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y Fr(\003)935 1449 y Fy(U)972 1455 y Fr(\003)1008 1449 y Fv(=)1078 1409 y Fo(X)1052 1500 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1172 1449 y Fp(J)1195 1455 y Fs(\013)1218 1449 y Fy(f)1233 1455 y Fs(\013)228 1583 y Fv(whic)o(h)i(is)f(a)h(diagonal)e(op)q(erator)i(on)g Fq(H)849 1589 y Fr(\003)p 530 1645 V 530 1679 a Fy(H)568 1685 y Fr(\003)600 1679 y Fq(j)o Fp(\015)r Fq(i)651 1691 y Fr(\003)717 1679 y Fv(=)818 1640 y Fo(X)791 1730 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)911 1679 y Fp(J)934 1685 y Fs(\013)965 1679 y Fv(\()p Fq(\000)p Fv(1\))1050 1658 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1142 1679 y Fq(j)p Fp(\015)r Fq(i)1193 1691 y Fr(\003)1230 1679 y Fv(=)e Fp(H)1309 1685 y Fs(\015)1336 1679 y Fq(j)p Fp(\015)r Fq(i)1388 1691 y Fr(\003)620 1800 y Fp(H)655 1806 y Fs(\015)717 1800 y Fv(=)818 1761 y Fo(X)791 1852 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)911 1800 y Fp(J)934 1806 y Fs(\013)965 1800 y Fv(\()p Fq(\000)p Fv(1\))1050 1779 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1184 1800 y Fq(8)p Fp(\015)i Fq(\022)e Fv(\003)-1087 b(\(2\))p 614 1888 V 614 1922 a Fy(H)651 1928 y Fr(\003)717 1922 y Fv(=)818 1882 y Fo(X)791 1973 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)911 1922 y Fp(H)946 1928 y Fs(\013)976 1922 y Fq(j)p Fp(\015)r Fq(i)8 b(h)p Fp(\015)r Fq(j)1086 1934 y Fr(\003)228 2056 y Fv(W)m(e)13 b(are)i(no)o(w)e(able)h(to)f (compute)h(the)g(partition)f(function)606 2152 y Fp(Z)637 2130 y Fr(\()p Fs(d)p Fr(\))634 2164 y(\003)690 2152 y Fv(\()p Fp(\014)r Fv(\))52 b(:)g(=)12 b Fp(tr)941 2158 y Fh(H)969 2162 y Fg(\003)999 2118 y Fo(\000)1018 2152 y Fp(e)1037 2135 y Fs(\014)q Fe(H)1086 2139 y Fg(\003)1110 2118 y Fo(\001)1140 2152 y Fv(=)1211 2112 y Fo(X)1184 2203 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1304 2152 y Fp(e)1323 2135 y Fh(\000)p Fs(\014)q(H)1396 2139 y Fd(\013)530 2273 y Fp(Z)561 2251 y Fr(\()p Fs(d)p Fr(\))558 2285 y(\003)613 2273 y Fv(\()q Fp(\014)i Fv(=)e(0\))41 b(=)h(2)884 2256 y Fh(j)o Fr(\003)p Fh(j)228 2363 y Fv(The)14 b(densit)o(y)g(op)q(erator)h(de\014ning)f (the)g(Gibbs)g(state)g(is)594 2469 y(\026)591 2470 y Fm(\032)617 2449 y Fr(\()p Fs(d)p Fr(\))617 2482 y(\003)669 2470 y Fv(\()p Fp(\014)r Fv(\))53 b(:)e(=)951 2442 y(1)p 891 2461 142 2 v 891 2509 a Fp(Z)922 2487 y Fr(\()p Fs(d)p Fr(\))919 2521 y(\003)975 2509 y Fv(\()p Fp(\014)r Fv(\))1071 2431 y Fo(X)1044 2522 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1165 2470 y Fp(e)1184 2453 y Fh(\000)p Fs(\014)q(H)1257 2457 y Fd(\013)1287 2470 y Fq(j)p Fp(\013)p Fq(i)7 b(h)p Fp(\013)p Fq(j)1403 2483 y Fr(\003)518 2590 y Fv(\026)515 2591 y Fm(\032)541 2570 y Fr(\()p Fs(d)p Fr(\))541 2603 y(\003)593 2591 y Fv(\()p Fp(\014)14 b Fv(=)e(0\))41 b(=)h Fy(I)860 2597 y Fr(\003)p eop %%Page: 4 4 bop 228 142 a Fx(4)588 b(M.)15 b(GIANFELICE)228 250 y Fv(and)f(the)g(Gibbs)f (a)o(v)o(erage)h(of)g(an)o(y)f(cylinder)h(observ)n(able)g Fm(\033)1165 256 y Fs(\015)1187 250 y Fv(;)f Fp(\015)h Fq(\022)e Fv(\003)i(is)228 387 y Fp(!)255 367 y Fs(\014)254 399 y Fr(\003)285 387 y Fv(\()p Fm(\033)332 393 y Fs(\015)353 387 y Fv(\))52 b(:)f(=)12 b Fp(!)555 369 y Fr(0)554 397 y(\003)586 340 y Fo(\020)610 387 y Fm(\032)636 365 y Fr(\()p Fs(d)p Fr(\))636 399 y(\003)688 387 y Fv(\()p Fp(\014)r Fv(\))c Fm(\033)783 393 y Fs(\015)804 340 y Fo(\021)841 387 y Fv(=)j Fp(tr)918 393 y Fh(H)946 397 y Fg(\003)976 316 y Fo( )1039 358 y Fp(e)1058 343 y Fs(\014)q Fe(H)1107 347 y Fg(\003)p 1014 377 142 2 v 1014 425 a Fp(Z)1045 404 y Fr(\()q Fs(d)p Fr(\))1042 438 y(\003)1098 425 y Fv(\()p Fp(\014)r Fv(\))1161 387 y Fm(\033)1191 393 y Fs(\015)1212 316 y Fo(!)1256 387 y Fv(=)410 532 y(=)489 467 y Fo(P)533 510 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)655 498 y Fp(e)674 483 y Fh(\000)p Fs(\014)q(H)747 487 y Fd(\013)778 498 y Fv(\()p Fq(\000)p Fv(1\))863 477 y Fh(j)p Fs(\015)r Fh(\\)p Fs(\013)p Fh(j)p 489 522 460 2 v 578 529 a Fo(P)622 573 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)744 560 y Fp(e)763 548 y Fh(\000)p Fs(\014)q(H)836 552 y Fd(\013)965 532 y Fv(=)h(1)d Fq(\000)g Fv(2)1106 467 y Fo(P)1150 510 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1272 498 y Fp(e)1291 483 y Fh(\000)p Fs(\014)q(H)1364 487 y Fd(\013)1388 498 y Fp(\016)g Fv(\()p Fq(j)o Fp(\015)j Fq(\\)d Fp(\013)p Fq(j)i Fv(=)h(2)p Fp(n)d Fv(+)g(1;)e Fp(n)k Fq(2)g Ft(N)p Fv(\))p 1106 522 758 2 v 1344 529 a Fo(P)1387 573 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1510 560 y Fp(e)1529 548 y Fh(\000)p Fs(\014)q(H)1602 552 y Fd(\013)228 661 y Fv(where)17 b Fp(\016)9 b Fv(\()p Fq(\003)p Fv(\))16 b(denotes)h(the)f(c)o(haracteristic)i(function)d(of)h(its)g(argumen) o(t.)23 b(Note)16 b(that)g(b)o(y)g(the)228 711 y(de\014nition)d(of)h Fq(C)482 717 y Fr(\003)520 711 y Fv(an)o(y)f(observ)n(able)i(is)e(just)h(a)g (linear)f(com)o(bination)f(of)h(cylinder)h(op)q(erators.)278 765 y(So)j(far)h(w)o(e)g(rewrote)h(kno)o(wn)f(quan)o(tities)f(making)f(use)i (of)g(this)g(formalism)c(to)j(sho)o(w)h(its)228 814 y(usefulness,)g(w)o(e)f (will)e(no)o(w)h(compute)g(explicitly)g(the)h(eigen)o(v)n(alue)f(of)g(the)h (Hamiltonian)d(op-)228 864 y(erators)j(of)f(some)f(w)o(ell)h(kno)o(wn)g(mo)q (del)f(of)g(in)o(teracting)i(spin)f(systems,)h(whic)o(h)f(app)q(ear)h(also) 228 914 y(in)d([QMIPS)g(I)q(I)q(].)21 b(F)m(or)14 b(sak)o(e)i(of)e(simplicit) o(y)m(,)e(in)j(the)h(follo)o(wing,)c(all)i(the)h(calculations)g(will)f(b)q(e) 228 964 y(carried)g(out)g(assuming)f(p)q(erio)q(dic)h(b)q(oundary)g (conditions)f(unless)i(di\013eren)o(tly)f(sp)q(eci\014ed.)228 1077 y Fy(Example)h(2.1.)21 b(Ising)14 b(mo)q(del)h(without)f(external)g (source.)695 1185 y(H)732 1191 y Fr(\003)769 1185 y Fv(=)d Fq(\000)878 1145 y Fo(X)851 1236 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)972 1185 y Fp(\016)e Fv(\()p Fp(d)e Fv(\()o Fp(\013)p Fv(\))12 b(=)g(1\))6 b Fm(\033)1231 1191 y Fs(\013)228 1185 y Fv(\(3\))228 1329 y Fn(wher)n(e)14 b Fp(d)7 b Fv(\()p Fp(\013)p Fv(\))14 b Fn(indic)n(ates)h(the)g(diameter)g(of)g(any)g(lattic)n(e)f(subset) h Fp(\013)p Fn(.)k(F)m(r)n(om)14 b(\()7 b Fv(2)p Fn(\))14 b(we)h(have)372 1435 y Fp(H)407 1441 y Fs(\015)469 1435 y Fv(=)42 b Fq(\000)609 1396 y Fo(X)582 1487 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)702 1435 y Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))7 b(\()p Fq(\000)p Fv(1\))1017 1414 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1114 1435 y Fv(=)-918 b(\(4\))469 1556 y(=)42 b Fq(\000)609 1517 y Fo(X)582 1608 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)702 1556 y Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))7 b([1)i Fq(\000)g Fv(2)p Fp(\016)g Fv(\()p Fq(j)o Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)j Fv(=)g(2)p Fp(n)d Fv(+)g(1;)e Fp(n)k Fq(2)g Ft(N)p Fv(\))n(])g(=)469 1665 y(=)42 b Fq(\000)p Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)i Fv(+)g(2)e Fq(j)p Fp(@)r(\015)r Fq(j)92 b(8)p Fp(\015)14 b Fq(\022)e Fv(\003)228 1763 y Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)12 b Fn(indic)n(ates)g(the)h(numb)n(er) g(of)f(dimers)h(\(i.e.)k(c)n(onne)n(cte)n(d)d(sets)e(made)h(by)g(two)f (lattic)n(e)g(p)n(oints\))228 1813 y(in)17 b Fv(\003)p Fn(,)g(and)g Fq(j)p Fp(@)r(\015)r Fq(j)g Fn(the)g(numb)n(er)g(of)g(ways)g(a)g(dimer)f(c)n (an)h(interse)n(ct,)g(with)f(a)h(single)g(site)f(only,)228 1862 y(sites)e(b)n(elonging)h(to)g Fp(\015)r Fn(.)k(These)c(sites)f(c)n (onstitutes)h(the)f(inner)h(b)n(oundary)g(of)g Fp(\015)r Fn(,)g(so)g Fq(j)p Fp(@)r(\015)r Fq(j)g Fn(is)f(the)228 1912 y(c)n(ar)n(dinality)h(of)g (the)g(hyp)n(ersurfac)n(e)h(b)n(ounding)g Fp(\015)r Fn(.)22 b(F)m(or)15 b(example,)g(if)g Fp(d)e Fv(=)f(2)p Fn(,)j(then)h Fq(j)p Fp(@)r(\015)r Fq(j)g Fn(is)f(the)228 1962 y(length)g(of)g(the)h(c)n (ontour)f(surr)n(ounding)h Fp(\015)r Fn(.)21 b(F)m(urther)14 b(r)n(esults)g(on)i(this)f(mo)n(del)g(wil)r(l)f(b)n(e)i(given)f(in)228 2012 y(the)g(next)g(se)n(ction.)228 2125 y Fy(Example)g(2.2.)21 b(Dobrushin)o(-Ger)o(tsi)o(k)13 b(mo)q(del)i Fn(\()7 b Fv([Ge)o(])p Fn(\).)18 b(The)d(lattic)n(e)f(is)h Ft(Z)1497 2110 y Fr(2)1513 2125 y Fn(.)316 2231 y Fy(H)353 2214 y Fs(D)q(G)353 2241 y Fr(\003)417 2231 y Fv(\()p Fp(h;)7 b(J)499 2237 y Fr(1)517 2231 y Fp(;)g(J)559 2237 y Fr(2)577 2231 y Fv(\))k(=)h Fq(\000)714 2191 y Fo(X)687 2282 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)808 2231 y Fv([)o Fp(J)842 2237 y Fr(1)861 2231 y Fp(\016)d Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))d(+)g Fp(J)1157 2237 y Fs(\013)1190 2231 y Fv(+)h Fp(h\016)f Fv(\()p Fp(d)e Fv(\()o Fp(\013)p Fv(\))12 b(=)g(0\)])6 b Fm(\033)1527 2237 y Fs(\013)357 2356 y Fp(J)380 2362 y Fs(\013)415 2356 y Fv(=)12 b Fp(J)482 2362 y Fr(2)500 2356 y Fp(\016)527 2310 y Fo(\020)552 2356 y Fp(\013)f Fv(=)h Fp(\013)661 2362 y Fr(1)689 2356 y Fq([)c Fp(\013)752 2362 y Fr(2)782 2356 y Fv(:)j Fp(d)c Fv(\()p Fp(\013)877 2362 y Fr(1)895 2356 y Fv(\))12 b(=)g Fp(d)7 b Fv(\()o Fp(\013)1038 2362 y Fr(2)1056 2356 y Fv(\))12 b(=)g(0)i(;)22 b Fp(dist)7 b Fv(\()p Fp(\013)1317 2362 y Fr(1)1335 2356 y Fp(;)g(\013)1381 2362 y Fr(2)1399 2356 y Fv(\))12 b(=)1471 2320 y Fq(p)p 1505 2320 21 2 v 1505 2356 a Fv(2)1526 2310 y Fo(\021)228 2467 y Fn(wher)n(e)i Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(0\))j Fn(me)n(ans)g Fp(\013)g Fn(to)f(b)n(e)h(a)g(single)g (lattic)n(e)f(site)h(and)g Fp(h;)7 b(J)1352 2473 y Fr(1)1370 2467 y Fp(;)g(J)1412 2473 y Fr(2)1442 2467 y Fq(2)k Ft(R)m Fn(.)16 b(Sinc)n(e)363 2533 y Fo(\014)363 2558 y(\014)363 2583 y(\014)377 2535 y(n)405 2581 y Fp(\013)11 b Fv(:)g Fp(\013)g Fv(=)h Fp(\013)575 2587 y Fr(1)603 2581 y Fq([)d Fp(\013)667 2587 y Fr(2)696 2581 y Fv(:)i Fp(d)c Fv(\()p Fp(\013)791 