b Fq(d\026)p Fr(\()p Fq(x)p Fr(\))p Fq(=dx)e Fi(has)d(the)g(form)k Fr(\(2)975 -5 y Ft(p)p 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Fr(is)k(o)q(dd,)-148 1841 y(then)g(the)f(cen)o(tered)i(traces)g(are)e (asymptotically)d(indep)q(enden)o(t)k(as)k Fq(n)12 b Ft(!)f(1)5 b Fr(.)-98 1913 y(Theorem)17 b(2)g(will)f(b)q(e)h(pro)o(v)o (ed)h(in)f Ft(xx)p Fr(4)g(and)g(5.)28 b(In)17 b Ft(x)q Fr(2)g(w)o(e)g(deriv)o(e)h(the)g(main)d(theorem)i(and)g(the)h (corollaries)e(from)-148 1963 y(Theorem)e(2,)f(the)h(latter)g(b)q(eing) g(tak)o(en)g(for)g(gran)o(ted.)-98 2012 y(The)h(results)g(of)e(the)h (presen)o(t)i(pap)q(er)f(are)f(also)f(v)n(alid)g(for)g(a)h(Wigner)f (ensem)o(ble)h(of)f(complex)g(self-adjoin)o(t)g(matrices.)-148 2062 y(This,)h(as)g(w)o(ell)f(as)h(other)g(remarks,)f(is)h(discussed)i (in)d Ft(x)p Fr(3.)-3 2157 y Ft(x)-2 2155 y(x)-1 2157 y(x)p Fs(2.)24 b(The)16 b(Deriv)m(ation)d(of)i(the)g(Main)g(Theorem)g (and)g(its)g(Corollaries)e(from)i(Theorem)f(2)-98 2232 y Fr(T)m(o)i(pro)o(v)o(e)h(the)h(main)d(theorem,)h(it)h(su\016ces)h(to) f(establish)g(the)g(con)o(v)o(ergence)i(of)d(the)i(Laplace)f (transforms)f(of)g(the)-148 2281 y(measures)j Fq(\026)60 2287 y Fp(n)88 2281 y Fr(,)13 b(that)h(is,)g(to)f(sho)o(w)h(that)137 2333 y Fm(Z)178 2343 y Fn(1)160 2427 y(\0001)228 2389 y Fq(e)247 2372 y Fn(\000)p Fp(c\022)307 2389 y Fq(d\026)354 2395 y Fp(n)376 2389 y Fr(\()p Fq(\022)q Fr(\))e(=)527 2361 y(1)p 489 2380 97 2 v 489 2428 a Fq(nr)534 2406 y Fo(3)p Fp(=)p Fo(2)533 2433 y Fp(n)618 2337 y(n)598 2350 y Fm(X)598 2439 y Fp(k)q Fo(=1)665 2389 y Fq(e)684 2372 y Fn(\000)p Fp(c\022)741 2376 y Fh(k)818 2366 y Ff(P)773 2389 y Ft(\000)-7 b(\000)e(\000)i(!)787 2418 y Fp(n)p Fn(!1)900 2333 y Fm(Z)941 2343 y Fo(+)p Fn(1)923 2427 y Fo(0)1009 2389 y Fq(e)1028 2372 y Fn(\000)p Fp(c\022)1099 2361 y Fr(2)1120 2327 y Ft(p)p 1155 2327 21 2 v 34 x Fr(2)p 1099 2380 77 2 v 1125 2418 a Fq(\031)1187 2352 y Ft(p)p 1222 2352 21 2 v 37 x Fq(\022)9 b(d\022)j Fr(=)1347 2324 y Fm(r)p 1389 2324 72 2 v 1414 2361 a Fr(2)p 1394 2380 62 2 v 1394 2418 a Fq(\031)q(c)1437 2406 y Fo(3)1463 2389 y Fq(;)201 b Fr(\(2.1\))-148 2503 y(where)20 b Fq(c)12 b(>)g Fr(0)5 b(.)1712 2603 y Fo(115)p eop %%Page: 116 3 116 2 bop -98 -71 a Fr(W)m(e)13 b(set)18 b Fq(s)60 -65 y Fp(n)95 -71 y Fr(=)12 b([\()p Fq(c=)p Fr(2\))t Fq(r)267 -86 y Fn(\000)p Fo(1)266 -60 y Fp(n)311 -71 y Fr(])17 b(and)h Fq(p)446 -65 y Fp(n)480 -71 y Fr(=)12 b(2)p Fq(s)564 -65 y Fp(n)604 -71 y Fr(and)h(in)o(tro)q(duce)h(the)f(follo)o(wing)d (renormalization)h(for)i(the)g(p)q(ositiv)o(e)g(and)-148 -21 y(negativ)o(e)h(eigen)o(v)n(alues:)550 33 y Fq(\025)574 39 y Fp(k)607 33 y Fr(=)d(1)e Ft(\000)h Fq(\022)741 39 y Fp(k)762 33 y Fq(r)781 39 y Fp(n)806 33 y Fq(;)113 b(\025)955 39 y Fp(k)987 33 y Ft(\025)12 b Fr(0)s Fq(;)553 95 y(\025)577 101 y Fp(j)607 95 y Fr(=)f Ft(\000)p Fr(1)f(+)f Fq(\034)772 101 y Fp(j)790 95 y Fq(r)809 101 y Fp(n)834 95 y Fq(;)88 b(\025)958 101 y Fp(j)987 95 y Fq(<)12 b Fr(0)7 b Fq(:)1676 68 y Fr(\(2.2\))-148 173 y(According)15 b(to)e(Theorem)h(2,)f(the)h(sequence)21 b Fq(n)595 158 y Fn(\000)p Fo(1)640 173 y Fq(r)660 151 y Fn(\000)p Fo(3)p Fp(=)p Fo(2)659 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Fh(n)841 685 y Fo(]+1)904 702 y Ft(\000)945 663 y Fm(X)967 751 y Fp(j)1005 702 y Fr(\(1)g Ft(\000)h Fq(\034)1111 708 y Fp(j)1128 702 y Fq(r)1147 708 y Fp(n)1170 702 y Fr(\))1186 685 y Fo(2[\()p Fp(c=)p Fo(2\))p Fp(r)1303 673 y Fe(\000)p Fg(1)1302 693 y Fh(n)1341 685 y Fo(]+1)1395 702 y Fq(:)269 b Fr(\(2.4\))-148 822 y(Note)18 b(that)g(if)k Ft(j)p Fq(\022)128 828 y Fp(k)149 822 y Ft(j)17 b(\024)h Fq(r)248 800 y Fn(\000)p Fo(1)p Fp(=)p Fo(3)247 827 y Fp(n)349 822 y Fr(and)k Ft(j)p Fq(\034)468 828 y Fp(j)485 822 y Ft(j)c(\024)g Fq(r)585 800 y Fn(\000)p Fo(1)p Fp(=)p Fo(3)584 827 y Fp(n)668 822 y Fr(,)g(then)g(the)h(corresp)q(onding)f (terms)g(in)f(b)q(oth)h(\(2.3\))f(and)g(\(2.4\))g(are)-148 881 y Fq(e)-129 866 y Fn(\000)p Fp(c\022)-72 870 y Fh(j)-54 881 y Fr(\(1)r(+)r Fq(o)p Fr(\()p Fq(r)75 860 y Fo(1)p Fp(=)p Fo(3)74 886 y Fp(n)127 881 y Fr(\)\))e(and)g Fq(e)275 866 y Fn(\000)p Fp(c\034)331 870 y Fh(j)349 881 y Fr(\(1)r(+)r Fq(o)p Fr(\()p Fq(r)478 860 y Fo(1)p Fp(=)p Fo(3)477 886 y 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b(and)g(hence)i(the)e(con)o(v)o(ergence)i(of)-148 1667 y(the)i(measures)23 b Fq(\026)139 1673 y Fp(n)184 1667 y Fr(in)17 b(the)g(main)f(theorem)h(also)f(o)q(ccurs)j(with)e (probabilit)o(y)k(1)5 b(.)28 b(It)17 b(also)g(follo)o(ws)f(from)f (Theorem)i(2)-148 1716 y(that)h(the)f(\015uctuations)h(of)f(the)g (random)f(v)n(ariables)711 1685 y Fm(P)755 1729 y Fp(k)783 1716 y Fq(e)802 1701 y Fn(\000)p Fp(c\022)859 1705 y Fh(k)901 1716 y Fr(are)i(of)e(the)i(order)g(of)e(a)h(constan)o(t)h(and) f(con)o(v)o(erge)h(in)-148 1770 y(distribution)c(to)f(the)i(normal)d (la)o(w)18 b Fq(N)5 b Fr(\(0)s Fq(;)j Fr(1)p Fq(=)p Fr(2)p Fq(\031)q Fr(\))19 b(as)g Fq(n)11 b Ft(!)g(1)19 b Fr(pro)o(vided)13 b(that)19 b Fq(r)1133 1776 y Fp(n)1167 1770 y Ft(\034)11 b Fq(n)1245 1755 y Fn(\000)p Fo(2)p Fp(=)p Fo(5)1328 1770 y Fr(.)