2587 y Fr(1)809 2581 y Fv(\))12 b(=)g Fp(d)7 b Fv(\()o Fp(\013)952 2587 y Fr(2)970 2581 y Fv(\))12 b(=)g(0;)7 b Fp(dist)g Fv(\()p Fp(\013)1202 2587 y Fr(1)1220 2581 y Fp(;)g(\013)1266 2587 y Fr(2)1283 2581 y Fv(\))12 b(=)1355 2544 y Fq(p)p 1389 2544 V 1389 2581 a Fv(2)1410 2535 y Fo(o)1438 2533 y(\014)1438 2558 y(\014)1438 2583 y(\014)1463 2581 y Fv(=)g(2)7 b Fq(j)o Fv(\003)p Fq(j)p eop %%Page: 5 5 bop 657 142 a Fx(QUANTUM)18 b(METHODS)f(F)o(OR)f(IPS'S)e(I)q(I)q(I)411 b(5)228 250 y Fq(8)p Fp(\015)14 b Fq(\022)e Fv(\003)j Fn(we)f(have)229 335 y Fp(H)267 317 y Fs(D)q(G)264 345 y(\015)329 335 y Fv(\()q Fp(h;)7 b(J)412 341 y Fr(1)430 335 y Fp(;)g(J)472 341 y Fr(2)490 335 y Fv(\))k(=)h(\()p Fq(\000)p Fv(2)7 b Fq(j)p Fv(\003)p Fq(j)h Fv(+)i(2)d Fq(j)o Fp(@)r(\015)r Fq(j)q Fv(\))g Fp(J)885 341 y Fr(1)913 335 y Fv(+)i(\()p Fq(\000)p Fv(2)e Fq(j)p Fv(\003)p Fq(j)h Fv(+)i(2)d Fq(j)o(6)12 b Fp(@)r(\015)r Fq(j)p Fv(\))7 b Fp(J)1289 341 y Fr(2)1317 335 y Fv(+)j(\()p Fq(\000)d(j)p Fv(\003)p Fq(j)h Fv(+)i(2)d Fq(j)o Fp(\015)r Fq(j)p Fv(\))g Fp(h)279 415 y Fq(j6)k Fp(@)r(\015)r Fq(j)h Fv(=)444 375 y Fo(X)417 466 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)537 415 y Fp(\016)564 369 y Fo(\020)589 415 y Fp(\013)f Fv(=)h Fp(\013)698 421 y Fr(1)725 415 y Fq([)d Fp(\013)789 421 y Fr(2)819 415 y Fv(:)i Fp(d)c Fv(\()p Fp(\013)914 421 y Fr(1)932 415 y Fv(\))12 b(=)f Fp(d)c Fv(\()p Fp(\013)1075 421 y Fr(2)1093 415 y Fv(\))12 b(=)g(0)i(;)22 b Fp(dist)7 b Fv(\()p Fp(\013)1354 421 y Fr(1)1372 415 y Fp(;)g(\013)1418 421 y Fr(2)1436 415 y Fv(\))12 b(=)1507 378 y Fq(p)p 1542 378 21 2 v 37 x Fv(2)1563 369 y Fo(\021)1597 415 y Fq(\002)1351 524 y(\002)p Fp(\016)d Fv(\()p Fq(j)p Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)j Fv(=)f(1\))228 610 y Fy(Example)k(2.3.)21 b(Ising)d(mo)q(del)h(with)f(second)h(neigh)o(b)q(ors)e(an)o(tiferro)o(magnet) o(ic)g(in-)228 660 y(teraction)c(in)i(an)h(external)e(source)p Fn(.)k(The)d(lattic)n(e)f(is)g Ft(Z)1185 645 y Fs(d)303 744 y Fy(H)340 750 y Fr(\003)372 744 y Fv(\()p Fp(h;)7 b(J)t Fv(\))41 b(=)h Fq(\000)655 705 y Fo(X)628 796 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)748 744 y Fv([)p Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))d(+)h Fp(J)1057 750 y Fs(\013)1090 744 y Fv(+)f Fp(h\016)g Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(0\)])6 b Fm(\033)1426 750 y Fs(\013)427 853 y Fp(J)450 859 y Fs(\013)515 853 y Fv(=)42 b Fq(\000)p Fp(J)t(\016)9 b Fv(\()q Fp(\013)i Fv(=)h Fp(\013)801 859 y Fr(1)828 853 y Fq([)d Fp(\013)892 859 y Fr(2)922 853 y Fv(:)i Fp(d)c Fv(\()o Fp(\013)1016 859 y Fr(1)1035 853 y Fv(\))k(=)h Fp(d)7 b Fv(\()p Fp(\013)1178 859 y Fr(2)1196 853 y Fv(\))12 b(=)f(0)k(;)21 b Fp(dist)7 b Fv(\()q Fp(\013)1457 859 y Fr(1)1475 853 y Fp(;)g(\013)1521 859 y Fr(2)1538 853 y Fv(\))12 b(=)g(2\))228 938 y Fp(h;)7 b(J)15 b(>)d Fv(0)p Fn(.)18 b(The)d(asso)n(ciate)n(d)g(eigenvalues)h(ar)n(e)e Fq(8)p Fp(\015)h Fq(\022)c Fv(\003)228 1022 y Fp(H)263 1028 y Fs(\015)291 1022 y Fv(\()p Fp(h;)c(J)t Fv(\))41 b(=)g(\()q Fq(\000)p Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)i Fv(+)g(2)e Fq(j)o Fp(@)r(\015)r Fq(j)q Fv(\))g(\(1)i Fq(\000)h Fp(J)t Fv(\))f Fq(\000)h Fp(h)d Fq(j)o Fv(\003)p Fq(j)h Fv(+)i(2)p Fp(h)d Fq(j)o Fp(\015)r Fq(j)j(\000)f Fv(2)p Fp(J)t(Y)1345 1028 y Fs(\015)347 1094 y Fp(Y)371 1100 y Fs(\015)434 1094 y Fv(=)534 1055 y Fo(X)507 1146 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)628 1094 y Fp(\016)g Fv(\()p Fp(\013)i Fv(=)h Fp(\013)780 1100 y Fr(1)807 1094 y Fq([)d Fp(\013)871 1100 y Fr(2)901 1094 y Fv(:)i Fp(d)c Fv(\()o Fp(\013)995 1100 y Fr(1)1014 1094 y Fv(\))k(=)h Fp(d)7 b Fv(\()p Fp(\013)1157 1100 y Fr(2)1175 1094 y Fv(\))12 b(=)f(0)k(;)21 b Fp(dist)7 b Fv(\()q Fp(\013)1436 1100 y Fr(1)1454 1094 y Fp(;)g(\013)1500 1100 y Fr(2)1518 1094 y Fv(\))k(=)h(2\))d Fq(\002)671 1203 y(\002)p Fp(\016)g Fv(\()p Fq(j)p Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)i Fv(=)h(1\))7 b([)p Fp(\016)i Fv(\()p Fp(\013)1047 1209 y Fr(1)1076 1203 y Fq(\032\032)k Fp(\015)r Fv(\))7 b Fp(\016)i Fv(\()p Fp(\013)1269 1209 y Fr(2)1297 1203 y Fq(\\)g Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))d(+)h Fp(\016)e Fv(\()q Fp(\013)1571 1209 y Fr(2)1600 1203 y Fq(\032\032)k Fp(\015)r Fv(\))c Fp(\016)h Fv(\()p Fp(\013)1793 1209 y Fr(1)1821 1203 y Fq(\\)g Fp(\015)14 b Fv(=)e Fq(;)p Fv(\)])228 1288 y Fn(wher)n(e)e Fp(\016)f Fv(\()p Fp(\013)411 1294 y Fs(i)436 1288 y Fq(\032\032)j Fp(\015)r Fv(\))g(=)g Fp(\016)d Fv(\()p Fp(\013)677 1294 y Fs(i)702 1288 y Fq(\032)j Fp(\015)r Fv(\))c Fp(\016)h Fv(\()p Fp(\013)863 1294 y Fs(i)886 1288 y Fq(\\)f Fp(@)r(\015)15 b Fv(=)d Fq(;)p Fv(\))49 b Fp(i)12 b Fv(=)g(1)p Fp(;)7 b Fv(2)p Fn(.)16 b(Ther)n(efor)n(e)10 b Fp(Y)1480 1294 y Fs(\015)1513 1288 y Fq(\024)h Fv(2)c Fq(j)p Fp(@)r(\015)r Fq(j)50 b(8)p Fp(\015)14 b Fq(\022)228 1337 y Fv(\003)p Fn(.)278 1429 y Fv(These)d(examples)f(sho)o(w)g(that)h(the)g(computation)d(of)i(the)h (eigen)o(v)n(alues)f(of)g(the)h(Hamiltonian)228 1478 y(op)q(erator)k(of)e(a)i (mo)q(del)d(is)j(reduced)h(to)e(a)g(com)o(binatorial)d(argumen)o(t.)19 b(Nev)o(ertheless,)d(ev)o(en)f(if)228 1528 y(the)e(explicit)f(form)f(of)h (suc)o(h)h(eigen)o(v)n(alues)f(is)h(kno)o(wn,)f(it)g(is)g(insu\016cien)o(t)h (for)f(the)h(computation)228 1578 y(of)j(the)i(partition)e(function.)27 b(This)17 b(requires)h(also)e(the)i(kno)o(wledge)e(of)h(the)g(m)o(ultiplicit) o(y)d(of)228 1628 y(suc)o(h)g(eigen)o(v)n(alues)g(whic)o(h)g(is)g(often)g(v)o (ery)g(hard)g(to)f(compute.)825 1736 y(3.)23 b Fu(Ising)15 b(model)278 1813 y Fv(Let's)20 b(consider)h(\014rst)g(the)g(Hamiltonian)c(op) q(erator)k(describing)g(the)g Fn(ne)n(ar)n(est)f(neighb)n(or)228 1862 y(inter)n(action)13 b Fv(Ising)h(mo)q(del)e(in)i(absence)h(of)e(an)o(y)h (external)g(source)h(from)e Fy(Example)i(2.1)645 1953 y(H)682 1959 y Fr(\003)748 1953 y Fv(=)42 b Fq(\000)888 1914 y Fo(X)861 2005 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)981 1953 y Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))12 b(=)f(1\))c Fm(\033)1241 1959 y Fs(\013)651 2062 y Fp(H)686 2068 y Fs(\015)748 2062 y Fv(=)42 b Fq(\000)p Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)i Fv(+)h(2)d Fq(j)o Fp(@)r(\015)r Fq(j)90 b(8)p Fp(\015)15 b Fq(\022)d Fv(\003)228 2146 y(The)i(partition)f(function)h(no)o(w)f(reads)547 2237 y Fp(Z)578 2215 y Fr(\()q Fs(d)p Fr(\))575 2249 y(\003)631 2237 y Fv(\()p Fp(\014)r Fv(\))42 b(=)830 2198 y Fo(X)804 2289 y Fh(;\022)p Fs(\015)r Fh(\022)p Fr(\003)922 2237 y Fp(e)941 2220 y Fs(\014)q Fr(\()q Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000)p Fr(2)p Fh(j)p Fs(@)r(\015)r Fh(j)o Fr(\))1162 2237 y Fv(=)-966 b(\(5\))730 2359 y(=)42 b Fp(e)823 2341 y Fs(\014)q(d)p Fh(j)q Fr(\003)p Fh(j)976 2319 y Fo(X)913 2410 y Fr(0)p Fh(\024j)o Fs(@)r(\015)r Fh(j\024)p Fs(d)p Fh(j)p Fr(\003)p Fh(j)1106 2359 y Fp(m)7 b Fv(\()q Fq(j)o Fp(@)r(\015)r Fq(j)q Fv(\))g Fp(e)1279 2341 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\015)r Fh(j)228 2492 y Fv(where)17 b Fp(m)7 b Fv(\()q Fq(j)o Fp(@)r(\015)r Fq(j)q Fv(\))16 b(is)g(the)h(m)o(ultipli)o(cit)o(y)c(of)j(the)h(Hamiltonia)o (n)d(op)q(erator's)i(eigen)o(v)n(alues,)g(that)228 2541 y(is)c(the)i(n)o(um)o (b)q(er)e(of)g(lattice's)h(subsets)h(whose)f Fn(b)n(oundary)i(hyp)n(ersurfac) n(e)e Fv(ha)o(v)o(e)f(the)i(same)e(cadi-)228 2591 y(nalit)o(y)m(.)j(The)d (maxim)o(um)7 b(v)n(alue)k(for)g Fq(j)o Fp(@)r(\015)r Fq(j)h Fv(is)g(reac)o(hed)h(considering)e(the)h Fn(chessb)n(o)n(ar)n(d)g Fv(set)h(whose)p eop %%Page: 6 6 bop 228 142 a Fx(6)588 b(M.)15 b(GIANFELICE)228 250 y Fv(asso)q(ciated)g (eigen)o(v)n(alue)g Fp(H)661 256 y Fs(\015)696 250 y Fv(has)g(m)o(ultiplicit) o(y)c(2.)21 b(The)15 b(expression)h(for)e(the)i(Gibbs)e(densit)o(y)228 300 y(op)q(erator)g(is)401 413 y(\026)399 414 y Fm(\032)424 392 y Fr(\()p Fs(d)p Fr(\))424 426 y(\003)476 414 y Fv(\()q Fp(\014)r Fv(\))42 b(=)715 386 y(1)p 655 404 142 2 v 655 452 a Fp(Z)686 431 y Fr(\()p Fs(d)p Fr(\))683 465 y(\003)738 452 y Fv(\()p Fp(\014)r Fv(\))834 374 y Fo(X)808 465 y Fh(;\022)p Fs(\015)r Fh(\022)p Fr(\003)926 414 y Fp(e)945 396 y Fs(\014)q Fr(\()p Fs(d)p Fh(j)q Fr(\003)p Fh(j)o(\000)p Fr(2)p Fh(j)p Fs(@)r(\015)r Fh(j)o Fr(\))1161 414 y Fq(j)p Fp(\015)r Fq(i)7 b(h)q Fp(\015)r Fq(j)1271 426 y Fr(\003)1307 414 y Fv(=)-1111 b(\(6\))576 557 y(=)675 528 y Fp(e)694 513 y Fs(\014)q(d)p Fh(j)q Fr(\003)p Fh(j)p 655 547 V 655 595 a Fp(Z)686 574 y Fr(\()p Fs(d)p Fr(\))683 608 y(\003)738 595 y Fv(\()p Fp(\014)r Fv(\))873 517 y Fo(X)808 608 y Fr(0)p Fh(\024j)o Fs(@)r(\013)p Fh(j)o(\024)p Fs(d)p Fh(j)p Fr(\003)p Fh(j)1102 517 y Fo(X)1004 608 y Fh(f)p Fs(\015)r Fh(\022)p Fr(\003:)p Fh(j)p Fs(@)r(\015)r Fh(j)p Fr(=)p Fh(j)p Fs(@)r(\013)p Fh(j)o(g)1267 557 y Fp(e)1286 539 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\015)r Fh(j)1417 557 y Fq(j)o Fp(\015)r Fq(i)8 b(h)p Fp(\015)r Fq(j)1526 569 y Fr(\003)228 699 y Fv(and)14 b(for)f(an)o(y)g(cylinder)i(observ)n(able)f(w)o (e)g(ha)o(v)o(e)402 835 y Fp(!)429 815 y Fs(\014)428 847 y Fr(\003)460 835 y Fv(\()p Fm(\033)506 841 y Fs(\015)527 835 y Fv(\))42 b(=)664 770 y Fo(P)708 813 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)830 801 y Fp(e)849 786 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)982 801 y Fv(\()p Fq(\000)p Fv(1\))1067 780 y Fh(j)p Fs(\015)r Fh(\\)p Fs(\013)p Fh(j)p 664 825 489 2 v 752 833 a Fo(P)796 877 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)918 864 y Fp(e)937 852 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)1169 835 y Fv(=)585 970 y(=)g(1)9 b Fq(\000)g Fv(2)756 904 y Fo(P)800 948 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)922 936 y Fp(e)941 920 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)1067 936 y Fp(\016)g Fv(\()p