-98 1819 y(Belo)o(w)j(w)o(e)g(pro)o(v)o(e)g(Corollaries)f (1)h(and)f(2.)-98 1868 y(Let)h(us)f(represen)o(t)j(the)d(maxim)o(um)8 b(eigen)o(v)n(alue)13 b(in)g(the)g(form)k Fq(\025)913 1874 y Fo(max)988 1868 y Fr(=)12 b(1)7 b(+)h Fq(\022)1119 1874 y Fo(max)1183 1868 y Fq(n)1208 1853 y Fn(\000)p Fo(2)p Fp(=)p Fo(3)1291 1868 y Fr(.)18 b(Supp)q(ose)c(that)f(there)h (exist)-148 1918 y(sequences)22 b Fq(n)71 1924 y Fp(i)96 1918 y Ft(!)11 b(1)18 b Fr(and)h Fq(L)323 1924 y Fp(i)349 1918 y Ft(!)11 b(1)18 b Fr(and)c(an)k Fq(")12 b(>)g Fr(0)19 b(suc)o(h)14 b(that)456 1997 y Fs(P)p Ft(f)p Fq(\025)534 2003 y Fo(max)597 1997 y Fr(\()p Fq(A)644 2003 y Fp(n)665 2007 y Fh(i)680 1997 y Fr(\))e Fq(>)f Fr(1)e(+)h Fq(n)848 1976 y Fn(\000)p Fo(2)p Fp(=)p Fo(3)848 2009 y Fp(i)926 1997 y Fq(L)954 2003 y Fp(i)968 1997 y Ft(g)h Fq(>)h(")g(>)g Fr(0)7 b Fq(:)517 b Fr(\(2.5\))-148 2080 y(Let)11 b(us)g(consider)16 b(T)m(r)7 b Fq(A)217 2055 y Fp(p)234 2059 y Fh(n)252 2066 y(i)217 2085 y Fp(n)238 2089 y Fh(i)268 2080 y Fr(,)k(where)16 b Fq(p)433 2086 y Fp(n)454 2090 y Fh(i)480 2080 y Fr(=)c(2)t([)p Fq(n)586 2065 y Fo(2)p Fp(=)p Fo(3)631 2080 y Fq(=L)680 2059 y Fo(1)p Fp(=)p Fo(2)680 2092 y Fp(i)732 2080 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b(let)h(us)g(consider)g(the)g(p)q(oin)o(t)g(random)e(\014eld)20 b Ft(f)p Fq(x)1423 2509 y Fo(1)1444 2503 y Fq(;)10 b(:)d(:)g(:)h(;)h(x) 1569 2509 y Fp(n)1591 2503 y Ft(g)21 b Fr(de\014ned)1712 2603 y Fo(117)p eop %%Page: 118 5 118 4 bop -148 -71 a Fr(in)14 b(\(3.1\),)g(c)o(ho)q(ose)h(a)g(sequence) 21 b Fq(R)390 -65 y Fp(n)425 -71 y Ft(!)13 b Fr(+)p Ft(1)19 b Fr(suc)o(h)c(that)20 b Fq(R)795 -65 y Fp(n)830 -71 y Ft(\034)12 b Fq(n)909 -86 y Fo(2)p Fp(=)p Fo(3)966 -71 y Fr(,)i(and)h(coun)o(t)f(the)i(p)q(oin)o(ts)j Fq(x)1415 -65 y Fp(i)1448 -71 y Fr(lying)13 b(to)i(the)g(righ)o(t)-148 -19 y(of)k Ft(\000)p Fq(R)-31 -13 y Fp(n)-9 -19 y Fr(:)24 b Fq(N)60 -13 y Fp(n)83 -19 y Fr(\()p Ft(\000)p Fq(R)163 -13 y Fp(n)186 -19 y Fr(\))12 b(=)g(#)p Ft(f)p Fq(x)338 -13 y Fp(i)363 -19 y Fq(>)h Ft(\000)p Fq(R)472 -13 y Fp(n)494 -19 y Ft(g)5 b Fr(.)19 b(By)14 b(de\014nition,)19 b Fq(N)851 -13 y Fp(n)874 -19 y Fr(\()p Ft(\000)p Fq(R)954 -13 y Fp(n)976 -19 y Fr(\))12 b(=)h(#)p Ft(f)p Fq(\025)1129 -13 y 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b(and)c(the)g(di\013erence) 23 b Fq(x)p Fr(\()p Fq(k)q Fr(\))10 b Ft(\000)h Fq(x)p Fr(\()p Fq(k)g Ft(\000)g Fr(1\))21 b(is)15 b(equal)h(to)k(1)h(or)f Ft(\000)p Fr(1)h(dep)q(ending)c(on)e(whether)i(the)22 b Fq(k)q Fr(th)-148 2503 y(instan)o(t)12 b(is)g(or)g(is)g(not)g(mark)o (ed.)k(Ob)o(viously)m(,)g Fq(x)p Fr(\()p Fq(t)p Fr(\))c Ft(\025)f Fr(0)17 b(for)12 b(all)k(0)11 b Ft(\024)h Fq(t)f Ft(\024)h Fq(p)1015 2509 y Fp(n)1043 2503 y Fr(,)g(and)g(w)o(e)g(ha)o (v)o(e)17 b Fq(x)p Fr(\()p Fq(p)1365 2509 y Fp(n)1387 2503 y Fr(\))12 b(=)g(0)k(for)c(ev)o(en)h(paths.)-148 2603 y Fo(118)p eop %%Page: 119 6 119 5 bop -98 -71 a Fs(De\014nition)9 b(2.)24 b Fr(An)12 b(ev)o(en)g(path)k Fc(P)h Fr(is)11 b(called)g(a)g Fi(p)n(ath)i(without) f(self-interse)n(ctions)i Fr(if,)c(for)h(an)o(y)g(t)o(w)o(o)g(distinct) h(mark)o(ed)-148 -20 y(instan)o(ts)19 b Fq(k)36 -35 y Fn(0)67 -20 y Fr(and)g Fq(k)176 -35 y Fn(0)o(0)202 -20 y Fr(,)13 b(one)h(has)19 b Fq(i)396 -14 y Fp(k)415 -22 y Fe(0)440 -20 y Ft(6)p Fr(=)11 b 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910 a Fm(p)p 827 910 68 2 v 827 945 a Fq(\031)q(s)871 933 y Fo(3)871 955 y Fp(n)904 911 y Fr(\(1)d(+)g Fq(o)p Fr(\(1\)\))p Fq(:)-98 1020 y Fr(In)14 b(what)g(follo)o(ws,)e(w)o(e)i (study)g(the)h(sum)e(o)o(v)o(er)g(the)i(paths)f(for)g(whic)o(h)f(w)o(e) i(alw)o(a)o(ys)d(ha)o(v)o(e)664 1078 y Fp(s)680 1082 y Fh(n)652 1090 y Fm(X)652 1180 y Fp(k)q Fo(=2)719 1130 y Fq(k)q(n)767 1136 y Fp(k)799 1130 y Fq(<)g Fr(10)896 1102 y Fq(s)915 1087 y Fo(2)915 1112 y Fp(n)p 896 1120 43 2 v 905 1158 a Fq(n)950 1130 y(:)714 b Fr(\(4.6\))-148 1251 y(Let)22 b Fq(M)d Fr(=)40 1219 y Fm(P)84 1230 y Fp(s)100 1234 y Fh(n)84 1263 y Fp(k)q Fo(=2)146 1251 y Fr(\()p Fq(k)12 b Ft(\000)f Fr(1\))t Fq(n)305 1257 y Fp(k)330 1251 y Fr(.)24 b(If)15 b(a)h(path)21 b Fc(P)g Fr(con)o(tains)15 b(only)g(simple)g(self-in)o(tersections,)i(then)k Fq(M)f Fr(=)15 b Fq(n)1546 1257 y Fo(2)1569 1251 y Fr(.)24 b(W)m(e)15 b(shall)-148 1304 y(sho)o(w)j(that)f(in)g(our)g(case)h(\()5 b Fq(s)312 1310 y Fp(n)352 1304 y Fr(=)18 b Fq(o)p Fr(\()p 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Fm(\023)1005 1499 y Fp(M)1042 1549 y Fr(\(1)k(+)h Fq(o)p Fr(\(1\)\))457 b(\(4.7\))-148 1654 y Fi(uniformly)15 b(with)f(r)n(esp)n(e)n(ct)g(to)20 b Fr(0)11 b Ft(\024)h Fq(M)k Ft(\024)c Fr(10)p Fq(s)559 1639 y Fo(2)559 1665 y Fp(n)582 1654 y Fq(=n)5 b Fi(.)-98 1729 y Fs(Remark)17 b(2.)24 b Fr(By)16 b(form)o(ula)c(\(4.7\),)j(the)h (n)o(um)o(b)q(er)j Fq(M)25 b Fr(of)15 b(self-in)o(tersections)i(for)d (t)o(ypical)h(paths)h(is)f(of)f(the)i(order)g(of)-148 1779 y Fq(s)-129 1764 y Fo(2)-129 1789 y Fp(n)-106 1779 y Fq(=n)j Fr(and)14 b(is)f(equal)h(to)19 b(\()p Fq(s)282 1764 y Fo(2)282 1789 y Fp(n)305 1779 y Fq(=)p Fr(2)p Fq(n)p Fr(\)\(1)9 b(+)g Fq(O)q Fr(\()524 1749 y Ft(p)p 559 1749 25 2 v 30 x Fq(n)o(=s)623 1785 y Fp(n)646 1779 y Fr(\)\))19 b(for)g Fq(s)785 1785 y Fp(n)820 1779 y Ft(\035)873 1749 y(p)p 907 1749 V 907 1779 a Fq(n)5 b Fr(.)-98 1854 y Fs(Pro)q(of)19 b(of)h(Prop)q(osition)d(1.)24 b Fr(Let)18 b(us)g(calculate)23 b Fq(Z)s Fr(\()p Fq(M)5 b Fr(\))23 b(as)17 b(follo)o(ws.)28 b(First,)19 b(w)o(e)f(c)o(ho)q(ose) 23 b Fq(s)1429 1860 y Fp(n)1470 1854 y Fr(=)18 b Fq(p)1541 1860 y Fp(n)1564 1854 y Fq(=)p Fr(2)k(mark)o(ed)-148 1903 y(instan)o(ts)e Fq(t)29 1909 y Fp(i)62 1903 y Fr(and)14 b(assume)g(that)19 b(0)12 b Ft(\024)h Fq(t)475 1909 y Fo(1)505 1903 y Fq(<)g(t)565 1909 y Fo(2)596 1903 y Fq(<)f Ft(\001)7 b(\001)g(\001)k Fq(<)h(t)760 1909 y Fp(s)776 1913 y Fh(n)811 1903 y Fq(<)g Fr(2)p Fq(s)895 1909 y Fp(n)923 1903 y Fr(.)19 b(Recall)14 b(that)g(to)g(an)o(y)g(c)o(hoice)h (of)e(mark)o(ed)g(instan)o(ts)-148 1953 y(there)k(corresp)q(onds)g(a)e (tra)r(jectory)21 b Fq(x)p Fr(\()p Fq(t)p Fr(\))5 b(,)20 b(0)13 b Ft(\024)h Fq(t)f Ft(\024)h Fr(2)p Fq(s)727 1959 y Fp(n)755 1953 y Fr(,)h(of)f(the)i(simplest)e(random)f(w)o(alk)h(on)h (the)h(p)q(ositiv)o(e)f(half-line,)-148 2003 y(namely)m(,)635 2094 y Fq(x)p Fr(\(0\))c(=)h Fq(x)p Fr(\(2)p Fq(s)847 2100 y Fp(n)870 2094 y Fr(\))f(=)h(0)s Fq(;)437 2159 y(x)p Fr(\()p Fq(t)492 2165 y Fp(j)509 2159 y Fr(\))e Ft(\000)f Fq(x)p Fr(\()p Fq(t)631 2165 y Fp(j)658 2159 y Ft(\000)g Fr(1\))j(=)f(1)s Fq(;)92 b(j)14 b Fr(=)e(1)s Fq(;)d(:)e(:)g(:)h(;)h(s)1137 2165 y Fp(n)1163 2159 y Fq(;)405 2223 y(x)p Fr(\()p Fq(t)p Fr(\))g Ft(\000)g Fq(x)p Fr(\()p Fq(t)g Ft(\000)h Fr(1\))h(=)h Ft(\000)p Fr(1)83 b(for)21 b Fq(t)11 b Ft(6)p Fr(=)h Fq(t)1016 2229 y Fo(1)1037 2223 y Fq(;)e(:)d(:)g(:)h(;)h(t)1153 2229 y Fp(s)1169 2233 y Fh(n)1198 2223 y Fq(:)-148 2304 y Fr(Then,)15 b(among)d(the)j(mark)o(ed)e(instan)o(ts)i(w)o(e)f(c)o(ho) q(ose)20 b Fq(M)k Fr(instan)o(ts)15 b(of)f(self-in)o(tersection)20 b Fq(t)1267 2310 y Fp(j)1281 2314 y Fg(1)1302 2304 y Fq(;)9 b(t)1338 2310 y Fp(j)1352 2314 y Fg(2)1373 2304 y Fq(;)g(:)e(:)g(:)h(;)h(t)1488 2310 y Fp(j)1502 2314 y Fh(M)1554 2304 y Fr(\(that)15 b(is,)e(w)o(e)-148 2354 y(c)o(ho)q(ose)h(indices)k Fq(j)140 2360 y Fo(1)162 2354 y Fq(;)9 b(:)e(:)g(:)h(;)h(j)279 2360 y Fp(M)334 2354 y Fr(suc)o(h)14 b(that)k(1)11 b Ft(\024)h Fq(j)614 2360 y Fo(1)644 2354 y Fq(<)g(j)705 2360 y Fo(2)735 2354 y Fq(<)g Ft(\001)7 b(\001)g(\001)j Fq(<)i(j)900 2360 y Fp(M)948 2354 y Ft(\024)g Fq(s)1011 2360 y Fp(n)1034 2354 y Fr(\).)18 b(After)c(this,)e(w)o(e)i(c)o(ho)q(ose)f(the)h(origin) e(of)g(the)-148 2403 y(path)g(and)g(the)h(v)o(ertices)h(o)q(ccurring)e (at)g(the)h(mark)o(ed)e(instan)o(ts.)18 b(\(The)12 b(origin)f(can)i(b)q (e)f(c)o(hosen)h(in)k Fq(n)g Fr(w)o(a)o(ys,)12 b(and)g(then)g(w)o(e) -148 2453 y(successiv)o(ely)h(c)o(ho)q(ose,)f(in)j Fq(n)s Ft(\000)s Fr(1)s Fq(;)9 b(n)s Ft(\000)s Fr(2)s Fq(;)g(:)e(:)g(:)h(;)i (n)s Ft(\000)s Fq(s)644 2459 y Fp(n)670 2453 y Fr(+)s Fq(M)21 b Fr(w)o(a)o(ys,)11 b(the)g(v)o(ertices)h(o)q(ccurring)g(at)e (the)i(mark)o(ed)e(instan)o(ts)h(that)-148 2503 y(are)j(not)e(instan)o (ts)h(of)f(self-in)o(tersection.\))19 b(A)o(t)13 b(the)g(instan)o(ts)g (of)f(self-in)o(tersection,)i(the)f(v)o(ertices)h(are)f(c)o(hosen)h(as) f(follo)o(ws.)