Fq(j)p Fp(\015)j Fq(\\)d Fp(\013)p Fq(j)h Fv(=)i(2)p Fp(n)d Fv(+)h(1;)d Fp(n)j Fq(2)h Ft(N)p Fv(\))p 756 960 787 2 v 994 968 a Fo(P)1038 1011 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1160 999 y Fp(e)1179 987 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)228 1099 y Fv(Adding)i(a)h(constan)o(t)h(external)f (magnetic)f(\014eld)h Fp(h)e Fq(2)f Ft(R)d Fv(on)14 b(the)g(whole)g(lattice,) g(from)e(\(3\))i(w)o(e)228 1149 y(get)472 1251 y Fy(H)509 1257 y Fr(\003)541 1251 y Fv(\()p Fp(h)p Fv(\))42 b(=)f Fq(\000)778 1212 y Fo(X)751 1303 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)872 1251 y Fv([)o Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))d(+)h Fp(h\016)f Fv(\()p Fp(d)e Fv(\()o Fp(\013)p Fv(\))12 b(=)g(0\)])6 b Fm(\033)1452 1257 y Fs(\013)478 1372 y Fp(H)513 1378 y Fs(\015)541 1372 y Fv(\()p Fp(h)p Fv(\))42 b(=)f Fq(\000)778 1333 y Fo(X)751 1424 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)872 1372 y Fv([)o Fp(\016)9 b Fv(\()p Fp(d)e Fv(\()p Fp(\013)p Fv(\))k(=)h(1\))d(+)h Fp(h\016)f Fv(\()p Fp(d)e Fv(\()o Fp(\013)p Fv(\))12 b(=)g(0\)])c Fq(\002)726 1481 y(\002)f Fv([1)i Fq(\000)g Fv(2)p Fp(\016)g Fv(\()p Fq(j)p Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)i Fv(=)h(2)p Fp(n)d Fv(+)h(1;)d Fp(n)j Fq(2)i Ft(N)p Fv(\))n(])639 1544 y(=)41 b Fq(\000)p Fp(d)7 b Fq(j)p Fv(\003)p Fq(j)h Fv(+)i(2)d Fq(j)o Fp(@)r(\015)r Fq(j)j(\000)f Fp(h)e Fq(j)p Fv(\003)p Fq(j)h Fv(+)i(2)p Fp(h)d Fq(j)o Fp(\015)r Fq(j)90 b(8)p Fp(\015)15 b Fq(\022)c Fv(\003)228 1640 y(whic)o(h)i(leads)g(to)g(the)g(follo)o(wing)e(expression)j(of)e(the)i (partition)e(function)g(and)h(Gibbs)g(densit)o(y)228 1690 y(op)q(erator)469 1793 y Fp(Z)500 1771 y Fr(\()p Fs(d)p Fr(\))497 1805 y(\003)553 1793 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))42 b(=)f Fp(e)787 1775 y Fs(\014)q Fr(\()q Fs(d)p Fr(+)p Fs(h)p Fr(\))q Fh(j)p Fr(\003)p Fh(j)973 1753 y Fo(X)948 1844 y Fh(;\022)p Fs(\015)r Fh(\022)p Fr(\003)1065 1793 y Fp(e)1084 1775 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\015)r Fh(j)p Fr(+)p Fs(h)p Fh(j)q Fs(\015)r Fh(j)p Fr(\))228 1793 y Fv(\(7\))477 1934 y(\026)475 1935 y Fm(\032)500 1914 y Fr(\()p Fs(d)p Fr(\))500 1948 y(\003)553 1935 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))42 b(=)779 1907 y Fp(e)798 1892 y Fs(\014)q Fr(\()q Fs(d)p Fr(+)p Fs(h)p Fr(\))q Fh(j)p Fr(\003)p Fh(j)p 773 1926 184 2 v 773 1974 a Fp(Z)804 1953 y Fr(\()q Fs(d)p Fr(\))801 1986 y(\003)857 1974 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))994 1896 y Fo(X)969 1987 y Fh(;\022)p Fs(\015)r Fh(\022)p Fr(\003)1087 1935 y Fp(e)1106 1918 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\015)r Fh(j)o Fr(+)p Fs(h)p Fh(j)q Fs(\015)r Fh(j)p Fr(\))1346 1935 y Fq(j)p Fp(\015)r Fq(i)h(h)p Fp(\015)r Fq(j)1456 1948 y Fr(\003)228 2077 y Fv(and)14 b(also)f Fq(8)p Fm(\033)445 2083 y Fs(\015)478 2077 y Fq(2)e(C)539 2083 y Fr(\003)564 2077 y Fv(;)i Fp(\015)h Fq(\022)e Fv(\003)332 2216 y Fp(!)359 2196 y Fs(\014)q(;h)358 2228 y Fr(\003)418 2216 y Fv(\()p Fm(\033)464 2222 y Fs(\015)486 2216 y Fv(\))41 b(=)622 2151 y Fo(P)666 2195 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)788 2182 y Fp(e)807 2167 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\013)p Fh(j)o Fr(+)p Fs(h)p Fh(j)q Fs(\013)p Fh(j)o Fr(\))1052 2182 y Fv(\()p Fq(\000)p Fv(1\))1137 2161 y Fh(j)p Fs(\015)r Fh(\\)p Fs(\013)p Fh(j)p 622 2207 601 2 v 711 2214 a Fo(P)754 2258 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)877 2245 y Fp(e)896 2233 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\013)p Fh(j)n Fr(+)p Fs(h)p Fh(j)q Fs(\013)p Fh(j)p Fr(\))1239 2216 y Fv(=)543 2351 y(=)h(1)9 b Fq(\000)g Fv(2)714 2286 y Fo(P)758 2329 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)880 2317 y Fp(e)899 2302 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\013)p Fh(j)o Fr(+)p Fs(h)p Fh(j)q Fs(\013)p Fh(j)o Fr(\))1137 2317 y Fp(\016)g Fv(\()p Fq(j)p Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)j Fv(=)g(2)p Fp(n)c Fv(+)i(1;)d Fp(n)k Fq(2)g Ft(N)p Fv(\))p 714 2341 899 2 v 952 2349 a Fo(P)996 2392 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1118 2380 y Fp(e)1137 2368 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\013)p Fh(j)o Fr(+)p Fs(h)p Fh(j)q Fs(\013)p Fh(j)o Fr(\))228 2486 y Fy(Remark)k(3.1.)21 b(One-dimensi)o(onal)11 b(Ising)h(mo)q(del)p Fn(.)17 b(Consider)d(the)f(lattic)n(e)g Ft(Z)1534 2471 y Fr(1)1549 2486 y Fn(.)19 b(By)14 b(sim-)228 2536 y(ple)i(c)n(ombinatorial)g(ar)n(guments)h(we)f(have)i Fp(m)7 b Fv(\()p Fq(j)p Fp(@)r(\015)r Fq(j)p Fv(\))15 b(=)g(2)1149 2502 y Fo(\000)1176 2517 y Fh(j)p Fr(\003)p Fh(j)1168 2551 y(j)p Fs(@)r(\015)r Fh(j)1226 2502 y Fo(\001)1262 2536 y Fn(\(the)i(r)n(e)n (ader)f(may)g(c)n(onvinc)n(e)228 2591 y(himself)e(c)n(omputing,)i(for)e (example,)i(the)f(numb)n(er)g(of)g(lattic)n(e)g(subsets)g(having)h(c)n(ar)n (dinality)e(of)p eop %%Page: 7 7 bop 657 142 a Fx(QUANTUM)18 b(METHODS)f(F)o(OR)f(IPS'S)e(I)q(I)q(I)411 b(7)228 250 y Fn(their)14 b(b)n(oundary)i(e)n(qual)f(2\).)k(F)m(r)n(om)14 b(\()7 b Fv(5)p Fn(\))14 b(we)h(get)493 374 y Fp(Z)524 353 y Fr(\(1\))521 387 y(\003)576 374 y Fv(\()p Fp(\014)r Fv(\))42 b(=)g Fp(e)768 357 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)849 319 y(j)p Fr(\003)p Fh(j)840 335 y Fo(X)840 424 y Fs(k)q Fr(=0)1006 335 y Fo(X)908 426 y Fh(f;\022)p Fs(\015)r Fh(\022)p Fr(\003:)p Fh(j)p Fs(@)r(\015)r Fh(j)o Fr(=)p Fs(k)q Fh(g)1171 374 y Fp(e)1190 357 y Fh(\000)p Fr(2)p Fs(\014)q(k)675 498 y Fv(=)g Fp(e)768 480 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)895 458 y Fo(X)840 549 y Fr(0)p Fh(\024j)p Fs(@)r(\015)r Fh(j)o(\024j)p Fr(\003)p Fh(j)1017 498 y Fp(m)7 b Fv(\()p Fq(j)p Fp(@)r(\015)r Fq(j)p Fv(\))g Fp(e)1189 480 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\015)r Fh(j)675 637 y Fv(=)42 b(2)p Fp(e)789 620 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)916 598 y Fo(X)861 689 y Fr(0)p Fh(\024j)o Fs(@)r(\015)r Fh(j\024j)p Fr(\003)p Fh(j)1037 579 y Fo(\022)1077 609 y Fq(j)p Fv(\003)p Fq(j)1068 666 y(j)o Fp(@)r(\015)r Fq(j)1139 579 y Fo(\023)1170 637 y Fp(e)1189 620 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\015)r Fh(j)675 766 y Fv(=)g(2)p Fp(e)789 749 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)861 732 y Fo(\000)880 766 y Fv(1)9 b(+)h Fp(e)971 749 y Fh(\000)p Fr(2)p Fs(\014)1036 732 y Fo(\001)1055 739 y Fh(j)p Fr(\003)p Fh(j)1111 766 y Fv(=)i(2)1176 749 y Fh(j)o Fr(\003)p Fh(j)p Fr(+1)1269 766 y Fv(\(cosh)c Fp(\014)r Fv(\))1412 745 y Fh(j)p Fr(\003)p Fh(j)228 854 y Fn(A)n(dding)13 b(an)g(external)g(\014eld,)g(sinc)n(e)f(the)h(numb)n(er)g(of) f(sets)h(in)g Fv(\003)f Fn(with)g(c)n(ar)n(dinality)g Fq(j)p Fp(\015)r Fq(j)h Fn(ar)n(e)1629 820 y Fo(\000)1648 835 y Fh(j)p Fr(\003)p Fh(j)1650 869 y(j)p Fs(\015)r Fh(j)1690 820 y Fo(\001)1709 854 y Fn(,)228 911 y(fr)n(om)h(\()7 b Fv(7)p Fn(\))14 b(we)h(have)271 998 y Fp(Z)302 976 y Fr(\()q(1\))299 1010 y(\003)354 998 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))42 b(=)g Fp(e)589 981 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)p Fr(\(1+)p Fs(h)p Fr(\))774 958 y Fo(X)748 1049 y Fh(;\022)p Fs(\015)r Fh(\022)p Fr(\003)866 998 y Fp(e)885 981 y Fh(\000)p Fr(2)p Fs(\014)q Fr(\()p Fh(j)p Fs(@)r(\015)r Fh(j)p Fr(+)p Fs(h)p Fh(j)p Fs(\015)r Fh(j)q Fr(\))1130 998 y Fv(=)496 1169 y(=)g Fp(e)589 1152 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)p Fr(\(1+)p Fs(h)p Fr(\))748 1084 y Fo(8)748 1121 y(<)748 1196 y(:)798 1096 y Ff(j)o Fg(\003)p Ff(j)p 798 1104 38 2 v 809 1120 a Fg(2)787 1129 y Fo(X)785 1217 y Fs(n)p Fr(=1)855 1169 y Fv(2)883 1110 y Fo(\022)934 1143 y Fq(j)o Fv(\003)p Fq(j)937 1193 y Fv(2)p Fp(n)1006 1110 y Fo(\023)1044 1169 y Fp(e)1063 1152 y Fh(\000)p Fr(4)p Fs(\014)q(n)1156 1113 y Fh(j)o Fr(\003)p Fh(j\000)p Fs(n)1170 1129 y Fo(X)1168 1219 y Fs(k)q Fr(=)p Fs(n)1251 1169 y Fp(e)1270 1152 y Fh(\000)p Fr(2)p Fs(\014)q(hk)1383 1169 y Fv(+)9 b(1)g(+)h Fp(e)1515 1152 y Fh(\000)p Fr(2)p Fs(\014)q(h)p Fh(j)p Fr(\003)p Fh(j)1642 1084 y Fo(9)1642 1121 y(=)1642 1196 y(;)228 1297 y Fy(3.1.)24 b(Geometric)10 b(dualit)o(y)g(and)i(p)q(ercolation) e(mapping.)19 b Fv(The)11 b(expression)i(\(5\))e(is)g(sim-)228 1347 y(ilar)k(to)h(the)h Fn(low)f(temp)n(er)n(atur)n(e)g Fv(\(L)m(T\))g (expansion)g(for)g(the)h(partition)e(function)h(of)g(the)g(Ising)228 1397 y(mo)q(del)9 b(obtained)h(b)o(y)h(geometric)e(dualit)o(y)h(argumen)o(ts) g([S)j(].)k(On)10 b(the)i(other)f(hand)f(the)h(compu-)228 1447 y(tation)f(of)h(\(5\))g(do)q(es)h(not)f(in)o(v)o(olv)o(e)f(an)o(y)h(argumen)o (t)f(of)h(phase)h(separation)f(in)g(the)h(con\014guration)228 1497 y(space)i(as)g(in)f(the)h(case)g(of)f(the)h(computation)e(of)h(the)h Fn(high)h(temp)n(er)n(atur)n(e)e Fv(\(HT\))h(expansion)h(of)228 1554 y Fp(Z)259 1533 y Fr(\()p Fs(d)p Fr(\))256 1567 y(\003)311 1554 y Fv(\()p Fp(\014)r Fv(\))q(.)26 b(In)16 b(the)h(follo)o(wing)d(w)o(e)j (will)e(sho)o(w)h(our)h(approac)o(h)f(to)h(giv)o(e)e(a)i(description)g(of)f (the)228 1604 y(con\014guration)10 b(space)i(of)e(t)o(w)o(o-state)h(spin)g (systems)f(in)h(terms)f(of)g(the)i(geometric)e(prop)q(erties)i(of)228 1654 y(the)h(underlying)g(lattice)f(structure)j(in)d(a)h(more)f(complete)g(w) o(a)o(y)g(than)h(that)f(giv)o(en)h(in)f([GHM)o(].)228 1712 y(This)k(will)f(b)q(e)h(done)h(deriving)e(the)i(expression)h(for)d Fp(Z)1101 1690 y Fr(\()q Fs(d)p Fr(\))1098 1724 y(\003)1154 1712 y Fv(\()p Fp(\014)r Fv(\))i(in)f(resp)q(ectiv)o(ely)h(the)g(HT,)f(L)m(T) 228 1762 y(and)e(F)m(ortuin-Kasteleyn)g(\(FK\))g(represen)o(tation)i(from)c (one)i(another.)228 1851 y(3.1.1.)19 b Fy(HT)14 b(expression)d(of)i Fp(Z)743 1829 y Fr(\()p Fs(d)p Fr(\))740 1863 y(\003)795 1851 y Fv(\()q Fp(\014)r Fv(\))p Fn(.)21 b Fv(Let)12 b(\001)994 1857 y Fr(\003)1030 1851 y Fv(:=)g Fq(f)o Fp(\015)j Fq(\022)d Fv(\003)f(:)g Fp(\016)e Fv(\()p Fp(d)e Fv(\()p Fp(\015)r Fv(\))12 b(=)g(1\))p Fq(g)g Fv(b)q(e)g(the)h(set)g(of)228 1900 y(dimers)f(in)g(\003,)h (then)g Fq(j)p Fv(\001)602 1906 y Fr(\003)626 1900 y Fq(j)e Fv(=)h Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)o Fv(.)18 b(W)m(e)12 b(de\014ne)i Fc(P)1026 1906 y Fr(\001)1053 1910 y Fg(\003)1090 1900 y Fv(as)f(the)g(set)h(of)e(collections)h(of)f(elemen)o(ts)228 1950 y(of)k(\001)313 1956 y Fr(\003)348 1950 y Fq([)11 b(;)16 b Fv(and)g Fq(8)p Fv(\000)g Fq(2)g Fc(P)650 1956 y Fr(\001)677 1960 y Fg(\003)717 1950 y Fq(j)o Fv(\000)p Fq(j)g Fv(will)f(denote)j(the)f(n) o(um)o(b)q(er)e(of)h(dimers)g(forming)e(\000.)26 b(Let's)228 2000 y(consider)13 b(also)e(the)h(maps)f(from)f Fc(P)775 2006 y Fr(\001)802 2010 y Fg(\003)838 2000 y Fv(to)h Fc(P)920 2006 y Fr(\003)957 2000 y Fv(\(the)i(set)g(of)e(the)i(parts)f(of)f(\003)h([QMIPS)i (I]\))e(giv)o(en)228 2050 y(b)o(y)h(the)i(symmetric)d(di\013erence)k(and)e (the)g(union)f(of)h(sets)h(\()p Fq(4)p Fp(;)7 b Fq([)p Fv(\))13 b(suc)o(h)i(that)625 2132 y Fq(8)p Fv(\000)c Fq(2)g Fc(P)758 2138 y Fr(\001)785 2142 y Fg(\003)857 2132 y Fv(\()q Fq(4)911 2138 y Fs(\015)r Fh(2)p Fr(\000)988 2132 y Fp(\015)r Fv(\))d Fp(;)f Fv(\()p Fq([)1098 2138 y Fs(\015)r Fh(2)p Fr(\000)1176 2132 y Fp(\015)r Fv(\))12 b Fq(2)f Fc(P)1300 2138 y Fr(\003)228 2211 y Fv(Then)427 2296 y Fp(Z)458 2274 y Fr(\()p Fs(d)p Fr(\))455 2308 y(\003)510 2296 y Fv(\()q Fp(\014)r Fv(\))42 b(=)710 2256 y Fo(X)684 2347 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)804 2296 y Fp(e)823 2271 y Fs(\014)849 2243 y Fo(P)893 2287 y Ff(;\022)p Fd(\015)q Ff(\022)p Fg(\003)997 2271 y Fs(\016)q Fr(\()p Fs(d)p Fr(\()p Fs(\015)r Fr(\)=1\)\()p Fh(\000)p 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y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)p Fr(+)p Fh(j)p Fs(\013)p Fh(\\)p Fs(\021)q Fh(j)876 257 y Fv(=)h(\()p Fq(\000)p Fv(1\))1010 236 y Fh(j)p Fs(\013)p Fh(\\)p Fr(\()p Fs(\015)r Fh(4)p Fs(\021)q Fr(\))p Fh(j)1186 257 y Fv(and)f Fp(\016)1287 264 y Fs(\015)r(;)p Fh(;)1352 257 y Fv(=)1401 226 y Fo(P)1445 269 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1572 237 y(\()p Fh(\000)p Fr(1\))1641 224 y Ff(j)o Fd(\013)p Ff(\\)p Fd(\015)q Ff(j)p 1572 247 146 2 v 1616 273 a Fr(2)1633 264 y Ff(j)o Fg(\003)p Ff(j)228 306 y Fv([QMIPS)e(I])f(than)h (\(8\))g(b)q(ecomes)233 520 y Fp(Z)264 498 y Fr(\()q Fs(d)p Fr(\))261 532 y(\003)317 520 y Fv(\()p Fp(\014)r Fv(\))42 b(=)g(\(cosh)8 b Fp(\014)r Fv(\))634 499 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)729 480 y Fo(X)702 571 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)823 436 y Fo(2)823 511 y(4)850 520 y Fv(1)h(+)939 480 y Fo(X)922 569 y Fr(\000)p Fh(\022)p Fr(\001)996 573 y Fg(\003)1023 520 y Fv(\()p Fq(\000)p Fv(1\))1109 499 y Fh(j)p Fs(\013)p Fh(\\4)1193 503 y Fd(\015)q Ff(2)p Fg(\000)1262 499 y Fs(\015)r Fh(j)1300 520 y Fv(\(tanh)e Fp(\014)r Fv(\))1448 499 y Fh(j)p Fr(\000)p Fh(j)1490 436 y Fo(3)1490 511 y(5)1529 520 y Fv(=)416 686 y(=)42 b(2)511 669 y Fh(j)p Fr(\003)p Fh(j)562 686 y Fv(\()q(cosh)7 b Fp(\014)r Fv(\))706 665 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)775 602 y Fo(2)775 677 y(4)802 686 y Fv(1)i(+)891 646 y Fo(X)874 735 y Fr(\000)p Fh(\022)p Fr(\001)948 739 y Fg(\003)975 686 y Fp(\016)g Fv(\()q Fq(4)1056 692 y Fs(\015)r Fh(2)p Fr(\000)1133 686 y Fp(\015)15 b Fv(=)d Fq(;)p Fv(\))6 b(\()q(tanh)g Fp(\014)r Fv(\))1404 665 y Fh(j)p Fr(\000)p Fh(j)1447 602 y Fo(3)1447 677 y(5)1486 686 y Fv(=)416 842 y(=)42 b(2)511 824 y Fh(j)p Fr(\003)p Fh(j)555 842 y Fp(e)574 824 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)701 802 y Fo(X)664 893 y Fr(\000)p Fh(\022)p Fr(\001)738 897 y Fg(\003)759 893 y Fh([;)805 842 y Fp(\016)9 b Fv(\()p Fq(4)885 848 y Fs(\015)r Fh(2)p Fr(\000)963 842 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))1086 783 y Fo(\022)1121 814 y Fv(1)d Fq(\000)h Fp(e)1212 798 y Fh(\000)p Fr(2)p Fs(\014)p 1121 832 156 2 v 1189 870 a Fv(2)1282 783 y Fo(\023)1313 789 y Fh(j)o Fr(\000)p Fh(j)1362 783 y Fo(\022)1397 814 y Fv(1)f(+)h Fp(e)1488 798 y Fh(\000)p Fr(2)p Fs(\014)p 1397 832 V 1465 870 a Fv(2)1558 783 y Fo(\023)1589 789 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)228 1068 y Fv(Setting)20 b Fp(p)398 1074 y Fr(HT)455 1068 y Fv(\()p Fp(\014)r Fv(\))i(=)588 1022 y Fo(\020)618 1052 y Fr(1)p Fh(\000)p Fs(e)677 1039 y Ff(\000)p Fg(2)p Fd(\014)p 618 1059 115 2 v 667 1082 a Fr(2)737 1022 y Fo(\021)782 1068 y Fv(and)e Fp(q)888 1074 y Fr(HT)944 1068 y Fv(\()p Fp(\014)r Fv(\))j(=)1078 1022 y Fo(\020)1107 1052 y Fr(1+)p Fs(e)1165 1039 y Ff(\000)p Fg(2)p Fd(\014)p 1107 1059 114 2 v 1156 1082 a Fr(2)1226 1022 y Fo(\021)1251 1068 y Fv(,)e(w)o(e)f(can)g(rewrite)h(the)f (last)228 1128 y(expression)14 b(ab)q(o)o(v)o(e)f(in)g(terms)g(of)g(a)g (Bernoulli)g(sum)g(on)g(those)h(sets)g(in)f Fc(P)1382 1134 y Fr(\001)1409 1138 y Fg(\003)1446 1128 y Fv(whose)h(image)e(in)228 1178 y Fc(P)262 1184 y Fr(\003)299 1178 y Fv(under)g(the)g(op)q(eration)g(of) f(symmetric)f(di\013erence)j(is)f(the)g(empt)o(yset,)g(that)g(is)f(their)h (image)228 1228 y(in)h Fc(P)310 1234 y Fr(\003)349 1228 y Fv(under)i(the)f (op)q(eration)g(of)f(union)h(of)f(sets)i(is)f(a)g(closed)g(lo)q(op)f(\(HT)h (graphs\).)19 b(Then)397 1404 y Fp(Z)428 1382 y Fr(\()q Fs(d)p Fr(\))425 1416 y(\003)481 1404 y Fv(\()p Fp(\014)r Fv(\))12 b(=)g(2)615 1386 y Fh(j)p Fr(\003)p Fh(j)660 1404 y Fp(e)679 1386 y Fs(d\014)q Fh(j)p Fr(\003)p Fh(j)805 1364 y Fo(X)768 1455 y Fr(\000)p Fh(\022)p Fr(\001)842 1459 y Fg(\003)863 1455 y Fh([;)909 1404 y Fp(\016)d Fv(\()p Fq(4)989 1410 y Fs(\015)r Fh(2)p Fr(\000)1067 1404 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))7 b Fp(p)1211 1382 y Fh(j)p Fr(\000)p Fh(j)1211 1416 y Fr(HT)1268 1404 y Fv(\()p Fp(\014)r Fv(\))h Fp(q)1353 1382 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)o Fr(\000)p Fh(j)1352 1416 y Fr(HT)1488 1404 y Fv(\()p Fp(\014)r Fv(\))-1317 b(\(9\))228 1611 y(The)14 b(Gibbs)g(densit)o(y)g(op)q(erator)g(can)g(no)o(w)g(b)q(e)g(rewritten)h(as) 295 1818 y(\026)293 1819 y Fm(\032)318 1797 y Fr(\()p Fs(d)p Fr(\))318 1831 y(\003)370 1819 y Fv(\()q Fp(\014)r Fv(\))d(=)510 1779 y Fo(X)484 1870 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)647 1753 y Fo(P)691 1797 y Fr(\000)p Fh(\022)p Fr(\001)765 1801 y Fg(\003)786 1797 y Fh([;)834 1785 y Fv(\()p Fq(\000)p Fv(1\))919 1764 y Fh(j)p Fs(\013)p Fh(\\4)1003 1768 y Fd(\015)q Ff(2)p Fg(\000)1073 1764 y Fs(\015)r Fh(j)1111 1785 y Fp(p)1132 1763 y Fh(j)o Fr(\000)p Fh(j)1132 1797 y Fr(HT)1189 1785 y Fv(\()p Fp(\014)r Fv(\))c Fp(q)1274 1763 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1273 1797 y Fr(HT)1408 1785 y Fv(\()q Fp(\014)r Fv(\))p 609 1809 896 2 v 609 1857 a(2)630 1845 y Fh(j)o Fr(\003)p Fh(j)681 1826 y Fo(P)725 1870 y Fr(\000)p Fh(\022)p Fr(\001)799 1874 y Fg(\003)820 1870 y Fh([;)868 1857 y Fp(\016)h Fv(\()p Fq(4)948 1863 y Fs(\015)r Fh(2)p Fr(\000)1026 1857 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))7 b Fp(p)1170 1836 y Fh(j)p Fr(\000)p Fh(j)1170 1870 y Fr(HT)1227 1857 y Fv(\()p Fp(\014)r Fv(\))h Fp(q)1312 1836 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)o Fr(\000)p Fh(j)1311 1870 y Fr(HT)1447 1857 y Fv(\()p Fp(\014)r Fv(\))1517 1819 y Fq(j)o Fp(\013)p Fq(i)f(h)p Fp(\013)p Fq(j)1632 1831 y Fr(\003)228 2026 y Fv(Then)14 b(for)g(an)o(y)f(cylinder)h (observ)n(able)g Fm(\033)868 2032 y Fs(\021)900 2026 y Fq(2)d(C)961 2032 y Fr(\003)986 2026 y Fv(;)i Fp(\021)g Fq(\022)f Fv(\003)228 2233 y Fp(!)255 2213 y Fs(\014)254 2245 y Fr(\003)285 2233 y Fv(\()p Fm(\033)332 2239 y Fs(\021)352 2233 y Fv(\))41 b(=)488 2168 y Fo(P)532 2212 y Fr(\000)p Fh(\022)p Fr(\001)606 2216 y Fg(\003)627 2212 y Fh([;)675 2199 y Fp(p)696 2178 y Fh(j)p Fr(\000)p Fh(j)696 2211 y Fr(HT)753 2199 y Fv(\()p Fp(\014)r Fv(\))8 b Fp(q)838 2178 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)837 2211 y Fr(HT)973 2199 y Fv(\()p Fp(\014)r Fv(\))1063 2183 y Fr(1)p 1043 2190 57 2 v 1043 2215 a(2)1060 2207 y Ff(j)o Fg(\003)p Ff(j)1111 2168 y Fo(P)1155 2212 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1277 2199 y Fv(\()p Fq(\000)p Fv(1\))1363 2178 y Fh(j)o Fs(\013)p Fh(\\4)1446 2182 y Fd(\015)q Ff(2)p Fg(\000)1516 2178 y Fs(\015)r Fh(j)1554 2199 y Fv(\()p Fq(\000)p Fv(1\))1639 2178 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\021)q Fh(j)p 488 2224 1235 2 v 694 2241 a Fo(P)737 2285 y Fr(\000)p Fh(\022)p Fr(\001)811 2289 y Fg(\003)832 2285 y Fh([;)881 2272 y Fp(\016)h Fv(\()p Fq(4)961 2278 y Fs(\015)r Fh(2)p Fr(\000)1039 2272 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))7 b Fp(p)1183 2250 y Fh(j)o Fr(\000)p Fh(j)1183 2284 y Fr(HT)1240 2272 y Fv(\()p Fp(\014)r Fv(\))h Fp(q)1325 2250 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1324 2284 y Fr(HT)1459 2272 y Fv(\()q Fp(\014)r Fv(\))1739 2233 y(=)409 2384 y(=)488 2319 y Fo(P)532 2362 y Fr(\000)p Fh(\022)p Fr(\001)606 2366 y Fg(\003)627 2362 y Fh([;)675 2350 y Fp(p)696 2328 y Fh(j)p Fr(\000)p Fh(j)696 2362 y Fr(HT)753 2350 y Fv(\()p Fp(\014)r Fv(\))g Fp(q)838 2328 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)837 2362 y Fr(HT)973 2350 y Fv(\()p Fp(\014)r Fv(\))1063 2334 y Fr(1)p 1043 2341 57 2 v 1043 2366 a(2)1060 2358 y Ff(j)o Fg(\003)p Ff(j)1111 2319 y Fo(P)1155 2362 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1277 2350 y Fv(\()p Fq(\000)p Fv(1\))1363 2329 y Fh(j)o Fs(\013)p Fh(\\)p Fr(\()p Fh(4)1459 2333 y Fd(\015)q Ff(2)p Fg(\000)1529 2329 y Fs(\015)r Fr(\))p Fh(4)p Fs(\021)q Fh(j)p 488 2374 1133 2 v 642 2392 a Fo(P)686 2435 y Fr(\000)p Fh(\022)p Fr(\001)760 2439 y Fg(\003)781 2435 y Fh([;)829 2423 y Fp(\016)h Fv(\()p Fq(4)909 2429 y