-148 2603 y Fo(120)p eop %%Page: 121 8 121 7 bop -148 -71 a Fr(A)o(t)17 b(the)h(\014rst)g(mark)o(ed)e(instan)o (t)22 b Fq(t)393 -65 y Fp(j)407 -61 y Fg(1)447 -71 y Fr(of)16 b(self-in)o(tersection,)i(w)o(e)g(can)f(c)o(ho)q(ose)h(one)f (of)f(the)i(v)o(ertices)g(that)f(o)q(ccur)h(in)f(the)-148 -19 y(path)d(b)q(efore)h(the)f(instan)o(t)19 b Fq(t)305 -13 y Fp(j)319 -9 y Fg(1)342 -19 y Fr(,)13 b(and)h(there)h(are)f (exactly)19 b Fq(j)788 -13 y Fo(1)816 -19 y Ft(\000)9 b Fr(1)19 b(suc)o(h)c(v)o(ertices)g(b)q(ecause)g(our)f(tra)r(jectory)20 b Fq(x)p Fr(\()p Fq(t)p Fr(\))f(of)13 b(the)-148 33 y(random)g(w)o(alk) g(made)18 b Fq(j)235 39 y Fo(1)263 33 y Ft(\000)10 b Fr(1)19 b(steps)c(to)f(the)h(righ)o(t)f(for)k(0)12 b Ft(\024)g Fq(t)g(<)h(t)905 39 y Fp(j)919 43 y Fg(1)942 33 y Fr(.)18 b(A)o(t)d(the)f(next)h(instan)o(t)f(of)g(self-in)o (tersection,)19 b Fq(t)1713 39 y Fp(j)1727 43 y Fg(2)1750 33 y Fr(,)-148 85 y(there)14 b(are)f(exactly)k Fq(j)188 91 y Fo(2)213 85 y Ft(\000)6 b Fr(2)18 b(p)q(ossibilities)12 b(of)f(c)o(ho)q(osing)h(a)g(v)o(ertex)i(\(as)e(long)g(as)g(w)o(e)h (deal)f(with)g(simple)f(self-in)o(tersections,)-148 137 y(the)i(v)o(ertex)g(c)o(hosen)g(at)f(the)h(instan)o(t)j Fq(t)453 143 y Fp(j)467 147 y Fg(1)502 137 y Fr(cannot)d(o)q(ccur)g(at) f(an)o(y)f(later)h(instan)o(t)g(of)g(self-in)o(tersection\).)18 b(Lik)o(ewise,)12 b(at)g(the)-148 189 y(last)j(mark)o(ed)f(instan)o(t)g (of)h(self-in)o(tersection,)20 b Fq(t)598 195 y Fp(j)612 199 y Fh(M)650 189 y Fr(,)14 b(there)j(are)j Fq(j)876 195 y Fp(M)923 189 y Ft(\000)10 b Fq(M)24 b Fr(p)q(ossibilities)15 b(of)f(c)o(ho)q(osing)g(the)i(next)f(v)o(ertex.)-148 241 y(F)m(or)d(a)g(path)g(with)f(self-in)o(tersections,)i(the)g(c)o (hoice)f(of)g(v)o(ertices)h(at)f(unmark)o(ed)f(instan)o(ts)h(\(the)h(c) o(hoice)f(of)f(the)i(\\bac)o(kw)o(ard)-148 293 y(tra)r(jectory"\))h(ma) o(y)d(b)q(e)j(am)o(biguous.)h(F)m(or)e(the)g(\\\014rst)h(return")g (from)d(a)h(v)o(ertex)i(of)f(simple)e(self-in)o(tersection,)j(one)f(of) f(the)-148 345 y(follo)o(wing)g(three)j(edges)g(can)f(b)q(e)g(c)o (hosen:)-98 407 y(\(a\))g(the)h(edge)f(used)h(to)f(arriv)o(e)f(at)h (the)h(v)o(ertex)f(for)g(the)h(\014rst)f(time;)-98 470 y(\(b\))g(the)h(edge)f(used)h(to)f(lea)o(v)o(e)g(the)g(v)o(ertex;)-98 532 y(\(c\))h(the)f(edge)h(used)g(to)e(arriv)o(e)h(at)g(the)g(v)o (ertex)h(for)f(the)g(second)h(time.)-148 595 y(W)m(e)h(shall)g(sho)o(w) g(b)q(elo)o(w)g(that)g(only)g(the)h(third)f(p)q(ossibilit)o(y)f(is)h (realized)h(for)f(t)o(ypical)f(paths)i(since,)g(b)o(y)f(the)h(instan)o (t)f(of)-148 647 y(self-in)o(tersection,)g(the)g(edges)g(\(a\))f(and)g (\(b\))g(ha)o(v)o(e)g(already)g(b)q(een)h(passed)g(t)o(wice.)22 b(Let)f Fq(Z)1279 653 y Fo(1)1298 647 y Fr(\()p Fq(M)5 b Fr(\))20 b(b)q(e)15 b(the)h(sum)e(o)o(v)o(er)h(the)-148 699 y(paths)e(with)k Fq(M)22 b Fr(simple)11 b(self-in)o(tersections)j (suc)o(h)f(that)f(the)h(edge)g(used)h(to)e(arriv)o(e)g(at)h(a)f(v)o (ertex)h(of)f(self-in)o(tersection)h(for)-148 751 y(the)i(last)e(time)g (is)h(alw)o(a)o(ys)f(c)o(hosen)h(for)g(returning.)19 b(If)13 b(eac)o(h)h(edge)h(of)j Fc(P)i Fr(is)13 b(passed)i(t)o(wice,)f (then)554 888 y Fs(E)589 830 y Fm(\022)627 836 y Fo(2)p Fp(s)660 840 y Fh(n)680 836 y Fn(\000)p Fo(1)648 849 y Fm(Y)648 938 y Fp(l)p Fo(=0)729 888 y Fq(\030)747 894 y Fp(i)759 898 y Fh(l)771 894 y Fp(i)783 898 y Fh(l)p Fg(+1)832 830 y Fm(\023)875 888 y Fr(=)918 830 y Fm(\022)954 860 y Fr(1)p 954 879 21 2 v 954 917 a(4)980 830 y Fm(\023)1010 838 y Fp(s)1026 842 y Fh(n)1048 888 y Fq(:)616 b Fr(\(4.8\))-148 1025 y(In)14 b(the)f(general)h(case,)g(some)e(edges)i(can)g(b)q(e)g (used)g(four)f(times.)k(Let)h Fq(m)h Fr(b)q(e)14 b(the)g(n)o(um)o(b)q (er)e(of)h(suc)o(h)h(edges.)19 b(In)13 b(this)g(case,)430 1163 y Fs(E)465 1104 y Fm(\022)503 1111 y Fo(2)p Fp(s)536 1115 y Fh(n)556 1111 y Fn(\000)p Fo(1)524 1124 y Fm(Y)524 1213 y Fp(l)p Fo(=0)605 1163 y Fq(\030)623 1169 y Fp(l)633 1173 y Fh(l)646 1169 y Fp(i)658 1173 y Fh(l)p Fg(+1)707 1104 y Fm(\023)749 1163 y Ft(\024)793 1104 y Fm(\022)829 1135 y Fr(1)p 829 1153 V 829 1191 a(4)854 1104 y Fm(\023)885 1113 y Fp(s)901 1117 y Fh(n)921 1113 y Fn(\000)p Fo(2)p Fp(m)995 1163 y Fr(\(const)1106 1169 y Fo(2)1125 1163 y Fr(\))1141 1146 y Fp(m)1172 1163 y Fq(:)492 b Fr(\(4.9\))-148 1303 y(Let)18 b(us)f(sho)o(w)g(that)h(the)f(main)e(con)o(tribution)i (to)22 b Fq(Z)694 1309 y Fo(1)713 1303 y Fr(\()p Fq(M)5 b Fr(\))22 b(is)17 b(due)g(to)g(paths)h(in)e(whic)o(h)h(eac)o(h)h(edge) g(is)f(passed)h(t)o(wice.)