Fs(\015)r Fh(2)p Fr(\000)987 2423 y Fp(\015)15 b Fv(=)c Fq(;)p Fv(\))c Fp(p)1131 2401 y Fh(j)p Fr(\000)p Fh(j)1131 2435 y Fr(HT)1188 2423 y Fv(\()q Fp(\014)r Fv(\))g Fp(q)1273 2401 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)1272 2435 y Fr(HT)1408 2423 y Fv(\()p Fp(\014)r Fv(\))1637 2384 y(=)409 2535 y(=)488 2469 y Fo(P)532 2513 y Fr(\000)p Fh(\022)p Fr(\001)606 2517 y Fg(\003)627 2513 y Fh([;)675 2501 y Fp(\016)i Fv(\()p Fq(4)755 2507 y Fs(\015)r Fh(2)p Fr(\000)833 2501 y Fp(\015)14 b Fv(=)e Fp(\021)q Fv(\))7 b Fp(p)978 2479 y Fh(j)p Fr(\000)p Fh(j)978 2513 y Fr(HT)1035 2501 y Fv(\()q Fp(\014)r Fv(\))g Fp(q)1120 2479 y Fs(d)p Fh(j)q Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1119 2513 y Fr(HT)1255 2501 y Fv(\()p Fp(\014)r Fv(\))p 488 2525 826 2 v 489 2542 a Fo(P)533 2586 y Fr(\000)p Fh(\022)p Fr(\001)607 2590 y Fg(\003)627 2586 y Fh([;)676 2573 y Fp(\016)i Fv(\()p Fq(4)756 2579 y Fs(\015)r Fh(2)p Fr(\000)834 2573 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))7 b Fp(p)978 2552 y Fh(j)o Fr(\000)p Fh(j)978 2586 y Fr(HT)1035 2573 y Fv(\()p Fp(\014)r Fv(\))h Fp(q)1120 2552 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1119 2586 y Fr(HT)1255 2573 y Fv(\()p Fp(\014)r Fv(\))p eop %%Page: 9 9 bop 657 142 a Fx(QUANTUM)18 b(METHODS)f(F)o(OR)f(IPS'S)e(I)q(I)q(I)411 b(9)228 251 y Fy(Remark)15 b(3.2.)21 b Fn(Sinc)n(e)13 b Fq(8)p Fp(\015)h Fq(2)d Fv(\001)744 257 y Fr(\003)769 251 y Fn(,)h Fq(8)p Fp(\013)g Fq(\022)f Fv(\003)h(\()q Fq(\000)p Fv(1\))1026 230 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1122 251 y Fv(=)g(1)s Fq(\000)s Fv(2)p Fp(\016)e Fv(\()p Fq(j)o Fp(\013)f Fq(\\)g Fp(\015)r Fq(j)j Fv(=)g(2)p Fp(n)d Fv(+)g(1;)e Fp(n)k Fq(2)g Ft(N)p Fv(\))228 301 y Fn(and)k Fp(q)327 307 y Fb(HT)385 301 y Fv(\()q Fp(\014)r Fv(\))10 b Fq(\000)f Fp(p)515 307 y Fb(HT)574 301 y Fv(\()p Fp(\014)r Fv(\))j(=)g Fp(e)706 286 y Fh(\000)p Fr(2)p Fs(\014)771 301 y Fn(,)j(we)f(have)280 399 y Fp(Z)311 382 y Fs(d)308 409 y Fr(\003)340 399 y Fv(\()p Fp(\014)r Fv(\))42 b(=)539 360 y Fo(X)513 451 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)651 360 y Fo(Y)633 449 y Fs(\015)r Fh(2)p Fr(\001)701 453 y Fg(\003)730 353 y Fo(h)749 399 y Fv(cosh)8 b Fp(\014)k Fv(+)d(\()p Fq(\000)p Fv(1\))997 378 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1089 399 y Fv(sinh)d Fp(\014)1194 353 y Fo(i)439 520 y Fv(=)539 481 y Fo(X)513 572 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)651 481 y Fo(Y)633 570 y Fs(\015)r Fh(2)p Fr(\001)701 574 y Fg(\003)730 520 y Fv([)o(cosh)i Fp(\014)k Fv(+)d(sinh)e Fp(\014)12 b Fq(\000)e Fv(2)p Fp(\016)e Fv(\()p Fq(j)p Fp(\013)h Fq(\\)g Fp(\015)r Fq(j)j Fv(=)f(1\))c(sinh)g Fp(\014)r Fv(])12 b(=)439 642 y(=)42 b Fp(e)532 624 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)648 602 y Fo(X)621 693 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)760 602 y Fo(Y)741 691 y Fs(\015)r Fh(2)p Fr(\001)809 695 y Fg(\003)838 608 y Fo(\002)855 642 y Fv(1)9 b Fq(\000)927 608 y Fo(\000)946 642 y Fv(1)g Fq(\000)g Fp(e)1036 624 y Fh(\000)p Fr(2)p Fs(\014)1102 608 y Fo(\001)1128 642 y Fp(\016)g Fv(\()p Fq(j)o Fp(\013)g Fq(\\)g Fp(\015)r Fq(j)j Fv(=)g(1\))1383 608 y Fo(\003)1411 642 y Fv(=)439 763 y(=)42 b Fp(e)532 746 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)648 723 y Fo(X)621 814 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)741 763 y Fp(e)760 746 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)898 763 y Fv(=)439 884 y(=)g Fp(e)532 867 y Fs(\014)q(d)p Fh(j)q Fr(\003)p Fh(j)648 845 y Fo(X)621 936 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)741 884 y Fv(\()p Fp(q)776 890 y Fb(HT)835 884 y Fv(\()p Fp(\014)r Fv(\))10 b Fq(\000)f Fp(p)964 890 y Fb(HT)1023 884 y Fv(\()p Fp(\014)r Fv(\))q(\))1097 863 y Fh(j)p Fs(@)r(\013)p Fh(j)1167 884 y Fv(\()p Fp(p)1204 890 y Fb(HT)1262 884 y Fv(\()p Fp(\014)r Fv(\))h(+)g Fp(q)1390 890 y Fb(HT)1448 884 y Fv(\()p Fp(\014)r Fv(\))q(\))1522 863 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)o Fs(@)r(\013)p Fh(j)228 1016 y Fn(which)k(is)h(exactly)g(\()7 b Fv(5)p Fn(\).)228 1109 y Fy(Remark)15 b(3.3.)21 b Fn(Consider)16 b(the)h Ft(Z)783 1094 y Fr(2)815 1109 y Fn(lattic)n(e.)23 b(Cho)n(ose)16 b(in)h(the)f(c)n (omputation)h(of)g(\()7 b Fv(9)p Fn(\))15 b(a)i(\014nite)228 1159 y(cub)n(e)g Fv(\003)e(:)g Fq(j)p Fv(\003)p Fq(j)f Fv(=)i Fp(L)536 1144 y Fr(2)572 1159 y Fn(with)g(fr)n(e)n(e)g(b.c.,)h(and)g(in)g (the)g(c)n(omputation)h(of)f(\()7 b Fv(5)p Fn(\))16 b(a)h(\014nite)h(cub)n(e) f Fv(\003)1684 1144 y Fh(0)1711 1159 y Fv(:)228 1213 y Fq(j)o Fv(\003)268 1198 y Fh(0)280 1213 y Fq(j)c Fv(=)h(\()p Fp(L)9 b Fq(\000)h Fv(1\))483 1192 y Fr(2)517 1213 y Fn(with)15 b(+)h(b.c.)22 b(Mor)n(e)n(over,)16 b(if)f(we)h(set)f Fv(sinh)7 b(2)p Fp(\014)j Fv(sinh)d(2)p Fp(\014)1370 1198 y Fh(0)1395 1213 y Fv(=)14 b(1)p Fn(,)h(fr)n(om)h(the)g(two)228 1282 y(expr)n(essions)g(for)f(the)g(p)n (artition)g(function)h(given)g(b)n(efor)n(e)f(we)g(get)1281 1259 y Fs(Z)1306 1243 y Fg(\()o(2\))q Fd(;f)r(r)q(ee)1304 1269 y Fg(\003)1413 1259 y Fr(\()p Fs(\014)q Fr(\))p 1281 1273 179 2 v 1342 1298 a(2)1359 1290 y Ff(j)o Fg(\003)p Ff(j)1477 1282 y Fv(=)e Fp(Z)1553 1261 y Fr(\(2\))p Fs(;)p Fr(+)1550 1294 y(\003)1573 1286 y Ff(0)1640 1282 y Fv(\()p Fp(\014)1681 1267 y Fh(0)1693 1282 y Fv(\))p Fn(.)228 1332 y(This)i(implies)f(that)h(the)g(set) g(of)h(HT)e(gr)n(aphs)i(in)f Fv(\003)g Fn(c)n(oincides)g(with)g(the)g(set)g (of)g(L)m(T)g(gr)n(aphs)g(in)228 1382 y Fv(\003)257 1367 y Fh(0)268 1382 y Fn(.)k(This)c(ar)n(gument)g(go)n(es)g(under)g(the)g(name)g (of)g(self-duality)j Fv([S)13 b(])p Fn(.)228 1483 y Fv(3.1.2.)19 b Fy(FK)f(expression)e(for)g Fp(Z)772 1461 y Fr(\()p Fs(d)p Fr(\))769 1495 y(\003)824 1483 y Fv(\()q Fp(\014)r Fv(\))p Fn(.)21 b Fv(F)m(ortuin-Kasteleyn)16 b(\(FK\))g(represen)o(tation)i([FK])228 1533 y(allo)o(w)e(the)i(description)g(of)f(t)o(w)o(o-state)h(systems)g (statistical)f(mec)o(hanics)g(in)g(terms)g(of)g Fn(b)n(ond)228 1583 y(p)n(er)n(c)n(olation)d(pr)n(o)n(c)n(esses)p Fv(.)k(Here)d(w)o(e)f (will)e(deriv)o(e)j(it)e(in)h(a)f(di\013eren)o(t)i(w)o(a)o(y)m(.)i(Consider)d (the)g(graph)228 1632 y(\(\003)p Fp(;)7 b Fv(\001)327 1638 y Fr(\003)351 1632 y Fv(\),)13 b(from)f(\(4\))i(and)g(\(5\))g(follo)o(ws)228 1738 y Fp(e)247 1721 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)414 1738 y Fv(=)515 1698 y Fo(X)488 1789 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)608 1738 y Fp(e)627 1711 y Fh(\000)p Fr(2)p Fs(\014)696 1683 y Fo(P)740 1726 y Fd(\015)q Ff(2)p Fg(\001)800 1733 y(\003)828 1711 y Fs(\016)q Fr(\()p Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)p Fr(=1\))1009 1738 y Fv(=)228 1870 y Fp(e)247 1853 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)414 1870 y Fv(=)42 b Fp(e)507 1843 y Fh(\000)p Fr(2)p Fs(\014)576 1815 y Fo(P)620 1859 y Fd(\015)q Ff(2)p Fg(\001)680 1866 y(\003)700 1859 y Ff(n)q Fg(\026)-15 b Fd(\015)740 1843 y Fs(\016)q Fr(\()p Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)p Fr(=1\))916 1837 y Fo(\002)933 1870 y Fv(\(1)9 b Fq(\000)h Fp(\016)f Fv(\()p Fq(j)p Fp(\013)g Fq(\\)h Fv(\026)-23 b Fp(\015)s Fq(j)11 b Fv(=)h(1\)\))d(+)h Fp(e)1362 1853 y Fh(\000)p Fr(2)p Fs(\014)1427 1870 y Fp(\016)f Fv(\()p Fq(j)p Fp(\013)g Fq(\\)h Fv(\026)-22 b Fp(\015)r Fq(j)11 b Fv(=)h(1\))1682 1837 y Fo(\003)1748 1870 y Fq(8)p Fp(\013)f Fq(\022)h Fv(\003)228 1980 y(where)k Fp(e)368 1953 y Fh(\000)p Fr(2)p Fs(\014)437 1925 y Fo(P)480 1969 y Fd(\015)q Ff(2)p Fg(\001)540 1976 y(\003)561 1969 y Ff(n)q Fg(\026)-15 b Fd(\015)601 1953 y Fs(\016)q Fr(\()p Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)p Fr(=1\))785 1980 y Fv(is)14 b Fp(e)846 1965 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)987 1980 y Fv(calculated)h(on)g(the)g(graph)g(\(\003)p Fp(;)7 b Fv(\001)1533 1986 y Fr(\003)1557 1980 y Fq(n)r Fv(\026)-23 b Fp(\015)s Fv(\).)21 b(Set-)228 2030 y(ting)609 2084 y Fo(\032)687 2117 y Fp(p)708 2123 y Fr(FK)763 2117 y Fv(\()q Fp(\014)r Fv(\))12 b(=)g(2)p Fp(p)919 2123 y Fr(HT)976 2117 y Fv(\()p Fp(\014)r Fv(\))g(=)1089 2084 y Fo(\000)1108 2117 y Fv(1)d Fq(\000)g Fp(e)1198 2102 y Fh(\000)p Fr(2)p Fs(\014)1264 2084 y Fo(\001)661 2168 y Fp(q)680 2174 y Fr(FK)735 2168 y Fv(\()p Fp(\014)r Fv(\))k(=)e Fp(q)867 2174 y Fr(HT)924 2168 y Fv(\()p Fp(\014)r Fv(\))f Fq(\000)g Fp(p)1054 2174 y Fr(HT)1111 2168 y Fv(\()p Fp(\014)r Fv(\))i(=)g Fp(e)1243 2152 y Fh(\000)p Fr(2)p Fs(\014)228 2143 y Fv(\(10\))228 2251 y(w)o(e)i(obtain)308 2350 y Fp(e)327 2333 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)464 2350 y Fv(=)e Fp(e)527 2324 y Fh(\000)p Fr(2)p Fs(\014)596 2295 y Fo(P)640 2339 y Fd(\015)q Ff(2)p Fg(\001)700 2346 y(\003)720 2339 y Ff(n)q Fg(\026)-15 b Fd(\015)760 2324 y Fs(\016)q Fr(\()p Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)p Fr(=1\))936 2350 y Fv([)o(\()q(1)9 b Fq(\000)g Fp(\016)g Fv(\()p Fq(j)p Fp(\013)g Fq(\\)h Fv(\026)-22 b Fp(\015)r Fq(j)11 b Fv(=)h(1\)\))7 b Fp(p)1334 2356 y Fr(FK)1390 2350 y Fv(\()p Fp(\014)r Fv(\))j(+)g Fp(q)1518 2356 y Fr(FK)1573 2350 y Fv(\()p Fp(\014)r Fv(\))q(])228 2436 y(Iterating)k(suc)o(h)g(expression)h(un)o(til)e(all)g(the)i(dimers)e(ha) o(v)o(e)g(b)q(een)i(deleted,)g(since)436 2579 y(\(1)9 b Fq(\000)g Fp(\016)g Fv(\()p Fq(j)p Fp(\013)g Fq(\\)g Fp(\021)q Fq(j)i Fv(=)h(1\)\))f(=)853 2490 y Fo(\020)878 2536 y Fv(1)e(+)g(\()q Fq(\000)p Fv(1\))1035 2515 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1120 2490 y Fo(\021)p 853 2570 292 2 v 988 2608 a Fv(2)1191 2579 y Fq(8)p Fp(\015)14 b Fq(2)e Fv(\001)1324 2585 y Fr(\003)1348 2579 y Fv(;)20 b Fq(8)p Fp(\013)12 b Fq(\022)f Fv(\003)p eop %%Page: 10 10 bop 228 142 a Fx(10)570 b(M.)15 b(GIANFELICE)228 250 y Fv(w)o(e)f(ha)o(v)o(e) 486 387 y Fp(e)505 370 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)642 387 y Fv(=)705 348 y Fo(Y)686 437 y Fs(\015)r Fh(2)p Fr(\001)754 441 y Fg(\003)783 304 y Fo(2)783 379 y(4)816 298 y(\020)840 344 y Fv(1)9 b(+)h(\()p Fq(\000)p Fv(1\))997 323 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1082 298 y Fo(\021)p 816 378 292 2 v 951 416 a Fv(2)1112 387 y Fp(p)1133 393 y Fr(FK)1189 387 y Fv(\()p Fp(\014)r Fv(\))g(+)f Fp(q)1316 393 y Fr(FK)1372 387 y Fv(\()p Fp(\014)r Fv(\))1430 304 y Fo(3)1430 379 y(5)228 387 y Fv(\(11\))228 523 y(De\014ning)16 b Fc(P)430 529 y Fr(\000)468 523 y Fq(\022)g Fc(P)550 529 y Fr(\001)577 533 y Fg(\003)617 523 y Fv(as)h(the)f(set)i(of)d(collection)h(of)g(dimers)g (in)f(\000)i(and)f(restricting)h Fq(4)p Fp(;)7 b Fq([)15 b Fv(:)228 573 y Fc(P)262 579 y Fr(\001)289 583 y Fg(\003)324 573 y Fq(!)c Fc(P)411 579 y Fr(\003)450 573 y Fv(to)j Fc(P)535 579 y Fr(\000)558 573 y Fv(,)f(suc)o(h)i(that)f Fq(8)p Fv(\000)d Fq(2)g Fc(P)900 579 y Fr(\001)927 583 y Fg(\003)951 573 y Fp(;)c Fq(8)p Fp(c)k Fq(2)g Fc(P)1095 579 y Fr(\000)1167 573 y Fv(\()p Fq(4)1220 579 y Fs(\015)r Fh(2)p Fs(c)1292 573 y Fp(\015)r Fv(\))i Fq(2)e Fc(P)1417 579 y Fr(\003)1442 573 y Fv(,)i(w)o(e)h(get)333 716 y Fp(e)352 699 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)490 716 y Fv(=)d Fp(q)553 695 y Fs(d)p Fh(j)q Fr(\003)p Fh(j)552 729 y Fr(FK)624 716 y Fv(+)683 677 y Fo(X)666 766 y Fr(\000)p Fh(\022)p Fr(\001)740 770 y Fg(\003)773 688 y Fp(p)794 666 y Fh(j)o Fr(\000)p Fh(j)794 700 y Fr(FK)849 688 y Fv(\()p Fp(\014)r Fv(\))d Fp(q)934 666 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)933 700 y Fr(FK)1069 688 y Fv(\()p Fp(\014)r Fv(\))p 773 707 355 2 v 918 746 a(2)939 734 y Fh(j)p Fr(\000)p Fh(j)1139 633 y Fo(0)1139 708 y(@)1175 716 y Fv(1)h(+)1247 677 y Fo(X)1247 766 y Fs(c)p Fh(\022)p Fr(\000)1315 716 y Fv(\()p Fq(\000)p Fv(1\))1400 696 y Fh(j)p Fs(\013)p Fh(\\4)1484 700 y Fd(\015)q Ff(2)p Fd(c)1550 696 y Fs(\015)r Fh(j)1581 633 y Fo(1)1581 708 y(A)228 852 y Fv(where)15 b Fp(c)c Fq(2)h Fc(P)451 858 y Fr(\000)473 852 y Fv(.)18 b(Then)228 987 y Fp(Z)259 966 y Fr(\()p Fs(d)p Fr(\))256 1000 y(\003)311 987 y Fv(\()p Fp(\014)r Fv(\))43 b(=)f Fp(e)504 970 y Fs(d\014)q Fh(j)p Fr(\003)p Fh(j)620 948 y Fo(X)593 1039 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)750 948 y Fo(X)713 1039 y Fr(\000)p Fh(\022)p Fr(\001)787 1043 y Fg(\003)808 1039 y Fh([;)859 959 y Fp(p)880 937 y Fh(j)p Fr(\000)p Fh(j)880 971 y Fr(FK)936 959 y Fv(\()p Fp(\014)r Fv(\))8 b Fp(q)1021 937 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1020 971 y Fr(FK)1156 959 y Fv(\()p Fp(\014)r Fv(\))p 859 978 V 1005 1017 a(2)1026 1005 y Fh(j)o Fr(\000)p Fh(j)1225 904 y Fo(0)1225 979 y(@)1262 987 y Fv(1)h(+)1355 948 y Fo(X)1333 1039 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1444 987 y Fv(\()p Fq(\000)p Fv(1\))1529 966 y Fh(j)p Fs(\013)p Fh(\\4)1613 970 y Fd(\015)q Ff(2)p Fd(c)1679 966 y Fs(\015)r Fh(j)1710 904 y Fo(1)1710 979 y(A)411 1153 y Fv(=)42 b(2)506 1136 y Fh(j)o Fr(\003)p Fh(j)550 1153 y Fp(e)569 1136 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)695 1114 y Fo(X)658 1205 y Fr(\000)p Fh(\022)p Fr(\001)732 1209 y Fg(\003)753 1205 y Fh([;)799 1153 y Fp(p)820 1132 y Fh(j)p Fr(\000)p Fh(j)820 1166 y Fr(FK)876 1153 y Fv(\()p Fp(\014)r Fv(\))8 b Fp(q)961 1132 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)o Fr(\000)p Fh(j)960 1166 y Fr(FK)1096 1153 y Fv(\()p Fp(\014)r Fv(\))g(2)1182 1136 y Fh(\000j)o Fr(\000)p Fh(j)1257 1070 y Fo(2)1257 1145 y(4)1284 1153 y Fv(1)h(+)1378 1114 y Fo(X)1356 1205 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1466 1153 y Fp(\016)g Fv(\()p Fq(4)1546 1159 y Fs(\015)r Fh(2)p Fs(c)1619 1153 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))1735 1070 y Fo(3)1735 1145 y(5)275 1153 y Fv(\(12\))228 1291 y(whic)o(h)i(is,)f(in)h(a) g(similar)d(fashion)i(to)h(\(9\),)g(a)g(sum)f(with)g(Bernoulli)h(w)o(eigh)o (ts)g(on)g(the)g(elemen)o(ts)228 1341 y(of)f Fc(P)309 1347 y Fr(\001)336 1351 y Fg(\003)374 1341 y Fv(whose)h(terms)g(in)o(v)o(olv)o(e)e (closed)j(lo)q(ops)e(in)h Fc(P)1064 1347 y Fr(\003)1089 1341 y Fv(.)228 1438 y Fy(Remark)h(3.4.)21 b Fn(T)m(o)14 b(obtain)g(the)g(ful)r(l) f(e)n(quivalenc)n(e)h(of)g(\()7 b Fv(12)p Fn(\))13 b(with)g(the)h(FK)g(r)n (epr)n(esentation)f(of)228 1488 y(the)f(mo)n(del)f(we)h(ne)n(e)n(d)g(to)g (give)f(some)h(mor)n(e)g(de\014nitions.)18 b Fq(8)p Fv(\000)12 b Fq(2)f Fc(P)1243 1494 y Fr(\001)1270 1498 y Fg(\003)1305 1488 y Fn(two)h(lattic)n(e)f(sites)g Fp(x;)c(y)13 b Fn(ar)n(e)228 1538 y(said)h(to)g(b)n(e)g(c)n(onne)n(cte)n(d)h(in)f Fv(\000)g Fn(if)f(either)g Fp(x;)7 b(y)13 b Fq(2)e Fp(\015)j Fv(:)d Fp(\015)k Fq(2)c Fv(\000)j Fn(or)g Fq(f)o Fp(x;)7 b(y)q Fq(g)12 b Fv(=)f Fq(4)1384 1544 y Fs(\015)r Fh(2)p Fs(c)1457 1538 y Fp(\015)16 b Fn(for)e(some)g Fp(c)e Fq(2)228 1588 y Fc(P)262 1594 y Fr(\000)285 1588 y Fn(.)17 b(We)c(wil)r(l)d(denote)i(by)g Fc(C)668 1594 y Fr(\000)703 1588 y Fn(the)f(set)h(of)f(c)n(onne)n(cte)n(d)i(c)n(omp)n (onents)g(of)e Fv(\000)h Fn(and)g(by)g Fp(C)1522 1594 y Fr(\000)1556 1588 y Fn(e)n(ach)g(c)n(on-)228 1637 y(ne)n(cte)n(d)k(c)n(omp)n(onent)h(of)e Fv(\000)h Fn(\()p Fv(\000)p Fn(-clusters\).)21 b(The)16 b(c)n(o)n(e\016cient) f Fv(2)1202 1622 y Fs(N)1228 1626 y Fg(\000)1265 1637 y Fn(app)n(e)n(aring)i (in)i Fv([FK])p Fn(,)c(wher)n(e)228 1700 y Fp(N)261 1706 y Fr(\000)296 1700 y Fn(is)c(the)i(numb)n(er)f(of)g(cluster)f(in)h Fv(\000)p Fn(,)g(is)g(then)h(just)e Fv(2)1055 1685 y Fh(j)p Fr(\003)p Fh(j\000j)o Fr(\000)p Fh(j)1173 1654 y Fo(h)1192 1700 y Fv(1)e(+)1264 1668 y Fo(P)1308 1712 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1420 1700 y Fp(\016)g Fv(\()q Fq(4)1501 1706 y Fs(\015)r Fh(2)p Fs(c)1574 1700 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))1690 1654 y Fo(i)1709 1700 y Fn(.)228 1762 y(This)i(c)n(an)i(b)n (e)f(e)n(asily)f(che)n(cke)n(d)i(c)n(omputing)f(\()7 b Fv(12)p Fn(\))14 b(for)g(simple)h(gr)n(aphs)g(in)g(b)n(oth)g(ways.)278 1859 y Fv(Since)382 1947 y Fo(X)345 2038 y Fr(\000)p Fh(\022)p Fr(\001)419 2042 y Fg(\003)439 2038 y Fh([;)486 1987 y Fp(p)507 1965 y Fh(j)p Fr(\000)p Fh(j)507 1999 y Fr(FK)562 1987 y Fv(\()q Fp(\014)r Fv(\))7 b Fp(q)647 1965 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)646 1999 y Fr(FK)782 1987 y Fv(\()p Fp(\014)r Fv(\))13 b(=)907 1931 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)907 1947 y Fo(X)896 2038 y Fh(j)o Fr(\000)p Fh(j)p Fr(=0)985 1928 y Fo(\022)1036 1962 y Fp(d)7 b Fq(j)o Fv(\003)p Fq(j)1052 2011 y(j)o Fv(\000)p Fq(j)1137 1928 y Fo(\023)1175 1987 y Fp(p)1196 1965 y Fh(j)o Fr(\000)p Fh(j)1196 1999 y Fr(FK)1251 1987 y Fv(\()q Fp(\014)r Fv(\))g Fp(q)1336 1965 y Fs(d)p Fh(j)p Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)1335 1999 y Fr(FK)1471 1987 y Fv(\()p Fp(\014)r Fv(\))13 b(=)e(1)228 2123 y(w)o(e)f(can)h(build)f(on)g(\001)550 2129 y Fr(\003)585 2123 y Fv(a)g Fn(b)n(ond)j(p)n(er)n(c)n(olation)e(pr)n(o)n (c)n(ess)g Fv(with)f Fn(b)n(ond)i(pr)n(ob)n(ability)d Fp(p)1455 2129 y Fr(FK)1511 2123 y Fv(\()p Fp(\014)r Fv(\))i(de\014ning)228 2173 y(the)j(exp)q(ectation)594 2271 y Fp(M)639 2254 y Fr(FK)634 2282 y Fs(\014)q(;)p Fr(\003)696 2271 y Fv([)p Fp(f)t Fv(])d(:=)848 2232 y Fo(X)811 2323 y Fr(\000)p Fh(\022)p Fr(\001)885 2327 y Fg(\003)906 2323 y Fh([;)952 2271 y Fp(p)973 2250 y Fh(j)p Fr(\000)p Fh(j)973 2284 y Fr(FK)1029 2271 y Fv(\()p Fp(\014)r Fv(\))c Fp(q)1113 2250 y Fs(d)p Fh(j)q Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1112 2284 y Fr(FK)1248 2271 y Fv(\()p Fp(\014)r Fv(\))h Fp(f)1333 2277 y Fr(\000)228 2404 y Fv(for)13 b(an)o(y)h(random)e(v)n (ariable)h Fp(f)j Fv(:)11 b(\001)772 2410 y Fr(\003)808 2404 y Fq(!)g Ft(R)m Fv(..)k(Th)o(us)425 2543 y Fp(Z)456 2522 y Fr(\()p Fs(d)p Fr(\))453 2556 y(\003)508 2543 y Fv(\()q Fp(\014)r Fv(\))d(=)g Fp(e)641 2526 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)723 2543 y Fp(M)768 2526 y Fr(FK)763 2554 y Fs(\014)q(;)p Fr(\003)825 2460 y Fo(2)825 2535 y(4)853 2543 y Fv(2)874 2526 y Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)991 2460 y Fo(2)991 2535 y(4)1019 2543 y Fv(1)d(+)1113 2504 y Fo(X)1091 2595 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1201 2543 y Fp(\016)g Fv(\()p Fq(4)1281 2549 y Fs(\015)r Fh(2)p Fs(c)1354 2543 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))1470 2460 y Fo(3)1470 2535 y(5)1497 2460 y(3)1497 2535 y(5)p eop %%Page: 11 11 bop 657 142 a Fx(QUANTUM)18 b(METHODS)f(F)o(OR)f(IPS'S)e(I)q(I)q(I)393 b(11)278 250 y Fv(\(6\))13 b(can)i(b)q(e)f(replaced)h(b)o(y)230 401 y(\026)228 402 y Fm(\032)253 381 y Fr(\()p Fs(d)p Fr(\))253 415 y(\003)305 402 y Fv(\()q Fp(\014)r Fv(\))42 b(=)505 363 y Fo(X)479 454 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)641 328 y Fo(P)685 372 y Fr(\000)p Fh(\022)p Fr(\001)759 376 y Fg(\003)780 372 y Fh([;)828 359 y Fp(p)849 338 y Fh(j)p Fr(\000)p Fh(j)849 371 y Fr(FK)905 359 y Fv(\()p Fp(\014)r Fv(\))8 b Fp(q)990 338 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)989 371 y Fr(FK)1125 359 y Fv(\()p Fp(\014)r Fv(\))f(2)1210 344 y Fh(\000j)p Fr(\000)p Fh(j)1285 313 y Fo(h)1305 359 y Fv(1)i(+)1376 328 y Fo(P)1420 372 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1533 359 y Fv(\()p Fq(\000)p Fv(1\))1618 338 y Fh(j)p Fs(\013)p Fh(\\4)1702 342 y Fd(\015)q Ff(2)p Fd(c)1768 338 y Fs(\015)r Fh(j)1799 313 y Fo(i)p 604 393 1253 2 v 604 446 a Fv(2)625 434 y Fh(j)o Fr(\003)p Fh(j)676 414 y Fo(P)720 458 y Fr(\000)p Fh(\022)p Fr(\001)794 462 y Fg(\003)815 458 y Fh([;)863 446 y Fp(p)884 424 y Fh(j)p Fr(\000)p Fh(j)884 458 y Fr(FK)940 446 y Fv(\()p Fp(\014)r Fv(\))e Fp(q)1024 424 y Fs(d)p Fh(j)q Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)1023 458 y Fr(FK)1159 446 y Fv(\()q Fp(\014)r Fv(\))g(2)1245 434 y Fh(\000j)p