-148 1354 y(In)e(the)g(App)q(endix,)g(w)o(e)g (study)g(the)h(follo)o(wing)c(c)o(haracteristic)k(of)j Fc(P)p Fr(:)i(the)16 b(maxim)n(um)c(n)o(um)o(b)q(er)j(of)g(v)o(ertices) i(that)e(can)-148 1406 y(b)q(e)i(visited)f(at)f(mark)o(ed)g(instan)o (ts)h(from)e(a)i(giv)o(en)f(v)o(ertex.)25 b(Let)16 b(us)g(denote)h (this)f(n)o(um)o(b)q(er)f(b)o(y)21 b Fq(\027)1390 1412 y Fp(n)1412 1406 y Fr(\()p Fc(P)p Fr(\))5 b(.)24 b(By)17 b(de\014nition,)-148 1458 y(eac)o(h)f(v)o(ertex)22 b Fq(i)e Fr(of)15 b(the)h(path)k Fc(P)h Fr(is)15 b(the)h(left)g(end)g(of) e(at)i(most)j Fq(\027)888 1464 y Fp(n)910 1458 y Fr(\()p Fc(P)p Fr(\))i(mark)o(ed)14 b(edges.)24 b(In)15 b(the)h(App)q(endix,)g (w)o(e)g(pro)o(v)o(e)-148 1510 y(that)22 b Fq(\027)-29 1516 y Fp(n)-6 1510 y Fr(\()p Fc(P)p Fr(\))g(cannot)17 b(gro)o(w)g(to)q(o)g(fast)g(for)f(t)o(ypical)h(paths;)h(for)f (instance,)h(it)f(gro)o(ws)g(no)f(faster)i(than)k Fq(s)1523 1495 y Fp(\015)1523 1520 y(n)1568 1510 y Fr(for)17 b(an)o(y)k Fq(\015)8 b Fr(,)-148 1562 y(0)k Fq(<)g(\015)i Ft(\024)e Fr(1)5 b(.)18 b(In)13 b(other)i(w)o(ords,)f(the)g(sum)f(o)o(v)o(er)h (paths)g(with)19 b Fq(\027)837 1568 y Fp(n)859 1562 y Fr(\()p Fc(P)p Fr(\))12 b Fq(>)g(s)993 1547 y Fp(\015)993 1572 y(n)1035 1562 y Fr(is)19 b Fq(o)p Fr(\(1\))g(\(see)c(Lemma)c(2\).) 18 b(In)c(what)g(follo)o(ws,)-148 1614 y(w)o(e)g(alw)o(a)o(ys)f(assume) h(that)686 1697 y Fq(\027)707 1703 y Fp(n)729 1697 y Fr(\()p Fc(P)q Fr(\))e Fq(<)g(s)864 1679 y Fo(1)p Fp(=)p Fo(4)864 1707 y Fp(n)916 1697 y Fq(:)728 b Fr(\(4.10\))-148 1780 y(Let)20 b Fq(j)-51 1786 y Fp(u)-31 1790 y Fg(1)-11 1780 y Fq(;)10 b(:)d(:)g(:)h(;)h(j)107 1786 y Fp(u)127 1790 y Fh(m)175 1780 y Fr(b)q(e)14 b(the)h(indices)f(of)f(the)i(instan) o(ts)f(of)f(self-in)o(tersection)h(corresp)q(onding)h(to)f(edges)h (that)e(are)i(passed)-148 1832 y(four)f(times.)j(In)d(this)g(case,)-106 1938 y Fq(Z)-78 1944 y Fo(1)-59 1938 y Fr(\()p Fq(M)5 b Fr(\))11 b(=)116 1898 y Fm(X)73 1989 y Fp(X)r Fo(=)p Fn(f)p Fp(x)p Fo(\()p Fp(t)p Fo(\))p Fn(g)338 1898 y Fm(X)238 1987 y Fo(1)p Fn(\024)p Fp(j)295 1991 y Fg(1)310 1987 y Fp(<)p Fn(\001\001\001)p Fp()h Fr(0)5 b(,)11 b(and)g(since)h(the)f(co)q(e\016cien)o(t)17 b(60)p Fq(s)901 655 y Fo(3)p Fp(=)p Fo(2)901 682 y Fp(n)947 677 y Fq(=n)e Fr(in)c(form)o(ula)d(\(2.22\))i(tends)i(to)f(zero)h(as)k Fq(n)c Ft(!)f(1)5 b Fr(,)-148 727 y(it)14 b(follo)o(ws)e(that)105 831 y(0)f Ft(\024)h Fs(E)216 837 y Fp(X)248 772 y Fm(\022)279 831 y Fr(exp)349 772 y Fm(\022)385 802 y Fr(60)p Fq(s)446 781 y Fo(3)p Fp(=)p Fo(2)446 807 y Fp(n)p 385 821 114 2 v 429 859 a Fq(n)538 831 y Fr(max)510 857 y Fo(0)p Fn(\024)p Fp(t)p Fn(\024)p Fo(2)p Fp(s)625 861 y Fh(n)658 802 y Fq(x)p Fr(\()p Fq(t)p Fr(\))p 655 821 77 2 v 655 832 a Ft(p)p 690 832 43 2 v 27 x Fq(s)709 865 y Fp(n)737 772 y Fm(\023)776 831 y Ft(\000)e Fr(1)839 772 y Fm(\023)881 831 y Ft(\024)i Fs(E)960 837 y Fp(X)992 772 y Fm(\022)1022 831 y Fr(exp)1093 772 y Fm(\022)1123 831 y Fq(")36 b Fr(max)1149 857 y Fo(0)p Fn(\024)p Fp(t)p Fn(\024)p Fo(2)p Fp(s)1264 861 y Fh(n)1298 802 y Fq(x)p Fr(\()p Fq(t)p Fr(\))p 1295 821 77 2 v 1295 832 a Ft(p)p 1329 832 43 2 v 1329 859 a Fq(s)1348 865 y Fp(n)1377 772 y Fm(\023)1416 831 y Ft(\000)10 b Fr(1)1479 772 y Fm(\023)137 936 y Ft(\024)181 890 y Fm(\020)206 936 y Fs(E)241 890 y Fm(\020)266 936 y Fr(exp)336 890 y Fm(\020)361 936 y Fq(")18 b Fr(max)387 963 y Fo(0)p Fn(\024)p Fp(t)p Fn(\024)p Fo(1)492 936 y Fq(b)p Fr(\()p Fq(t)p Fr(\))569 889 y Fm(\014)569 914 y(\014)569 938 y(\014)594 936 y Fq(b)p Fr(\(1\))11 b(=)h(0)741 890 y Fm(\021\021)800 936 y Ft(\000)d Fr(1)862 890 y Fm(\021)887 936 y Fr(\(1)g(+)g Fq(")p Fr(\))647 b(\(4.23\))-148 1034 y(for)17 b(su\016cien)o(tly)g(large)k Fq(n)5 b Fr(.)27 b(By)17 b(c)o(ho)q(osing)22 b Fq(")g Fr(to)17 b(b)q(e)g(su\016cien)o (tly)g(small,)e(w)o(e)i(can)g(ensure)i(that)e(the)g(righ)o(t-hand)f (side)-148 1084 y(of)e(\(4.23\))f(is)g(as)h(close)h(to)e(zero)i(as)f (desired.)19 b(Hence,)649 1162 y Fq(Z)677 1168 y Fo(2)707 1162 y Fr(=)12 b Fq(o)p Fr(\()p Fq(Z)815 1168 y Fo(1)834 1162 y Fr(\))e(+)f Fq(Z)932 1145 y Fn(0)929 1172 y Fo(2)951 1162 y Fq(;)693 b Fr(\(4.24\))-148 1240 y(where)20 b Fq(Z)8 1225 y Fn(0)5 1250 y Fo(2)43 1240 y Fr(is)14 b(the)h(subsum)e (of)19 b Fq(Z)388 1246 y Fo(2)425 1240 y Fr(o)o(v)o(er)14 b(the)h(paths)f(in)g(whic)o(h)g(at)g(least)g(one)g(edge)h(is)e(used)i (four)f(times.)j(The)e(sum)j Fq(Z)1746 1225 y Fn(0)1743 1250 y Fo(2)-148 1290 y Fr(can)e(b)q(e)h(analyzed)e(in)h(the)g(same)f (w)o(a)o(y)g(as)21 b Fq(Z)567 1274 y Fn(0)564 1300 y Fo(1)588 1290 y Fr(,)16 b(and)f(w)o(e)h(\014nally)f(obtain)20 b Fq(Z)1056 1274 y Fn(0)1053 1300 y Fo(2)1086 1290 y Fr(=)15 b Fq(o)p Fr(\()p Fq(Z)1197 1296 y Fo(1)1217 1290 y Fr(\))5 b(.)23 b(This,)16 b(together)h(with)e(\(4.24\),)-148 1339 y(yields)k Fq(Z)2 1345 y Fo(2)33 1339 y Fr(=)11 b Fq(o)p Fr(\()p Fq(Z)140 1345 y Fo(1)160 1339 y Fr(\))5 b(.)-98 1389 y(No)o(w)11 b(let)g(us)h(consider)g(the)g(sum)j Fq(Z)448 1395 y Fo(3)484 1389 y Fr(o)o(v)o(er)c(the)h(paths)f(that)h (admit)d(nonsimple)h(self-in)o(tersections.)18 b(Using)11 b(the)h(ab)q(o)o(v)o(e)-148 1439 y(notation,)h(w)o(e)h(can)g(write)g (out)g(the)h(follo)o(wing)c(estimate)i(for)19 b Fq(Z)857 1445 y Fo(3)876 1439 y Fr(:)33 1566 y Fq(Z)61 1572 y Fo(3)91 1566 y Ft(\024)178 1526 y Fm(X)135 1617 y Fp(X)r Fo(=)p Fn(f)p Fp(x)p Fo(\()p Fp(t)p Fo(\))p Fn(g)288 1509 y Fo(10)p Fp(s)338 1497 y Fg(2)338 1518 y Fh(n)357 1509 y Fp(=n)311 1526 y Fm(X)303 1615 y Fp(M)s Fo(=1)561 1526 y Fm(X)402 1615 y Fp(n)423 1619 y Fg(2)438 1615 y Fo(+2)p Fp(n)501 1619 y Fg(3)517 1615 y Fo(+)p Fn(\001\001\001)p Fo(+)p Fp(M)s(n)653 1619 y Fh(M)r Fg(+1)719 1615 y Fo(=)p Fp(M)810 1514 y(n)831 1518 y Fg(2)798 1526 y Fm(X)799 1614 y Fp(r)q Fo(=0)980 1526 y Fm(X)877 1616 y Fo(1)p Fn(\024)p Fp(j)934 1620 y Fg(1)949 1616 y Fp(<)p Fn(\001\001\001)o Fp()f Fr(2\),)g(whic)o(h)-148 731 y(results)17 b(in)e(the)h(app)q (earance)g(of)f(the)h(factor)k(\(4)7 b(const)h Ft(\001)f Fq(l)q Fr(\))769 716 y Fp(l)802 731 y Fr(on)15 b(the)h(righ)o(t-hand)f (side)g(in)g(\(4.28\),)g(then)h(w)o(e)f(can)h(reduce)-148 780 y(the)e(estimate)e(for)17 b Fq(W)194 786 y Fp(n)234 780 y Fr(on)c(the)g(righ)o(t-hand)f(side)h(in)f(\(4.26\))f(b)o(y)i(a)f (factor)h(of)k Fq(l)q Fr(!)g(b)q(ecause)d(the)g(\\returns")f(o)q(ccur) 19 b Fq(l)g Fr(times)-148 830 y(along)13 b(the)i(same)e(edge.)18 b(Th)o(us,)50 942 y Fs(E)85 883 y Fm(\022)123 889 y Fo(2)p Fp(s)156 893 y Fh(n)175 889 y Fn(\000)p Fo(1)144 902 y Fm(Y)139 990 y Fp(u)p Fo(=0)225 942 y Fq(\030)243 948 y Fp(i)255 952 y Fh(u)274 948 y Fp(i)286 952 y Fh(u)p Fg(+1)343 883 y Fm(\023)374 942 y Fq(W)413 948 y Fp(n)447 942 y Ft(\024)491 883 y Fm(\022)526 913 y Fr(1)p 526 932 V 526 970 a(4)552 883 y Fm(\023)583 892 y Fp(s)599 896 y Fh(n)639 902 y Fm(Y)628 993 y Fn(f)p Fp(i)r(;)r(j)r Fn(g)699 904 y(0)711 942 y Fr(\(4)7 b(const)g Ft(\001)g Fq(l)q Fr(\()p Ft(f)p Fq(i)s(;)i(j)r Ft(g)p Fr(\)\))1035 924 y Fp(l)p Fo(\()p Fn(f)p Fp(i)r(;)s(j)r Fn(g)p Fo(\))1229 913 y Fr(1)p 1162 932 157 2 v 1162 970 a Fq(l)q Fr(\()p Ft(f)p Fq(i)s(;)g(j)r Ft(g)p Fr(\)!)1343 890 y Fp(M)1334 902 y Fm(Y)1330 991 y Fp(k)q Fo(=2)1390 942 y Fr(\(2)p Fq(k)q Fr(\))1466 924 y Fp(k)q(n)1506 928 y Fh(k)1525 942 y Fr(3)1546 924 y Fp(r)447 1097 y Ft(\024)491 1038 y Fm(\022)526 1069 y Fr(1)p 526 1087 21 2 v 526 1125 a(4)552 1038 y Fm(\023)583 1047 y Fp(s)599 1051 y Fh(n)639 1058 y Fm(Y)628 1149 y Fn(f)p Fp(i)r(;)r(j)r Fn(g)699 1060 y(0)711 1097 y Fr(\(const)822 1103 y Fo(1)840 1097 y Fr(\))856 1080 y Fp(l)p Fo(\()p Fn(f)p Fp(i)r(;)s(j)r Fn(g)p Fo(\))990 1045 y Fp(M)981 1058 y Fm(Y)978 1147 y Fp(k)q Fo(=3)1038 1097 y Fr(\(2)p Fq(k)q Fr(\))1114 1080 y Fp(k)q(n)1154 1084 y Fh(k)1173 1097 y Fr(3)1194 1080 y Fp(r)447 1253 y Ft(\024)491 1194 y Fm(\022)526 1224 y Fr(1)p 526 1243 V 526 1281 a(4)552 1194 y Fm(\023)583 1203 y Fp(s)599 1207 y Fh(n)621 1253 y Fr(\(const)731 1259 y Fo(1)750 1253 y Fr(\))766 1214 y Fb(P)801 1223 y Fh(M)801 1245 y(k)p Fg(=3)861 1235 y Fp(k)q(n)901 1239 y Fh(k)939 1201 y Fp(M)930 1213 y Fm(Y)927 1302 y Fp(k)q Fo(=3)987 1253 y Fr(\(2)p Fq(k)q Fr(\))1063 1235 y Fp(k)q(n)1103 1239 y Fh(k)1122 1253 y Fr(3)1143 1235 y Fp(r)1161 1253 y Fq(:)483 b Fr(\(4.29\))-148 1371 y(This)13 b(is)f(just)h(the)g (estimate)f(from)f(Lemma)f(1.)17 b(In)c(\(4.29\),)e(the)i(pro)q(duct) 1006 1340 y Fm(Q)1046 1351 y Fn(0)1046 1384 y(f)p Fp(i)r(;)r(j)r Fn(g)1141 1371 y Fr(is)f(tak)o(en)h(o)o(v)o(er)f(the)i(edges)k Ft(f)p Fq(i)s(;)9 b(j)r Ft(g)18 b Fr(suc)o(h)-148 1426 y(that)h Fq(l)q Fr(\()p Ft(f)p Fq(i)s(;)10 b(j)r Ft(g)p Fr(\))i Ft(\025)f Fr(4)5 b(.)-98 1476 y(The)17 b(subsequen)o(t)g (estimates)f(for)21 b Fq(Z)488 1482 y Fo(3)527 1476 y Fr(are)c(similar)c(to)j(those)h(for)j Fq(Z)999 1482 y Fo(2)1039 1476 y Fr(and)h Fq(Z)1155 1482 y Fo(1)1179 1476 y Fr(.)j(Namely)m(,)14 b(in)i(\(4.25\))f(w)o(e)h(consider)-148 1525 y(the)f(subsum)232 1557 y Fm(X)129 1646 y Fo(1)p Fn(\024)p Fp(j)186 1650 y Fg(1)201 1646 y Fp(<)p Fn(\001\001\001)p Fp()f Fr(2)p Fi(,)1676 783 y Fr(\(5.8\))-148 924 y Fi(wher)n(e)j(the)g(pr)n(o)n(duct)190 893 y Fm(Q)230 903 y Fn(\003)269 924 y Fi(in)j Fr(\(5.8\))c Fi(is)g(taken)i(over)f (the)g(p)n(aths)g(forming)f(a)h(cluster.)