Fr(\000)p Fh(j)1320 399 y Fo(h)1340 446 y Fv(1)i(+)1411 414 y Fo(P)1455 458 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1568 446 y Fp(\016)g Fv(\()p Fq(4)1648 452 y Fs(\015)r Fh(2)p Fs(c)1720 446 y Fp(\015)15 b Fv(=)c Fq(;)p Fv(\))1836 399 y Fo(i)1868 402 y Fq(j)o Fp(\013)p Fq(i)c(h)p Fp(\013)p Fq(j)1983 415 y Fr(\003)2020 402 y Fv(=)405 580 y(=)505 541 y Fo(X)479 632 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)626 537 y Fp(M)671 522 y Fr(FK)666 549 y Fs(\014)q(;)p Fr(\003)728 491 y Fo(h)748 537 y Fv(2)769 522 y Fh(\000j)p Fr(\000)p Fh(j)844 491 y Fo(h)864 537 y Fv(1)i(+)935 506 y Fo(P)979 549 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1092 537 y Fv(\()p Fq(\000)p Fv(1\))1177 516 y Fh(j)p Fs(\013)p Fh(\\4)1261 520 y Fd(\015)q Ff(2)p Fd(c)1326 516 y Fs(\015)r Fh(j)1357 491 y Fo(ii)p 604 571 816 2 v 604 623 a Fp(M)649 609 y Fr(FK)644 636 y Fs(\014)q(;)p Fr(\003)706 577 y Fo(h)726 623 y Fv(2)747 611 y Fh(j)o Fr(\003)p Fh(j\000j)p Fr(\000)p Fh(j)864 577 y Fo(h)884 623 y Fv(1)f(+)955 592 y Fo(P)999 636 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1112 623 y Fp(\016)h Fv(\()p Fq(4)1192 629 y Fs(\015)r Fh(2)p Fs(c)1264 623 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\))1380 577 y Fo(ii)1431 580 y Fq(j)p Fp(\013)p Fq(i)6 b(h)q Fp(\013)p Fq(j)1547 593 y Fr(\003)228 730 y Fv(and)14 b Fq(8)p Fm(\033)362 736 y Fs(\021)393 730 y Fq(2)e(C)455 736 y Fr(\003)479 730 y Fv(;)i Fp(\021)e Fq(\022)g Fv(\003)i(w)o(e)g(get)228 884 y Fp(!)255 864 y Fs(\014)254 896 y Fr(\003)285 884 y Fv(\()p Fm(\033)332 890 y Fs(\021)352 884 y Fv(\))41 b(=)488 810 y Fo(P)532 853 y 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Fs(\015)r Fh(2)p Fs(c)1577 1019 y Fp(\015)15 b Fv(=)c Fp(\021)q Fv(\))1695 973 y Fo(i)p 488 1052 1227 2 v 511 1074 a(P)555 1117 y Fr(\000)p Fh(\022)p Fr(\001)629 1121 y Fg(\003)650 1117 y Fh([;)698 1105 y Fp(p)719 1083 y Fh(j)p Fr(\000)p Fh(j)719 1117 y Fr(FK)775 1105 y Fv(\()p Fp(\014)r Fv(\))d Fp(q)860 1083 y Fs(d)p Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)859 1117 y Fr(FK)995 1105 y Fv(\()p Fp(\014)r Fv(\))f(2)1080 1093 y Fh(\000j)p Fr(\000)p Fh(j)1155 1059 y Fo(h)1175 1105 y Fv(1)i(+)1247 1074 y Fo(P)1290 1117 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1403 1105 y Fp(\016)g Fv(\()p Fq(4)1483 1111 y Fs(\015)r Fh(2)p Fs(c)1555 1105 y Fp(\015)15 b Fv(=)d Fq(;)p Fv(\))1671 1059 y Fo(i)1731 1062 y Fv(=)409 1240 y(=)488 1196 y Fp(M)533 1181 y Fr(FK)528 1208 y Fs(\014)q(;)p Fr(\003)590 1150 y Fo(h)610 1196 y Fv(2)631 1181 y Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)p Fh(j)748 1150 y Fo(h)768 1196 y Fp(\016)786 1203 y Fs(\021)q(;)p Fh(;)843 1196 y Fv(+)884 1165 y Fo(P)928 1209 y Fh(;6)p 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1499 y Fb(FK)946 1493 y Fv(\()p Fp(\014)r Fv(\))q Fn(.)k(Ther)n(efor)n(e)443 1598 y Fp(Z)474 1576 y Fr(\()p Fs(d)p Fr(\))471 1610 y(\003)526 1598 y Fv(\()p Fp(\014)r Fv(\))43 b(=)e(2)720 1581 y Fh(j)p Fr(\003)p Fh(j)765 1598 y Fp(e)784 1581 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)866 1598 y Fp(M)911 1581 y Fb(HT)906 1608 y Fs(\014)q(;)p Fr(\003)969 1598 y Fv([)p Fp(\016)9 b Fv(\()p Fq(4)1061 1604 y Fs(\015)r Fh(2)p Fr(\000)1140 1598 y Fp(\015)14 b Fv(=)e Fq(;)p Fv(\)])451 1720 y(\026)449 1721 y Fm(\032)474 1700 y Fr(\()p Fs(d)p Fr(\))474 1734 y(\003)526 1721 y Fv(\()p Fp(\014)r Fv(\))43 b(=)726 1682 y Fo(X)699 1773 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)851 1678 y Fp(M)896 1663 y Fb(HT)891 1690 y Fs(\014)q(;)p Fr(\003)955 1632 y Fo(h)974 1678 y Fv(\()p Fq(\000)p Fv(1\))1060 1657 y Fh(j)o Fs(\013)p Fh(\\4)1143 1661 y Fd(\015)q Ff(2)p Fg(\000)1214 1657 y Fs(\015)r Fh(j)1245 1632 y Fo(i)p 825 1712 467 2 v 825 1751 a Fv(2)846 1739 y Fh(j)o Fr(\003)p Fh(j)890 1751 y Fp(M)935 1736 y Fb(HT)930 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2071 y Fo(Y)228 2160 y Fs(\015)r Fh(2)p Fr(\001)296 2164 y Fg(\003)325 2065 y Fo(h)o(\020)369 2111 y Fv(1)9 b(+)g(\()q Fq(\000)p Fv(1\))526 2090 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)611 2065 y Fo(\021)642 2111 y Fp(p)663 2117 y Fb(HT)722 2111 y Fv(\()p Fp(\014)r Fv(\))h(+)g Fp(q)850 2117 y Fb(HT)908 2111 y Fv(\()p Fp(\014)r Fv(\))g Fq(\000)f Fp(p)1037 2117 y Fb(HT)1096 2111 y Fv(\()p Fp(\014)r Fv(\))1154 2065 y Fo(i)1185 2111 y Fv(=)1247 2071 y Fo(Y)1229 2160 y Fs(\015)r Fh(2)p Fr(\001)1297 2164 y Fg(\003)1325 2065 y Fo(h)1345 2111 y Fv(\()p Fq(\000)p Fv(1\))1430 2090 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\015)r Fh(j)1522 2111 y Fp(p)1543 2117 y Fb(HT)1602 2111 y Fv(\()p Fp(\014)r Fv(\))h(+)f Fp(q)1729 2117 y Fb(HT)1787 2111 y Fv(\()p Fp(\014)r Fv(\))1845 2065 y Fo(i)228 2258 y Fn(multiplying)14 b(by)h Fp(e)516 2243 y Fs(d\014)q Fh(j)q Fr(\003)p Fh(j)614 2258 y Fn(and)g(summing)g(over)g(al)r(l) f Fq(;)e(\022)g Fp(\013)f Fq(\022)h Fv(\003)i Fn(we)h(get)g(\()7 b Fv(9)p Fn(\).)278 2370 y Fv(The)14 b(discussion)g(carried)h(on)f(in)f(this) h(section)h(can)f(b)q(e)g(summarized)e(b)o(y)i(the)g(follo)o(wing)228 2481 y Fy(Theorem)h(3.1.)21 b Fn(If)11 b Fv(\003)g Fn(is)g(any)h(\014nite)g (subset)g(of)f Ft(Z)1023 2466 y Fs(d)1039 2481 y Fn(,)h(assuming)g(p)n(erio)n (dic)f(b.c.,)g(we)g(c)n(an)h(derive)228 2531 y(HT,)j(L)m(T)g(and)h(FK)g(r)n (epr)n(esentation)f(for)g(the)h(statistic)n(al)e(me)n(chanics)i(of)g(the)f (Ising)h(mo)n(del)g(on)228 2581 y Ft(Z)258 2566 y Fr(\003)258 2591 y(2)280 2581 y Fn(,)f(in)g(absenc)n(e)g(of)g(external)g(magnetic)g (\014elds,)g(one)g(fr)n(om)g(the)g(other.)p eop %%Page: 12 12 bop 228 142 a Fx(12)570 b(M.)15 b(GIANFELICE)228 250 y Fv(3.1.3.)k Fn(Pr)n(o)n(of.)h Fv(F)m(rom)8 b Fy(Remark)j(3.2)e Fv(\(9\))h(=)-7 b Fq(\))10 b Fv(\(5\);)g(\(5\))g(=)-7 b Fq(\))10 b Fv(\(12\))f(follo)o(ws)f (from)g(\(11\).)17 b(Finally)228 300 y(from)12 b Fy(Remark)j(3.5)f Fv(\(12\))g(=)-7 b Fq(\))14 b Fv(\(9\).)p 829 300 31 31 v 278 369 a(So)21 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Fs(\014)q(J)1449 1181 y Fd(\015)1471 1192 y Fv(,)f Fp(p)1515 1170 y Fh(j)p Fr(\000)p Fh(j)1515 1204 y Fr(HT)1572 1192 y Fv(\()p Fp(\014)r Fv(\))h(with)228 1232 y Fo(Q)267 1276 y Fs(\015)r Fh(2)p Fr(\000)338 1218 y Fo(\020)368 1247 y Fr(1)p Fh(\000)p Fs(e)427 1235 y Ff(\000)p Fg(2)p Fd(\014)q(J)496 1239 y(\015)p 368 1254 150 2 v 434 1278 a Fr(2)522 1218 y Fo(\021)561 1264 y Fv(and)h Fp(p)662 1242 y Fh(j)p Fr(\000)p Fh(j)662 1276 y Fr(FK)718 1264 y Fv(\()p Fp(\014)r Fv(\))h(with)883 1232 y Fo(Q)923 1276 y Fs(\015)r Fh(2)p Fr(\000)994 1230 y Fo(\000)1013 1264 y Fv(1)9 b Fq(\000)g Fp(e)1103 1249 y Fh(\000)p Fr(2)p Fs(\014)q(J)1184 1253 y Fd(\015)1206 1230 y Fo(\001)1238 1264 y Fv(resp)q(ectiv)o(ely)15 b(in)e(the)i(L)m(T,)d(HT)228 1338 y(and)18 b(FK)h(expressions)i(for)d Fp(Z)712 1317 y Fr(\()p Fs(d)p Fr(\))709 1350 y(\003)765 1338 y Fv(\()p Fp(\014)r Fv(\),)i Fm(\032)879 1348 y Fr(\003)911 1338 y Fv(\()p Fp(\014)r Fv(\))g(and)e Fp(!)1100 1318 y Fs(\014)1099 1350 y Fr(\003)1131 1338 y Fv(\()p Fm(\033)1177 1344 y Fs(\013)1200 1338 y Fv(\))h Fq(8)p Fm(\033)1288 1344 y Fs(\013)1332 1338 y Fq(2)g(C)1401 1344 y Fr(\003)1425 1338 y Fv(;)7 b Fp(\013)19 b Fq(\022)h Fv(\003)e(already)228 1388 y(computed)f(for)h(the)g(Ising)g(mo)q(del.)28 b(As)18 b(a)g(further)g(example)f(of)g(the)h(applicabilit)o(y)e(of)h(this)228 1438 y(framew)o(ork)f(w)o(e)i(compute)f(FK)h(and)g(HT)g(expansions)g(for)f (the)i(partition)e(function)g(of)h(the)228 1488 y(Ising)c(mo)q(del)f(in)i(a)f (constan)o(t)h(magnetic)e(\014eld)i(\(7\).)21 b(W)m(e)14 b(b)q(egin)h(in)o (tro)q(ducing)f(the)h Fn(ghost)h(spin)228 1537 y(\(site\))d Fv([Gr)o(])e([FK])g([S)j(])d Fq(f)p Fv(0)p Fq(g)g Fv(in)o(teracting)h(with)f (all)g Fp(x)g Fq(2)g Fv(\003)h(trough)g(the)g(Hamiltonian)d(op)q(erator)733 1744 y Fy(H)770 1727 y Fr(0)770 1754 y(\003)802 1744 y Fv(\()p Fp(h)p Fv(\))j(:=)f Fq(\000)p Fp(h)990 1704 y Fo(X)988 1793 y Fs(x)p Fh(2)p Fr(\003)1059 1744 y Fm(\033)1089 1751 y Fh(f)p Fs(x)p Fh(g[f)p Fr(0)p Fh(g)228 1983 y Fv(Then)g(w)o(e)f(de\014ne)h(the)g (graph)f Fp(G)721 1989 y Fr(0)751 1983 y Fv(:=)807 1950 y Fo(\000)826 1983 y Fv(\003)f Fq([)g(f)o Fv(0)p Fq(g)e Fp(;)g Fv(\001)1024 1968 y Fr(0)1024 1995 y(\003)1048 1950 y Fo(\001)1077 1983 y Fv(where)k(\001)1228 1968 y Fr(0)1228 1995 y(\003)1264 1983 y Fv(:=)h(\001)1355 1989 y Fr(\003)1381 1983 y Fq([)r(f)o(f)p Fv(0)p Fp(;)7 b(x)p Fq(g)f Fv(;)20 b Fq(8)p Fp(x)11 b Fq(2)h Fv(\003)p Fq(g)228 2033 y Fv(and)h(\001)343 2018 y Fr(0)343 2044 y(\003)379 2033 y Fq(3)e Fp(\015)439 2039 y Fr(0)470 2033 y Fv(:=)g Fp(\015)g Fq([)d(f)p Fv(0)p Fp(;)f(x)p Fq(g)e Fv(;)20 b Fq(8)p Fp(x)11 b Fq(2)h Fv(\003)p Fp(;)19 b(\015)14 b Fq(2)e Fv(\001)1004 2039 y Fr(\003)1028 2033 y Fv(.)18 b(Therefore)c(an)o(y)f (collection)g(of)g(dimers)g(in)228 2083 y(\001)263 2068 y Fr(0)263 2094 y(\003)298 2083 y Fv(is)e(no)o(w)g(giv)o(en)f(b)o(y)h(\000)607 2089 y Fr(0)637 2083 y Fv(=)h(\000)s Fq([)s(ff)p Fv(0)p Fp(;)7 b(x)p Fq(g)e Fv(;)21 b Fq(8)p Fp(x)11 b Fq(2)g Fp(\021)q Fq(g)g(8)p Fv(\000)g Fq(2)h Fv(\001)1192 2089 y Fr(\003)1216 2083 y Fp(;)17 b(\021)c Fq(\022)f Fv(\003)f(with)f Fq(j)p Fv(\000)1492 2089 y Fr(0)1510 2083 y Fq(j)i Fv(=)f Fq(j)p Fv(\000)p Fq(j)s Fv(+)s Fq(j)p Fp(\021)q Fq(j)p Fv(.)228 2142 y(In)g(the)g(same)f(fashion)g(to)h Fp(M)678 2120 y Fr(FK\(HT\))673 2154 y Fs(\014)q(;)p Fr(\003)812 2142 y Fv(w)o(e)g(de\014ne)h Fp(M)1032 2120 y Fr(FK\(HT\))1027 2154 y Fs(\014)q(;h;)p Fr(\003)1165 2142 y Fv(as)f(the)h(exp)q(ectation)f (with)g(resp)q(ect)228 2198 y(to)j(the)g Fn(b)n(ond)i(p)n(er)n(c)n(olation)d Fv(pro)q(cess)j(de\014ned)f(on)e(\001)1036 2183 y Fr(0)1036 2209 y(\003)1061 2198 y Fv(.)18 b(Since)244 2430 y Fp(e)263 2413 y Fh(\000)p Fr(2)p Fs(\014)q(h)p Fh(j)q Fs(\013)p Fh(j)431 2430 y Fv(=)41 b Fp(e)523 2408 y Fh(\000)p