-98 993 y Fr(The)h(pro)q(of)f (of)g(Lemma)e(2)i(is)g(giv)o(en)g(in)g([14].)21 b(Note)16 b(that)g(the)g(case)21 b Fq(l)16 b Fr(=)e(2)20 b(corresp)q(onds)e(to)d (Prop)q(osition)g(3)g(of)g(the)-148 1042 y(presen)o(t)e(pap)q(er.)18 b(The)11 b(case)18 b Fq(l)12 b(>)g Fr(2)k(can)11 b(b)q(e)h(treated)g (in)f(a)g(similar)d(w)o(a)o(y)m(.)16 b(T)m(o)11 b(eac)o(h)g(cluster)i (of)i Fq(l)i Fr(paths)12 b(w)o(e)f(assign)g(an)g(ev)o(en)-148 1091 y(path)i(of)f(length)18 b Fq(l)137 1097 y Fp(p)154 1101 y Fh(n)183 1091 y Ft(\000)7 b Fq(q)f Fr(,)13 b(where)18 b Fq(l)13 b Ft(\024)f Fq(q)h Ft(\024)e Fr(2)p Fq(l)6 b Fr(.)18 b(The)13 b(n)o(um)o(b)q(er)f(of)g(preimages)g(under)h(this)g (corresp)q(ondence)j(is)c(b)q(ounded)-148 1143 y(ab)q(o)o(v)o(e)i(b)o (y)k Fq(K)70 1128 y Fp(l)p Fn(\000)p Fo(1)67 1153 y Fp(n)130 1143 y Fr(,)13 b(where)20 b Fq(K)315 1149 y Fp(n)356 1143 y Fr(is)14 b(the)g(n)o(um)o(b)q(er)e(of)h(instan)o(ts)19 b Fq(t)842 1149 y Fp(i)861 1143 y Fr(,)f Fq(i)12 b Fr(=)f(1)s Fq(;)e(:)e(:)g(:)h(;)h(K)1119 1149 y Fp(n)1147 1143 y Fr(,)k(suc)o(h)i(that)j Fq(t)1375 1149 y Fp(i)1400 1143 y Ft(\024)12 b Fr(\()p Fq(l)e Ft(\000)f Fr(1\))t Fq(p)1585 1149 y Fp(n)1616 1143 y Ft(\000)f Fq(q)20 b Fr(and)-148 1192 y Fq(x)p Fr(\()p Fq(t)p Fr(\))12 b Ft(\025)g Fq(x)p Fr(\()p Fq(t)34 1198 y Fp(i)47 1192 y Fr(\))19 b(for)g Fq(t)166 1198 y Fp(i)191 1192 y Ft(\024)12 b Fq(t)g Ft(\024)f Fq(t)320 1198 y Fp(i)343 1192 y Fr(+)f Fq(p)406 1198 y Fp(n)438 1192 y Ft(\000)f Fr(1)c(.)18 b(The)c(corresp)q(onding)h (analog)d(of)i(\(5.2\))f(is)h(giv)o(en)f(b)o(y)h(the)g(inequalit)o(y) 562 1269 y Fs(E)592 1275 y Fp(X)624 1269 y Fq(K)662 1252 y Fp(l)p Fn(\000)p Fo(1)659 1279 y Fp(n)729 1269 y Ft(\024)d Fr(const)867 1275 y Fp(l)887 1269 y Ft(\001)c Fq(p)927 1252 y Fo(\()p Fp(l)p Fn(\000)p Fo(1\))p Fp(=)p Fo(2)927 1279 y Fp(n)1041 1269 y Fq(:)705 1357 y Fs(App)q(endix)-98 1432 y Fr(In)13 b(this)h(App)q(endix,)f(w)o(e)h(pro)o(v)o(e)f(that)g (the)h(sum)e(o)o(v)o(er)i(paths)f(in)g(whic)o(h)g(some)g(edge)h(is)f (passed)h(at)f(least)h(four)e(times)h(is)-148 1481 y(small)f(compared)h (with)h(the)g(total)f(sum)g(\(4.1\).)-98 1530 y(Let)21 b Fc(P)14 b Fr(=)f Ft(f)p Fq(i)104 1536 y Fo(0)136 1530 y Ft(!)g Fq(i)205 1536 y Fo(1)238 1530 y Ft(!)g(\001)7 b(\001)g(\001)12 b(!)h Fq(i)424 1536 y Fo(2)p Fp(s)457 1540 y Fh(n)492 1530 y Fr(=)h Fq(i)552 1536 y Fo(0)571 1530 y Ft(g)5 b Fr(,)14 b(and)h(let)20 b Fq(\027)792 1536 y Fp(n)814 1530 y Fr(\()p Fc(P)q Fr(\))g(b)q(e)c(the)f(maxim)o(um) 10 b(n)o(um)o(b)q(er)15 b(of)f(v)o(ertices)i(in)o(to)f(whic)o(h)-148 1579 y(one)f(can)g(get)g(at)f(mark)o(ed)f(instan)o(ts)i(from)e(a)h(giv) o(en)g(v)o(ertex.)19 b(By)14 b(de\014nition,)f(eac)o(h)h(v)o(ertex)19 b Fq(i)g Fr(of)13 b(the)h(path)k Fc(P)h Fr(is)14 b(the)g(left)-148 1629 y(end)g(of)e(at)h(most)j Fq(\027)152 1635 y Fp(n)174 1629 y Fr(\()p Fc(P)q Fr(\))i(mark)o(ed)11 b(edges.)19 b(In)13 b(what)g(follo)o(ws,)e(w)o(e)i(sho)o(w)f(that,)h(for)f(t)o (ypical)g(paths,)18 b Fq(\027)1435 1635 y Fp(n)1457 1629 y Fr(\()p Fc(P)q Fr(\))g(gro)o(ws)13 b(slo)o(w)o(er)-148 1678 y(than)h(an)o(y)g(p)q(ositiv)o(e)f(p)q(o)o(w)o(er)h(of)19 b Fq(s)374 1684 y Fp(n)402 1678 y Fr(.)-98 1747 y Fs(Lemma)h(3.)k Fi(F)m(or)18 b(e)n(ach)23 b Fq(\015)8 b Fi(,)24 b Fr(0)18 b Fq(<)g(\015)i(<)e Fr(1)5 b Fi(,)19 b(the)f(subsum)h(in)j Fr(\(4.1\))17 b Fi(over)i(the)f(p)n(aths)23 b Fc(P)h Fi(such)19 b(that)k Fq(\027)1552 1753 y Fp(n)1574 1747 y Fr(\()p Fc(P)q Fr(\))18 b Fq(>)g(s)1721 1732 y Fp(\015)1721 1757 y(n)1749 1747 y Fi(,)-148 1796 y Fr(0)12 b Fq(<)f(\015)k(<)d Fr(1)5 b Fi(,)14 b(is)19 b Fq(o)p Fr(\(1\))h Fi(and)c(henc)n(e)g(is)e (ne)n(gligibly)g(smal)r(l)h(c)n(omp)n(ar)n(e)n(d)g(with)f(the)h(total)f (sum.)-98 1865 y Fr(T)m(o)i(understand)i(wh)o(y)j Fq(\027)302 1871 y Fp(n)324 1865 y Fr(\()p Fc(P)q Fr(\))g(cannot)c(b)q(e)g(large)f (for)g(t)o(ypical)g(paths,)h(let)g(us)g(consider)g(a)f(path)h(with)f (simple)f(self-)-148 1914 y(in)o(tersections)h(and)f(without)f (nonclosed)h(v)o(ertices.)21 b(In)15 b(this)f(case,)i(if)d(a)i(v)o (ertex)g(is)g(the)g(left)f(end)h(of)k Fq(\027)1463 1920 y Fp(n)1505 1914 y Fr(mark)o(ed)13 b(edges,)-148 1964 y(then,)i(for)f(the)h(corresp)q(onding)g(random)e(w)o(alk)18 b Fq(x)p Fr(\()p Fq(t)p Fr(\))5 b(,)19 b(0)12 b Ft(\024)h Fq(t)f Ft(\024)h Fr(2)p Fq(s)914 1970 y Fp(n)942 1964 y Fr(,)h(on)g(the)h(p)q(ositiv)o(e)f(half-line,)e(there)k(exists)f(a)f (time)-148 2013 y(in)o(terv)n(al)21 b([)p Fq(t)37 2019 y Fo(1)57 2013 y Fq(;)10 b(t)94 2019 y Fo(2)112 2013 y Fr(])21 b(on)16 b(whic)o(h)g(the)g(tra)r(jectory)23 b Fq(x)p Fr(\()p Fq(t)p Fr(\))e(falls)e Fq(\027)807 2019 y Fp(n)830 2013 y Fq(=)p Fr(2)h(times)15 b(in)o(to)h(the)g(lev)o(el)21 b Fq(x)p Fr(\()p Fq(t)1322 2019 y Fo(1)1341 2013 y Fr(\))g(but)16 b(nev)o(er)h(falls)e(b)q(elo)o(w)-148 2062 y(this)e(lev)o(el.)k (Indeed,)c(to)f(mak)o(e)e(eac)o(h)j(new)g(step)g(from)d(the)j(v)o (ertex)18 b Fq(i)913 2068 y Fp(t)926 2072 y Fg(1)949 2062 y Fr(,)12 b(w)o(e)g(m)o(ust)g(\014rst)g(return)i(to)e(this)g(v)o (ertex)h(along)e(the)-148 2111 y(tra)r(jectory)m(,)j(and)f(the)h(co)q (e\016cien)o(t)20 b(1)p Fq(=)p Fr(2)d(of)h Fq(\027)557 2117 y Fp(n)598 2111 y Fr(resp)q(onds)c(to)g(the)g(fact)f(that)19 b Fq(i)1081 2117 y Fp(t)1094 2121 y Fg(1)1130 2111 y Fr(can)14 b(b)q(e)g(a)f(v)o(ertex)h(of)f(self-in)o(tersection.)-148 2161 y(Ob)o(viously)m(,)f(the)i(probabilit)o(y)e(of)h(suc)o(h)i(tra)r (jectories)g(of)e(the)h(random)e(w)o(alk)h(deca)o(ys)h(exp)q(onen)o (tially)f(as)18 b Fq(\027)1527 2167 y Fp(n)1561 2161 y 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2467 y Fr(\(8)7 b(const)q(\))1075 2450 y Fo(2)p Fp(\021)1109 2454 y Fh(n)1131 2467 y Fq(:)1712 2603 y Fo(129)p eop %%Page: 130 17 130 16 bop -148 -71 a Fr(F)m(or)19 b(a)h(giv)o(en)k Fq(\027)113 -65 y Fp(n)140 -71 y Fr(,)c(w)o(e)g(can)f(readily)g(write)h(out)f(an)h (upp)q(er)g(b)q(ound)g(b)q(oth)f(for)g(the)h(n)o(um)o(b)q(er)f(of)g (edges)h(passed)h(four)-148 -21 y(times)15 b(and)g(for)g(the)h(quan)o (tit)o(y)j(\(8)7 b(const)q(\))512 -36 y Fo(2)p Fp(\021)546 -32 y Fh(n)573 -21 y Fr(.)23 b(W)m(e)15 b(still)f(denote)i(the)g(total) f(n)o(um)o(b)q(er)g(of)f(instan)o(ts)i(of)f(self-in)o(tersection)-148 29 y(b)o(y)22 b Fq(M)5 b Fr(.)25 b(As)17 b(w)o(as)g(sho)o(wn)f(in)g Ft(x)q Fr(4,)g(it)h(su\016ces)g(to)g(consider)g(the)g(case)23 b Fq(M)e Ft(\024)16 b Fr(10)p Fq(s)1140 14 y Fo(2)1140 39 y Fp(n)1162 29 y Fq(=n)5 b Fr(.)26 b(F)m(or)16 b(eac)o(h)h(of)f(the) 22 b Fq(M)k Fr(instan)o(ts)-148 79 y(of)17 b(self-in)o(tersection,)i 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y(tra)r(jectory)d(falls)d(at)h(least)16 b Fq(\027)296 792 y Fp(n)318 786 y Fq(=)p Fr(\()p Fq(N)388 792 y Fp(n)415 786 y Fr(+)t(1\))h(times)10 b(in)o(to)h(the)g(lev)o(el)16 b Fq(x)p Fr(\()p Fq(t)915 792 y Fo(1)934 786 y Fr(\))g(but)c(nev)o(er)g (falls)e(b)q(elo)o(w)h(it.)17 b(The)11 b(fraction)g(of)g(suc)o(h)-148 838 y(tra)r(jectories)21 b Fq(x)p Fr(\()7 b Fa(\001)g Fr(\))19 b(do)q(es)14 b(not)g(exceed)21 b(\(2)p Fq(s)539 844 y Fp(n)562 838 y Fr(\))578 822 y Fo(2)596 838 y Fq(e)615 822 y Fn(\000)6 b Fo(const)723 826 y Fg(7)740 822 y Fn(\001)p Fp(\027)767 826 y Fh(n)787 822 y Fp(=)p Fo(\()p Fp(N)843 826 y Fh(n)863 822 y Fo(+1\))939 838 y Fr(\(cf.)18 b(\(A1\)\),)c(and)g (b)o(y)f(carrying)h(out)g(computations)-148 887 y(similar)e(to)i(those) h(used)g(in)f Ft(x)q Fr(4)g(for)19 b Fq(Z)444 893 y Fo(1)468 887 y Fr(,)f Fq(Z)526 893 y Fo(2)550 887 y Fr(,)c(and)19 b Fq(Z)690 893 y Fo(3)714 887 y Fr(,)14 b(w)o(e)h(\014nd)f(that)g(the)h (subsum)f(o)o(v)o(er)g(the)h(paths)g(with)k Fq(\027)1594 893 y Fp(n)1628 887 y Fq(>)13 b(s)1692 872 y Fp(\015)1692 898 y(n)1734 887 y Fr(is)-148 937 y(b)q(ounded)i(ab)q(o)o(v)o(e)e(b)o (y)152 1048 y Fq(Z)180 1054 y Fo(1)208 1048 y Ft(\001)229 990 y Fm(\022)298 996 y Fp(s)314 1000 y Fh(n)286 1009 y Fm(X)267 1101 y Fp(\027)284 1105 y Fh(n)304 1101 y Fo(=)p Fp(s)345 1087 y Fh(\015)345 1106 y(n)398 996 y Fp(s)414 1000 y Fh(n)386 1009 y Fm(X)372 1098 y Fp(N)398 1102 y Fh(n)419 1098 y Fo(=0)468 990 y Fm(\022)503 1020 y Fr(const)598 1026 y Fo(5)624 1020 y Ft(\001)7 b Fq(s)662 1026 y Fp(n)p 503 1039 181 2 v 555 1078 a Fq(n)580 1066 y Fo(2)p Fp(=)p Fo(3)689 990 y Fm(\023)720 998 y Fp(N)746 1002 y Fh(n)768 1048 y Fq(e)787 1031 y Fn(\000)f Fo(const)895 1035 y Fg(7)912 1031 y Fn(\001)p Fp(\027)939 1035 y Fh(n)959 1031 y Fp(=)p Fo(\()p Fp(N)1015 1035 y Fh(n)1035 1031 y Fo(+1\))1092 1048 y Fr(\(8)h(const)q(\))1263 1014 y Fh(\027)1278 1018 y(n)p 1252 1021 58 2 v 1252 1024 a Fe(p)p 1275 1024 35 2 v 13 x Fh(s)1289 1041 y(n)1325 1010 y Fg(20)p Fh(s)1367 998 y Fg(3)p Fh(=)p Fg(2)1367 1018 y Fh(n)p 1325 1021 88 2 v 1360 1037 a(n)1420 990 y Fm(\023)1450 1048 y Fq(:)216 b Fr(\(A2\))-148 1174 y(Elemen)o(tary)12 b(computations)f(sho)o(w)h(that)g(the)h(second)g (factor)f(in)g(\(A2\))g(is)17 b Fq(o)p Fr(\(1\))5 b(.)18 b(The)12 b(pro)q(of)g(of)f(Lemma)f(3)i(is)g(complete.)-98 1224 y(The)17 b(sum)e(o)o(v)o(er)h(the)g(paths)h(with)e(edges)i(of)f(m) o(ultiplici)o(t)o(y)i Ft(\025)9 b Fr(4)20 b(and)c(with)21 b Fq(\027)1136 1230 y Fp(n)1173 1224 y Fq(<)15 b(s)1239 1209 y Fp(\015)1239 1234 y(n)1283 1224 y Fr(w)o(as)h(studied)h(in)f Ft(x)p Fr(4)g(\(see)h(the)-148 1274 y(estimates)d(for)19 b Fq(Z)135 1259 y Fn(0)132 1284 y Fo(1)169 1274 y Fr(and)g Fq(Z)286 1259 y Fn(0)283 1284 y Fo(2)302 1274 y Fr(\),)13 b(where)j(it)d(w)o(as)h(sho)o(wn)g(that)g(this)g(sum)e(is)i(also)k Fq(o)p Fr(\()p Fq(Z)1163 1280 y Fo(1)1183 1274 y Fr(\))5 b(.)695 1365 y Fs(References)-130 1457 y Fr(1.)20 b(E.)c(Wigner,)g 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