Fr(2)p Fs(\014)q(h)612 2380 y Fo(P)655 2424 y Fd(x)p Ff(2)p Fg(\003)718 2408 y Fs(\016)q Fr(\()p Fs(x)p Fh(2)p Fs(\013)p Fr(\))837 2430 y Fv(=)886 2391 y Fo(Y)880 2480 y Fs(x)p Fh(2)p Fr(\003)951 2359 y Fo(")980 2402 y Fv(1)9 b(+)h(\()p Fq(\000)p Fv(1\))1137 2381 y Fh(j)p Fs(\013)p Fh(\\f)p Fs(x)p Fh(gj)p 980 2421 276 2 v 1108 2459 a Fv(2)1261 2430 y Fp(p)1282 2436 y Fr(FK)1338 2430 y Fv(\()p Fp(\014)r(h)p Fv(\))g(+)f Fp(q)1489 2436 y Fr(FK)1544 2430 y Fv(\()q Fp(\014)r(h)p Fv(\))1626 2359 y Fo(#)1662 2430 y Fv(=)431 2546 y(=)529 2507 y Fo(X)504 2598 y Fh(;\022)p Fs(\021)q Fh(\022)p Fr(\003)621 2546 y Fp(p)642 2525 y Fh(j)p Fs(\021)q Fh(j)642 2559 y Fr(FK)698 2546 y Fv(\()p Fp(\014)r(h)p Fv(\))f Fp(q)807 2525 y Fh(j)o Fr(\003)p Fh(j\000j)p Fs(\021)q Fh(j)806 2559 y Fr(FK)922 2546 y Fv(\()p Fp(\014)r(h)p Fv(\))1041 2507 y Fo(X)1011 2598 y Fh(;\022)p Fs(\021)1071 2602 y Fg(1)1087 2598 y Fh(\022)p Fs(\021)1138 2546 y Fv(\()p Fq(\000)p Fv(1\))1223 2526 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\021)1294 2530 y Fg(1)1310 2526 y Fh(j)p eop %%Page: 13 13 bop 657 142 a Fx(QUANTUM)18 b(METHODS)f(F)o(OR)f(IPS'S)e(I)q(I)q(I)393 b(13)228 250 y Fv(from)12 b(\(12\))i(w)o(e)g(ha)o(v)o(e)267 347 y Fp(Z)298 326 y Fr(\()q Fs(d)p Fr(\))295 360 y(\003)351 347 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))42 b(=)f(2)587 330 y Fh(j)p Fr(\003)p Fh(j)632 347 y Fp(e)651 330 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)p Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))831 319 y Fv(1)p 809 338 66 2 v 809 377 a(2)830 365 y Fh(j)p Fr(\003)p Fh(j)913 308 y Fo(X)886 399 y Fh(;\022)p Fs(\013)p Fh(\022)p Fr(\003)1006 347 y Fp(e)1025 330 y Fh(\000)p Fr(2)p Fs(\014)q Fh(j)p Fs(@)r(\013)p Fh(j)o(\000)p Fr(2)p Fs(\014)q(h)p Fh(j)p Fs(\013)p Fh(j)1286 347 y Fv(=)493 510 y(=)g(2)587 493 y Fh(j)p Fr(\003)p Fh(j)632 510 y Fp(e)651 493 y Fs(\014)q Fh(j)p Fr(\003)p Fh(j)p Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))804 510 y Fp(M)849 493 y Fr(FK)844 521 y Fs(\014)q(;)p Fr(\003)906 427 y Fo(2)906 502 y(4)959 471 y(X)934 562 y Fh(;\022)p Fs(\021)q Fh(\022)p Fr(\003)1051 510 y Fp(p)1072 489 y Fh(j)o Fs(\021)q Fh(j)1072 522 y Fr(FK)1127 510 y Fv(\()p Fp(\014)r(h)p Fv(\))8 b Fp(q)1236 489 y Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fs(\021)q Fh(j)1235 522 y Fr(FK)1351 510 y Fv(\()p Fp(\014)r(h)p Fv(\))g(2)1461 493 y Fh(\000)p Fr(\()p Fh(j)p Fr(\000)p Fh(j)o Fr(+)p Fh(j)p Fs(\021)q Fh(j)p Fr(\))1633 510 y Fq(\002)580 676 y(\002)624 593 y Fo(0)624 668 y(@)691 637 y(X)661 728 y Fh(;\022)p Fs(\021)721 732 y Fg(1)737 728 y Fh(\022)p Fs(\021)788 676 y Fp(\016)806 683 y Fs(\021)823 687 y Fg(1)840 683 y Fs(;)p Fh(;)878 676 y Fv(+)941 637 y Fo(X)920 728 y Fh(;6)p Fr(=)p Fs(c)p Fh(\022)p Fr(\000)1060 637 y Fo(X)1030 728 y Fh(;\022)p Fs(\021)1090 732 y Fg(1)1106 728 y Fh(\022)p Fs(\021)1157 676 y Fp(\016)h Fv(\()q Fq(4)1238 682 y Fs(\015)r Fh(2)p Fs(c)1310 676 y Fp(\015)14 b Fv(=)e Fp(\021)q Fv(\))1427 593 y Fo(1)1427 668 y(A)1464 593 y(3)1464 668 y(5)493 842 y Fv(=)41 b Fp(e)585 825 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)p Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))739 842 y Fp(M)784 825 y Fr(FK)779 853 y Fs(\014)q(;h;)p Fr(\003)870 759 y Fo(2)870 834 y(4)898 842 y Fv(2)919 825 y Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fr(\000)1018 829 y Fg(0)1034 825 y Fh(j)1052 759 y Fo(0)1052 834 y(@)1089 842 y Fv(1)9 b(+)1198 803 y Fo(X)1160 894 y Fh(;6)p Fr(=)p Fs(c)1217 898 y Fg(0)1234 894 y Fh(\022)p Fr(\000)1281 898 y Fg(0)1303 842 y Fp(\016)g Fv(\()p Fq(4)1383 848 y Fs(\015)1400 852 y Fg(0)1417 848 y Fh(2)p Fs(c)1454 852 y Fg(0)1486 842 y Fp(\015)1507 848 y Fr(0)1538 842 y Fv(=)j Fq(;)p Fv(\))1619 759 y Fo(1)1619 834 y(A)1655 759 y(3)1655 834 y(5)228 970 y Fv(In)i(the)g(same)f(w)o(a)o(y)g(w)o(e)h(can)h(compute)e (the)h(HT)g(expansion)g(for)g(\(7\))352 1063 y Fp(e)371 1041 y Fs(\014)q(h)417 1013 y Fo(P)461 1057 y Fd(x)p Ff(2)p Fg(\003)518 1041 y Fr(\()p Fh(\000)p Fr(1\))586 1028 y Ff(j)p Fd(\013)p Ff(\\f)p Fd(x)p Ff(gj)737 1063 y Fv(=)816 1024 y Fo(Y)810 1113 y Fs(x)p Fh(2)p Fr(\003)881 1017 y Fo(h)901 1063 y Fv(cosh)7 b(\()q Fp(\014)r(h)p Fv(\))i(+)h(\()p Fq(\000)p Fv(1\))1204 1042 y Fh(j)p Fs(\013)p Fh(\\f)p Fs(x)p Fh(gj)1330 1063 y Fv(sinh)d(\()p Fp(\014)r(h)p Fv(\))1493 1017 y Fo(i)1524 1063 y Fv(=)737 1179 y(=)41 b Fp(e)829 1162 y Fs(\014)q(h)p Fh(j)q Fr(\003)p Fh(j)946 1140 y Fo(X)921 1231 y Fh(;\022)p Fs(\021)q Fh(\022)p Fr(\003)1038 1179 y Fp(p)1059 1158 y Fh(j)o Fs(\021)q Fh(j)1059 1192 y Fr(HT)1116 1179 y Fv(\()p Fp(\014)r(h)p Fv(\))7 b Fp(q)1224 1158 y Fh(j)p Fr(\003)p Fh(j\000j)p Fs(\021)q Fh(j)1223 1192 y Fr(HT)1340 1179 y Fv(\()p Fp(\014)r(h)p Fv(\))g(\()q Fq(\000)p Fv(1\))1514 1158 y Fh(j)p Fs(\013)p Fh(\\)p Fs(\021)q Fh(j)228 1304 y Fv(th)o(us)14 b(from)e(\(9\))228 1432 y Fp(Z)259 1410 y Fr(\()p Fs(d)p Fr(\))256 1444 y(\003)311 1432 y Fv(\()p Fp(\014)r(;)7 b(h)p Fv(\))42 b(=)g(2)548 1415 y Fh(j)p Fr(\003)p Fh(j)592 1432 y Fp(e)611 1415 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)o Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))765 1432 y Fp(M)810 1415 y Fr(FK)805 1442 y Fs(\014)q(;)p Fr(\003)867 1348 y Fo(2)867 1423 y(4)919 1392 y(X)894 1483 y Fh(;\022)p Fs(\021)q Fh(\022)p Fr(\003)1011 1432 y Fp(p)1032 1410 y Fh(j)p Fs(\021)q Fh(j)1032 1444 y Fr(HT)1089 1432 y Fv(\()p Fp(\014)r(h)p Fv(\))8 b Fp(q)1198 1410 y Fh(j)p Fr(\003)p Fh(j)o(\000j)p Fs(\021)q Fh(j)1197 1444 y Fr(HT)1313 1432 y Fv(\()p Fp(\014)r(h)p Fv(\))g Fp(\016)h Fv(\()p Fq(4)1482 1438 y Fs(\015)r Fh(2)p Fr(\000)1560 1432 y Fp(\015)14 b Fv(=)e Fp(\021)q Fv(\))1677 1348 y Fo(3)1677 1423 y(5)1717 1432 y Fv(=)453 1561 y(=)42 b(2)548 1543 y Fh(j)p Fr(\003)p Fh(j)592 1561 y Fp(e)611 1543 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)o Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))765 1561 y Fp(M)810 1543 y Fr(FK)805 1571 y Fs(\014)q(;)p Fr(\003)867 1515 y Fo(h)886 1561 y Fp(p)907 1538 y Fh(j4)947 1542 y Fd(\015)q Ff(2)p Fg(\000)1016 1538 y Fs(\015)r Fh(j)907 1573 y Fr(HT)1054 1561 y Fv(\()p Fp(\014)r(h)p Fv(\))8 b Fp(q)1163 1538 y Fh(j)p Fr(\003)p Fh(j\000j)o(4)1271 1542 y Fd(\015)q Ff(2)p Fg(\000)1341 1538 y Fs(\015)r Fh(j)1162 1573 y Fr(HT)1379 1561 y Fv(\()p Fp(\014)r(h)p Fv(\))1461 1515 y Fo(i)1492 1561 y Fv(=)453 1643 y(=)42 b(2)548 1626 y Fh(j)p Fr(\003)p Fh(j)592 1643 y Fp(e)611 1626 y Fs(\014)q Fh(j)q Fr(\003)p Fh(j)o Fr(\()p Fs(d)p Fr(+)p Fs(h)p Fr(\))765 1643 y Fp(M)810 1626 y Fr(FK)805 1653 y Fs(\014)q(;h;)p Fr(\003)896 1643 y Fv([)p Fp(\016)8 b Fv(\()q Fq(4)988 1649 y Fs(\015)1005 1653 y Fg(0)1021 1649 y Fh(2)p Fr(\000)1064 1653 y Fg(0)1096 1643 y Fp(\015)1117 1649 y Fr(0)1148 1643 y Fv(=)k Fq(;)p Fv(\))o(])818 1737 y(4.)23 b Fu(Conclusions)278 1813 y Fv(Through)12 b(all)f(this)i(w)o (ork)f(w)o(e)h(ha)o(v)o(e)f(assumed)g(the)h(lattice)f(to)h(b)q(e)g Ft(Z)1333 1798 y Fs(d)1349 1813 y Fv(,)f(but)h(it)f(is)g(clear,)h(from)228 1862 y(the)j(analysis)f(carried)h(on)f(up)h(to)f(no)o(w,)g(that)h(this)f(c)o (hoice)i(a\013ects)f(only)f(com)o(binatorial)e(cal-)228 1912 y(culations.)18 b(Th)o(us)c(similar)e(computations)h(can)h(b)q(e)h(p)q (erformed)f(considering)g(other)h(kinds)f(of)228 1962 y(lattices,)e(b)o(y)f (simply)e(mo)q(difying)f(the)k(com)o(binatorics.)k(It)11 b(is)g(also)g(clear) h(that)f(this)h(framew)o(ork)228 2012 y(can)i(b)q(e)g(extended,)h(with)f (some)f(mo)q(di\014cation,)e(to)j(higher)g(spin)g(systems.)753 2106 y(5.)22 b Fu(A)o(ckno)o(wledgments)278 2181 y Fv(W)m(e)13 b(thank)h(F.)f(Guerra)i(and)e(M.)h(Isopi)f(for)h(helpful)f(discussions)i(and) e(commen)o(ts.)854 2276 y Fu(References)228 2342 y Fx([FK])111 b Fa(C.)16 b(M.)g(F)m(ortuin,)f(P)m(.)g(W.)g(Kasteleyn)f Fl(On)h(the)f(R)n (andom-Cluster)g(Mo)n(del)i(I.)d(Intr)n(o)n(duction)409 2384 y(and)h(r)n(elation)f(to)g(other)h(Mo)n(dels)f Fx(Ph)o(ysica)d Fa(57)j Fx(\(1972\))c(536-564)228 2425 y([Ge])117 b Fa(H-O.)13 b(Georgii)g Fl(Gibbs)f(Me)n(asur)n(e)i(and)f(Phase)h(T)m(r)n(ansition)e Fx(W)m(alter)f(de)g(Gruyter)e(1988)228 2467 y([GHM])75 b Fa(C.Grub)q(er,)10 b(A.)f(Hin)o(terman,)e(D.)j(Merlini)f Fl(Gr)n(oup)h(A)o(nalysis)f(of)h (Classic)n(al)f(L)n(attic)n(e)h(Systems)409 2508 y(\()p Fx(Lecture)g(Notes)h (In)g(Ph)o(ysics)g(60\))f(Springer)f(V)m(erlagh)i(1977)228 2550 y([Gr])119 b Fa(G.)14 b(Grimmett)d Fl(Per)n(c)n(olation)j Fx(Springer)9 b(V)m(erlagh)h(1989)228 2591 y([M])129 b Fa(P)m(.-A.)11 b(Ma)o(y)o(er)h Fl(Quantum)g(Pr)n(ob)n(ability)h(for)f(Pr)n(ob)n(abilists)g Fx(LNM)f(1538)d(Springer)g(V)m(erlagh)h(1995)p eop %%Page: 14 14 bop 228 142 a Fx(14)570 b(M.)15 b(GIANFELICE)228 250 y([QMIPS)c(I])21 b Fa(M.)15 b(Isopi)f Fl(Quantum)f(Metho)n(ds)h(in)f(Particle)g(Systems)g(I)e Fx(preprin)o(t)228 291 y([QMIPS)g(I)q(I])17 b Fa(M.)h(Gianfelice,)g(M.)f (Isopi)g Fl(Quantum)f(Metho)n(ds)g(in)f(Particle)i(Systems)e(II,)g(Glaub)n (er)h(dy-)409 333 y(namics)e(for)e(Ising)h(Mo)n(dels)f Fx(submitted)d(to)i (Mark)o(o)o(v)g(Pro)q(cesses)f(and)g(related)g(Fields)228 374 y([QMIPS)h(IV])18 b Fa(M.)12 b(Gianfelice)f Fl(Quantum)g(Metho)n(ds)g(in)g (Particle)g(Systems)g(IV,)g(Swendsen-Wang)g(dynam-)409 416 y(ics)i(for)g(the)g(Ising)g(Mo)n(del)f Fx(preprin)o(t)228 458 y([S)f(])130 b Fa(B.)18 b(Simon)d Fl(The)h(Statistic)n(al)g(Me)n(chanics)h (of)e(lattic)n(e)i(Gases)e(I)g Fx(Princeton)d(Univ)o(ersit)o(y)h(Press)409 499 y(1996)278 590 y Fw(c)p Fj(n)o Fw(o)f(M.)g(Isopi,)g(Dip)n(ar)n(timento)e (di)i(Ma)n(tema)n(tica,)e(Universit)1172 587 y(\022)1171 590 y(a)j(di)e(R)o(oma)h("La)h(Sapienza)n(",)f(p.le)f(A.)228 632 y(Mor)o(o)i(5,)f(00185)j(R)o(oma)d(IT)m(AL)m(Y)278 673 y Fl(E-mail)h(addr)n (ess)s Fx(:)i Fl(gianfeli)p Fx(@)p Fl(mat.unir)n(oma1.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF