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Fo(\026)3051 5249 y Fj(1)3091 5235 y Fo(f)p 2900 5275 V 2968 5359 a(\026f)3156 5195 y Fm(i)3214 5296 y Fo(:)474 b Fp(\(2)p Fo(:)p Fp(5\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([3])1390 b Fp(2.1)p eop %%Page: 4 4 4 3 bop 0 91 a Fp(The)27 b(\014rst)g(term)g(of)h(\(2.5\))h(is)e(equal)g (to)h Fo(\026)p Fp(\(En)m(t)1567 105 y Fj(1)1608 91 y Fp(\()p Fo(f)10 b Fp(\)\).)40 b(The)27 b(second)h(term)f(is)g(less)f (than)i(or)f(equal)g(to)h Fo(\026)p Fp(\(En)m(t)3674 105 y Fj(2)3714 91 y Fp(\()p Fo(f)10 b Fp(\)\),)0 218 y(since,)30 b(in)f(general)h(one)h(has)f(\(Prop)s(osition)f(2.2)i(in)e ([Le1]\))1281 419 y(En)m(t\()p Fo(f)10 b Fp(\))25 b(=)g(sup)o Fi(f)p Fo(\026)p Fp(\()p Fo(f)10 b(g)s Fp(\))26 b(:)g Fo(\026)p Fp(\()p Fo(e)2289 381 y Fh(g)2331 419 y Fp(\))g Fi(\024)f Fp(1)p Fi(g)15 b Fo(:)0 620 y Fp(Finally)34 b(the)i(last)f(term)h(is)f(zero,)j(b)s(ecause)e Fi(F)1605 634 y Fj(1)1646 620 y Fp(,)h Fi(F)1773 634 y Fj(2)1850 620 y Fp(are)f(indep)s(enden)m(t,)g(and)f(th)m(us)g Fo(\026)2997 634 y Fj(2)3038 620 y Fo(\026)3093 634 y Fj(1)3133 620 y Fo(f)44 b Fp(=)34 b Fo(\026f)45 b Fp(a.s.)57 b(Hence)0 747 y(\(2.3\))32 b(follo)m(ws.)0 921 y(It)j(is)f(natural)g(to)h(guess)g (that)h(inequalities)c(\(2.2\),)38 b(\(2.3\))e(are)g(stable)e(against)h (appropriate)f(\\p)s(erturbations")0 1048 y(of)29 b(the)h(h)m(yp)s (othesis)e(of)h(indep)s(endence)e(of)j(the)f Fo(\033)s Fp({algebras)h Fi(F)2146 1062 y Fj(1)2187 1048 y Fp(,)g Fi(F)2307 1062 y Fj(2)2348 1048 y Fp(.)40 b(The)29 b(indep)s(endence)e (assumption)h(can)i(b)s(e)0 1175 y(stated)k(b)m(y)f(sa)m(ying)h(that)g Fo(\026)942 1189 y Fj(2)982 1175 y Fp(\()p Fo(f)10 b Fp(\))33 b(is)g(a.s.)50 b(equal)33 b(to)h Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))33 b(whenev)m(er)g Fo(f)42 b Fp(is)33 b(measurable)f(w.r.t.)50 b Fi(F)3375 1189 y Fj(1)3416 1175 y Fp(.)f(Hence)35 b(one)0 1302 y(ma)m(y)c(lo)s(ok)f(for)g(a)h(\\w) m(eak)h(dep)s(endence")d(condition)g(of)i(the)f(form)761 1503 y Fi(j)p Fo(\026)841 1517 y Fj(2)882 1503 y Fp(\()p Fo(f)10 b 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Fo(f)2695 4418 y Fj(2)2735 4455 y Fi(k)2780 4469 y Fj(1)2847 4455 y Fi(\024)f Fo(")p Fi(k)p Fo(f)3084 4418 y Fj(2)3125 4455 y Fi(k)3170 4469 y Fj(1)0 4683 y Fd(i.e.)999 4810 y Fp(\(1)d Fi(\000)f Fo(")p Fp(\))15 b Fo(\026)p Fp(\()p Fo(f)1428 4772 y Fj(2)1469 4810 y Fp(\))25 b Fi(\024)g Fo(\026)1680 4824 y Fj(2)1721 4810 y Fo(\026)1776 4824 y Fj(1)1816 4810 y Fp(\()p Fo(f)1906 4772 y Fj(2)1947 4810 y Fp(\))g Fi(\024)g Fp(\(1)c(+)f Fo(")p Fp(\))15 b Fo(\026)p Fp(\()p Fo(f)2532 4772 y Fj(2)2573 4810 y Fp(\))61 b Fo(\026)p Fp({a.s.)813 b(\(3)p Fo(:)p Fp(3\))0 4997 y(If)30 b Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))25 b(=)g(0,)31 b(from)f(\(3.3\))i(w)m(e)f(get)974 5291 y Fo(\026)1029 5190 y Fm(h)1072 5291 y Fo(f)1127 5254 y Fj(2)1182 5291 y Fp(log)1326 5230 y Fo(\026)1381 5244 y Fj(2)1421 5230 y Fo(\026)1476 5244 y Fj(1)1517 5230 y Fp(\()p Fo(f)1607 5197 y Fj(2)1647 5230 y Fp(\))p 1326 5270 V 1394 5354 a Fo(\026)p Fp(\()p Fo(f)1539 5327 y Fj(2)1579 5354 y Fp(\))1693 5190 y Fm(i)1762 5291 y Fi(\024)25 b Fp(log)q(\(1)c(+)f Fo(")p Fp(\))15 b Fo(\026)p Fp(\()p Fo(f)2404 5254 y Fj(2)2445 5291 y Fp(\))26 b Fi(\024)f Fo(")15 b Fp(V)-8 b(ar)q(\()p Fo(f)10 b Fp(\))787 b(\(3)p Fo(:)p Fp(4\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([5])1390 b Fp(3.1)p eop %%Page: 6 6 6 5 bop 0 91 a Fp(and)27 b(\(3.2\))i(holds)c(with)h Fo(#)p Fp(\()p Fo(")p Fp(\))g(=)f Fo(")p Fp(.)40 b(In)27 b(order)f(to)i(pro)m (v)m(e)g(\(3.2\))h(when)e Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))25 b Fi(6)p Fp(=)g(0,)j(w)m(e)g(pro)s(ceed)f(as)g(in)f(the)h(pro) s(of)g(of)0 217 y(Rothaus)e(inequalit)m(y)e([Ro],)k(giv)m(en)e(in)e ([DeSt,)k(p.246].)41 b(Without)24 b(loss)g(of)h(generalit)m(y)g(w)m(e)h (can)f(assume)f Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))25 b(=)g(1,)0 344 y(and)e(w)m(e)h(write)f Fo(f)35 b Fp(=)25 b(1)7 b(+)g Fo(t)862 358 y Fj(0)903 344 y Fo(g)27 b Fp(where)d Fo(g)j Fp(is)c(a)h(function)e(with)h(zero)h(mean)g(and)g(unit)e(v)-5 b(ariance,)25 b(while)d Fo(t)3421 358 y Fj(0)3487 344 y Fp(:=)j(V)-8 b(ar)q(\()p Fo(f)10 b Fp(\).)0 471 y(F)-8 b(or)25 b Fo(t)g Fi(\025)g Fp(0)g(w)m(e)g(let)f Fo(f)679 485 y Fh(t)736 471 y Fp(:=)h(1)8 b(+)g Fo(tg)s Fp(.)39 b(Since)23 b Fo(f)1418 438 y Fj(2)1408 493 y Fh(t)1483 471 y Fp(has)h(no)g(reason)g(to)h(satisfy)f(a)h(lo)m(w)m(er)f(b)s(ound) f Fo(f)3050 438 y Fj(2)3040 493 y Fh(t)3115 471 y Fi(\025)i Fo(b)3250 438 y Fe(\000)p Fj(1)3348 471 y Fp(,)g(w)m(e)g(in)m(tro)s (duce)0 597 y(a)31 b(regularizing)d(parameter)j Fo(\015)5 b Fp(,)31 b(with)e Fo(\015)h Fi(2)25 b Fp(\(0)p Fo(;)15 b(")p Fp(])32 b(and)e(de\014ne)570 835 y Fo(')629 849 y Fh(\015)676 835 y Fp(\()p Fo(t)p Fp(\))c(:=)f Fo(\026)981 734 y Fm(h)1024 835 y Fo(f)1079 797 y Fj(2)1069 857 y Fh(t)1134 835 y Fp(log)1278 773 y Fo(\026)1333 787 y Fj(2)1373 773 y Fo(\026)1428 787 y Fj(1)1469 773 y Fp(\()p Fo(f)1559 740 y Fj(2)1549 796 y Fh(t)1599 773 y Fp(\))c(+)f Fo(\015)p 1278 814 521 4 v 1427 897 a(\026)p Fp(\()p Fo(f)1572 866 y Fj(2)1562 920 y Fh(t)1613 897 y Fp(\))1809 734 y Fm(i)1877 835 y Fp(=)25 b Fo(\026)2028 734 y Fm(h)2070 835 y Fo(f)2125 797 y Fj(2)2115 857 y Fh(t)2181 835 y Fp(log)2324 773 y Fo(\026)2379 787 y Fj(2)2420 773 y Fo(\026)2475 787 y Fj(1)2515 773 y Fp(\()p Fo(f)2605 740 y Fj(2)2595 796 y Fh(t)2646 773 y Fp(\))20 b(+)g Fo(\015)p 2324 814 V 2469 897 a Fp(1)h(+)f Fo(t)2659 871 y Fj(2)2855 734 y Fm(i)3080 835 y Fo(t)25 b Fi(2)g Ff(R)18 b Fo(:)0 1068 y Fp(W)-8 b(e)32 b(claim)d(that,)i(for)f(all)g Fo(\015)g Fi(2)25 b Fp(\(0)p Fo(;)15 b(")p Fp(])1320 1236 y Fo(')1379 1250 y Fh(\015)1426 1236 y Fp(\(0\))26 b Fi(\024)f Fo(\015)2003 b Fp(\(3)p Fo(:)p Fp(5\))1320 1390 y Fo(')1379 1353 y Fe(0)1379 1413 y Fh(\015)1426 1390 y Fp(\(0\))26 b(=)f(0)2005 b(\(3)p Fo(:)p Fp(6\))1332 1544 y Fo(')1391 1506 y Fe(00)1391 1566 y Fh(\015)1438 1544 y Fp(\()p Fo(t)p Fp(\))26 b Fi(\024)f Fp(168)15 b Fo(")g Fp(\(1)23 b Fi(\000)d Fo(")p Fp(\))2141 1506 y Fe(\000)p Fj(2)2330 1544 y Fi(8)p Fo(t)25 b Fi(\025)g Fp(0)1133 b(\(3)p Fo(:)p Fp(7\))0 1714 y(Giv)m(en)40 b(\(3.5\),)k(\(3.6\),)g(\(3.7\))e(the)e(pro)s(of)f(of)h(\(3.2\))h (easily)e(follo)m(ws.)68 b(In)39 b(fact)i Fo(')2753 1728 y Fh(\015)2800 1714 y Fp(\()p Fo(t)2868 1728 y Fj(0)2909 1714 y Fp(\))g Fi(\024)g Fo(\015)32 b Fp(+)26 b(84)15 b Fo(")p Fp(\(1)29 b Fi(\000)d Fo(")p Fp(\))3703 1681 y Fe(\000)p Fj(2)3801 1714 y Fo(t)3834 1681 y Fj(2)3834 1736 y(0)3875 1714 y Fp(.)0 1840 y(Moreo)m(v)m(er,)33 b(since)980 2030 y Fo(b)15 b Fp(log)r(\()p Fo(b)1226 1992 y Fe(\000)p Fj(2)1323 2030 y Fp(\))26 b Fi(\024)f Fo(f)1535 1992 y Fj(2)1590 2030 y Fp(log)1734 1968 y Fo(\026)1789 1982 y Fj(2)1829 1968 y Fo(\026)1884 1982 y Fj(1)1925 1968 y Fp(\()p Fo(f)2015 1935 y Fj(2)2055 1968 y Fp(\))c(+)f Fo(\015)p 1734 2009 V 1884 2092 a(\026)p Fp(\()p Fo(f)2029 2066 y Fj(2)2069 2092 y Fp(\))2290 2030 y Fi(\024)25 b Fo(b)15 b Fp(log)r([)p Fo(b)p Fp(\()p Fo(b)20 b Fp(+)g Fo(\015)5 b Fp(\)])0 2249 y(w)m(e)31 b(ha)m(v)m(e,)h(b)m(y)e(dominated)f(con)m(v)m(ergence,)755 2481 y Fo(\026)810 2380 y Fm(h)853 2481 y Fo(f)908 2444 y Fj(2)963 2481 y Fp(log)1107 2420 y Fo(\026)1162 2434 y Fj(2)1202 2420 y Fo(\026)1257 2434 y Fj(1)1298 2420 y Fp(\()p Fo(f)1388 2387 y Fj(2)1428 2420 y Fp(\))p 1107 2460 357 4 v 1175 2544 a Fo(\026)p Fp(\()p Fo(f)1320 2517 y Fj(2)1360 2544 y Fp(\))1474 2380 y Fm(i)1543 2481 y Fp(=)56 b(lim)1639 2536 y Fh(n)p Fe(!1)1844 2481 y Fo(')1903 2498 y Fj(1)p Fh(=n)2026 2481 y Fp(\()p Fo(t)2094 2495 y Fj(0)2135 2481 y Fp(\))26 b Fi(\024)f Fp(84)15 b Fo(")p Fp(\(1)22 b Fi(\000)e Fo(")p Fp(\))2709 2444 y Fe(\000)p Fj(2)2838 2481 y Fp(V)-8 b(ar\()p Fo(f)10 b Fp(\))15 b Fo(:)0 2714 y Fp(In)25 b(the)g(rest)h(of)g(this)e(section) i(w)m(e)g(sho)m(w)f(that)h(\(3.5\),)j(\(3.6\),)f(\(3.7\))f(hold,)f (completing)f(the)g(pro)s(of)g(of)h(the)f(theorem.)0 2841 y(The)35 b(\014rst)g(inequalit)m(y)f(\(3.5\))k(is)d(trivial.)55 b(Thanks)34 b(to)j(the)f(parameter)g Fo(\015)41 b Fp(w)m(e)36 b(can)g(safely)g(di\013eren)m(tiate)f(under)0 2967 y(the)c(exp)s (ectations,)g(so,)f(using)f Fo(\026)p Fp(\()p Fo(g)s Fp(\))e(=)e(0)30 b(and)g Fo(\026)p Fp(\()p Fo(g)1764 2934 y Fj(2)1806 2967 y Fp(\))25 b(=)g(1,)31 b(w)m(e)g(get)703 3205 y Fo(')762 3167 y Fe(0)762 3227 y Fh(\015)809 3205 y Fp(\()p Fo(t)p Fp(\))26 b(=)f(2)p Fo(\026)1134 3104 y Fm(h)1177 3205 y Fo(f)1222 3219 y Fh(t)1254 3205 y Fo(g)18 b Fp(log)1459 3143 y Fo(\026)1514 3157 y Fj(2)1554 3143 y Fo(\026)1609 3157 y Fj(1)1650 3143 y Fp(\()p Fo(f)1740 3110 y Fj(2)1730 3166 y Fh(t)1780 3143 y Fp(\))j(+)f Fo(\015)p 1459 3184 521 4 v 1604 3267 a Fp(1)h(+)e Fo(t)1793 3241 y Fj(2)1990 3104 y Fm(i)2053 3205 y Fp(+)h(2)p Fo(\026)2244 3104 y Fm(h)2287 3205 y Fo(f)2342 3167 y Fj(2)2332 3227 y Fh(t)2461 3143 y Fo(\026)2516 3157 y Fj(2)2556 3143 y Fo(\026)2611 3157 y Fj(1)2652 3143 y Fp(\()p Fo(f)2732 3157 y Fh(t)2764 3143 y Fo(g)s Fp(\))p 2393 3184 V 2393 3267 a Fo(\026)2448 3281 y Fj(2)2489 3267 y Fo(\026)2544 3281 y Fj(1)2584 3267 y Fp(\()p Fo(f)2674 3236 y Fj(2)2664 3290 y Fh(t)2715 3267 y Fp(\))g(+)g Fo(\015)2924 3104 y Fm(i)2987 3205 y Fi(\000)g Fp(2)p Fo(t)15 b(:)517 b Fp(\(3)p Fo(:)p Fp(8\))0 3432 y(Hence)864 3558 y Fo(')923 3521 y Fe(0)923 3581 y Fh(\015)970 3558 y Fp(\(0\))26 b(=)f(2)31 b(log)q(\(1)21 b(+)f Fo(\015)5 b Fp(\))15 b Fo(\026)p Fp(\()p Fo(g)s Fp(\))22 b(+)e(2)15 b(\(1)22 b(+)e Fo(\015)5 b Fp(\))2318 3521 y Fe(\000)p Fj(1)2416 3558 y Fo(\026)p Fp([)p Fo(\026)2551 3572 y Fj(2)2591 3558 y Fo(\026)2646 3572 y Fj(1)2687 3558 y Fp(\()p Fo(g)s Fp(\)])27 b(=)e(0)15 b Fo(:)0 3719 y Fp(Di\013eren)m(tiating)30 b(again)g(\(3.8\))j(w)m(e)d(obtain)543 3957 y Fo(')602 3919 y Fe(00)602 3979 y Fh(\015)650 3957 y Fp(\()p Fo(t)p Fp(\))25 b(=)g(2)p Fo(\026)974 3856 y Fm(h)1017 3957 y Fo(g)1063 3919 y Fj(2)1120 3957 y Fp(log)1264 3895 y Fo(\026)1319 3909 y Fj(2)1359 3895 y Fo(\026)1414 3909 y Fj(1)1455 3895 y Fp(\()p Fo(f)1545 3862 y Fj(2)1535 3918 y Fh(t)1585 3895 y Fp(\))c(+)f Fo(\015)p 1264 3936 V 1408 4019 a Fp(1)h(+)f Fo(t)1598 3993 y Fj(2)1795 3856 y Fm(i)1858 3957 y Fp(+)g(2)p Fo(\026)2049 3856 y Fm(h)2092 3957 y Fo(f)2147 3919 y Fj(2)2137 3979 y Fh(t)2283 3895 y Fo(\026)2338 3909 y Fj(2)2379 3895 y Fo(\026)2434 3909 y Fj(1)2474 3895 y Fp(\()p Fo(g)2555 3862 y Fj(2)2597 3895 y Fp(\))p 2198 3936 V 2198 4019 a Fo(\026)2253 4033 y Fj(2)2293 4019 y Fo(\026)2348 4033 y Fj(1)2389 4019 y Fp(\()p Fo(f)2479 3988 y Fj(2)2469 4042 y Fh(t)2519 4019 y Fp(\))h(+)f Fo(\015)2729 3856 y Fm(i)804 4241 y Fp(+)g(8)p Fo(\026)995 4140 y Fm(h)1038 4241 y Fo(g)s(f)1129 4255 y Fh(t)1239 4180 y Fo(\026)1294 4194 y Fj(2)1335 4180 y Fo(\026)1390 4194 y Fj(1)1430 4180 y Fp(\()p Fo(g)s(f)1556 4194 y Fh(t)1589 4180 y Fp(\))p 1172 4220 V 1172 4303 a Fo(\026)1227 4317 y Fj(2)1267 4303 y Fo(\026)1322 4317 y Fj(1)1363 4303 y Fp(\()p Fo(f)1453 4272 y Fj(2)1443 4326 y Fh(t)1493 4303 y Fp(\))h(+)f Fo(\015)1703 4140 y Fm(i)1766 4241 y Fi(\000)g Fp(4)p Fo(\026)1957 4140 y Fm(h)2000 4241 y Fo(f)2055 4204 y Fj(2)2045 4264 y Fh(t)2174 4180 y Fp(\()p Fo(\026)2264 4194 y Fj(2)2304 4180 y Fo(\026)2359 4194 y Fj(1)2400 4180 y Fp(\()p Fo(g)s(f)2526 4194 y Fh(t)2559 4180 y Fp(\)\))2629 4147 y Fj(2)p 2106 4220 632 4 v 2106 4303 a Fp(\()p Fo(\026)2196 4317 y Fj(2)2237 4303 y Fo(\026)2292 4317 y Fj(1)2332 4303 y Fp(\()p Fo(f)2422 4272 y Fj(2)2412 4326 y Fh(t)2463 4303 y Fp(\))g(+)g Fo(\015)5 b Fp(\))2696 4277 y Fj(2)2748 4140 y Fm(i)2812 4241 y Fi(\000)20 b Fp(4)3037 4180 y Fo(t)3070 4147 y Fj(2)p 2959 4220 231 4 v 2959 4303 a Fp(1)h(+)e Fo(t)3148 4277 y Fj(2)3220 4241 y Fi(\000)h Fp(2)3713 4096 y(\(3)p Fo(:)p Fp(9\))0 4469 y(In)29 b(order)g(to)h (conclude)f(the)h(pro)s(of)f(w)m(e)h(observ)m(e)g(that)g(if)f(w)m(e)h (replace)f Fo(\026)2456 4483 y Fj(2)2497 4469 y Fo(\026)2552 4483 y Fj(1)2622 4469 y Fp(with)f Fo(\026)i Fp(and)f(tak)m(e)i Fo(\015)f Fp(=)25 b(0,)31 b(the)e(RHS)0 4595 y(of)i(\(3.9\))h(is)d (iden)m(tically)g(zero.)41 b(By)31 b(consequence)g(w)m(e)g(can)f(write) 164 4833 y Fo(')223 4796 y Fe(00)223 4856 y Fh(\015)270 4833 y Fp(\()p Fo(t)p Fp(\))c(=)f(2)p Fo(\026)595 4732 y Fm(h)638 4833 y Fo(g)684 4796 y Fj(2)741 4833 y Fp(log)884 4772 y Fo(\026)939 4786 y Fj(2)980 4772 y Fo(\026)1035 4786 y Fj(1)1075 4772 y Fp(\()p Fo(f)1165 4739 y Fj(2)1155 4794 y Fh(t)1206 4772 y Fp(\))20 b(+)g Fo(\015)p 884 4812 521 4 v 1029 4895 a Fp(1)h(+)f Fo(t)1219 4869 y Fj(2)1415 4732 y Fm(i)1478 4833 y Fp(+)g(2)p Fo(\026)1669 4732 y Fm(h)1712 4833 y Fo(f)1767 4796 y Fj(2)1757 4856 y Fh(t)1807 4732 y Fm(\020)1958 4772 y Fo(\026)2013 4786 y Fj(2)2054 4772 y Fo(\026)2109 4786 y Fj(1)2149 4772 y Fp(\()p Fo(g)2230 4739 y Fj(2)2272 4772 y Fp(\))p 1873 4812 V 1873 4895 a Fo(\026)1928 4909 y Fj(2)1968 4895 y Fo(\026)2023 4909 y Fj(1)2064 4895 y Fp(\()p Fo(f)2154 4864 y Fj(2)2144 4918 y Fh(t)2194 4895 y Fp(\))h(+)f Fo(\015)2424 4833 y Fi(\000)2529 4772 y Fo(\026)p Fp(\()p Fo(g)2665 4739 y Fj(2)2707 4772 y Fp(\))p 2526 4812 221 4 v 2526 4895 a Fo(\026)p Fp(\()p Fo(f)2671 4864 y Fj(2)2661 4918 y Fh(t)2711 4895 y Fp(\))2757 4732 y Fm(\021i)424 5117 y Fp(+)g(8)p Fo(\026)615 5016 y Fm(h)658 5117 y Fo(g)s(f)749 5131 y Fh(t)781 5016 y Fm(\020)914 5056 y Fo(\026)969 5070 y Fj(2)1010 5056 y Fo(\026)1065 5070 y Fj(1)1105 5056 y Fp(\()p Fo(g)s(f)1231 5070 y Fh(t)1264 5056 y Fp(\))p 847 5096 521 4 v 847 5180 a Fo(\026)902 5194 y Fj(2)942 5180 y Fo(\026)997 5194 y Fj(1)1038 5180 y Fp(\()p Fo(f)1128 5148 y Fj(2)1118 5202 y Fh(t)1168 5180 y Fp(\))h(+)f Fo(\015)1398 5117 y Fi(\000)1500 5056 y Fo(\026)p Fp(\()p Fo(g)s(f)1681 5070 y Fh(t)1713 5056 y Fp(\))p 1500 5096 249 4 v 1514 5180 a Fo(\026)p Fp(\()p Fo(f)1659 5148 y Fj(2)1649 5202 y Fh(t)1699 5180 y Fp(\))1759 5016 y Fm(\021i)1877 5117 y Fi(\000)f Fp(4)p Fo(\026)2067 5016 y Fm(h)2111 5117 y Fo(f)2166 5080 y Fj(2)2156 5140 y Fh(t)2206 5016 y Fm(\020)2338 5056 y Fp(\()p Fo(\026)2428 5070 y Fj(2)2469 5056 y Fo(\026)2524 5070 y Fj(1)2565 5056 y Fp(\()p Fo(g)s(f)2691 5070 y Fh(t)2723 5056 y Fp(\)\))2793 5023 y Fj(2)p 2271 5096 632 4 v 2271 5180 a Fp(\()p Fo(\026)2361 5194 y Fj(2)2402 5180 y Fo(\026)2457 5194 y Fj(1)2497 5180 y Fp(\()p Fo(f)2587 5148 y Fj(2)2577 5202 y Fh(t)2628 5180 y Fp(\))h(+)g Fo(\015)5 b Fp(\))2861 5153 y Fj(2)2934 5117 y Fi(\000)3035 5056 y Fp(\()p Fo(\026)p Fp(\()p Fo(g)s(f)3251 5070 y Fh(t)3284 5056 y Fp(\)\))3354 5023 y Fj(2)p 3035 5096 361 4 v 3049 5180 a Fp(\()p Fo(\026)p Fp(\()p Fo(f)3229 5148 y Fj(2)3219 5202 y Fh(t)3270 5180 y Fp(\)\))3340 5153 y Fj(2)3406 5016 y Fm(\021)q(i)399 5345 y Fp(=)25 b(2)p Fo(\026)p Fp(\()p Fo(g)676 5307 y Fj(2)718 5345 y Fp(\001)794 5359 y Fj(0)834 5345 y Fp(\))c(+)f(2)p Fo(\026)p Fp(\()p Fo(f)1171 5307 y Fj(2)1161 5367 y Fh(t)1211 5345 y Fp(\001)1287 5359 y Fj(1)1328 5345 y Fp(\))h(+)e(8)p Fo(\026)p Fp(\()p Fo(g)s(f)1700 5359 y Fh(t)1733 5345 y Fp(\001)1809 5359 y Fj(2)1850 5345 y Fp(\))h(+)g(4)p Fo(\026)p Fp(\()p Fo(f)2186 5307 y Fj(2)2176 5367 y Fh(t)2227 5345 y Fp(\001)2303 5359 y Fj(3)2343 5345 y Fp(\))3668 5060 y(\(3)p Fo(:)p Fp(10\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([6])1390 b Fp(3.2)p eop %%Page: 7 7 7 6 bop 0 91 a Fp(where)25 b(\001)334 105 y Fj(0)375 91 y Fo(;)15 b(:)g(:)g(:)h(;)f Fp(\001)652 105 y Fj(3)719 91 y Fp(are)25 b(implicitly)d(de\014ned)i(in)h(the)g(last)h(equalit)m (y)-8 b(.)39 b(The)25 b(estimate)h(of)g Fo(')3067 58 y Fe(00)3067 113 y Fh(\015)3139 91 y Fp(will)d(follo)m(w)i(from)g(the)0 218 y(fact)k(that)g(for)f(all)g(functions)e Fo(h)p Fp(,)k Fo(\026)1185 232 y Fj(2)1225 218 y Fo(\026)1280 232 y Fj(1)1321 218 y Fp(\()p Fo(h)p Fp(\))f(is)f(close)g(to)h Fo(\026)p Fp(\()p Fo(f)10 b Fp(\))28 b(in)f Fo(L)2260 185 y Fe(1)2366 218 y Fp(sense.)40 b(More)29 b(precisely)e(from)g (assumption)0 346 y(\(2.8\),)32 b(using)d(also)i Fi(k)p Fo(g)s Fi(k)799 360 y Fj(1)866 346 y Fi(\024)25 b(k)p Fo(g)s Fi(k)1098 360 y Fj(2)1166 346 y Fp(=)g(1,)31 b(it)f(follo)m(ws) 918 595 y(\(1)21 b Fi(\000)f Fo(")p Fp(\))15 b(\(1)22 b(+)e Fo(t)1428 557 y Fj(2)1469 595 y Fp(\))25 b Fi(\024)g Fo(\026)1680 609 y Fj(2)1721 595 y Fo(\026)1776 609 y Fj(1)1816 595 y Fp(\()p Fo(f)1906 557 y Fj(2)1896 617 y Fh(t)1947 595 y Fp(\))g Fi(\024)g Fp(\(1)c(+)f Fo(")p Fp(\))15 b(\(1)22 b(+)e Fo(t)2613 557 y Fj(2)2654 595 y Fp(\))61 b Fo(\026)p Fp({a.s.)1058 749 y Fo(t)20 b Fi(\000)g Fo(")p Fp(\(1)h(+)f Fo(t)p Fp(\))25 b Fi(\024)g Fo(\026)1680 763 y Fj(2)1721 749 y Fo(\026)1776 763 y Fj(1)1816 749 y Fp(\()p Fo(g)s(f)1942 763 y Fh(t)1975 749 y Fp(\))g Fi(\024)g Fo(t)20 b Fp(+)g Fo(")p Fp(\(1)i(+)e Fo(t)p Fp(\))60 b Fo(\026)p Fp({a.s.)3668 671 y(\(3)p Fo(:)p Fp(11\))0 992 y(The)28 b(remaining)f(rather)h(detailed)g(and)f (straigh)m(tforw)m(ard)i(computations)f(ha)m(v)m(e)i(the)e(main)g(purp) s(ose)f(of)h(\014nding)0 1120 y(the)34 b(explicit)e(v)-5 b(alue)34 b(for)f(the)h(function)f Fo(#)h Fp(stated)g(in)f(the)h (theorem,)h(whic)m(h)e(is)g(ho)m(w)m(ev)m(er)i(not)f(to)h(b)s(e)e(tak)m (en)i(to)s(o)0 1247 y(seriously)-8 b(.)39 b(F)-8 b(rom)31 b(\(3.10\))i(w)m(e)e(get)482 1486 y Fi(j)p Fo(')566 1448 y Fe(00)566 1508 y Fh(\015)613 1486 y Fp(\()p Fo(t)p Fp(\))p Fi(j)26 b(\024)f Fp(2)p Fi(k)p Fp(\001)1029 1500 y Fj(0)1070 1486 y Fi(k)1115 1500 y Fe(1)1213 1486 y Fp(+)20 b(2)p Fi(k)p Fp(\001)1470 1500 y Fj(1)1512 1486 y Fi(k)1557 1500 y Fe(1)1634 1486 y Fp(\(1)h(+)f Fo(t)1859 1448 y Fj(2)1900 1486 y Fp(\))g(+)g(8)p Fi(k)p Fp(\001)2212 1500 y Fj(2)2253 1486 y Fi(k)2298 1500 y Fe(1)2376 1486 y Fp(\(1)h(+)f Fo(t)p Fp(\))g(+)g(4)p Fi(k)p Fp(\001)2913 1500 y Fj(3)2955 1486 y Fi(k)3000 1500 y Fe(1)3077 1486 y Fp(\(1)h(+)f Fo(t)3302 1448 y Fj(2)3343 1486 y Fp(\))15 b Fo(:)250 b Fp(\(3)p Fo(:)p Fp(12\))0 1725 y(W)-8 b(e)32 b(\014nally)c(estimates)j(\001)908 1739 y Fj(0)948 1725 y Fo(;)15 b(:)g(:)g(:)i(;)e Fp(\001)1226 1739 y Fj(3)1267 1725 y Fp(,)30 b(using)f Fo(\015)i Fi(2)25 b Fp(\(0)p Fo(;)15 b(")p Fp(])32 b(and)e(\(3.11\))976 1964 y Fi(k)p Fp(\001)1097 1978 y Fj(0)1138 1964 y Fi(k)1183 1978 y Fe(1)1285 1964 y Fi(\024)25 b Fp(max)q Fi(f)p Fp(log)q(\(1)c(+)f Fo(")p Fp(\))p Fo(;)15 b Fi(\000)g Fp(log)s(\(1)21 b Fi(\000)f Fo(")p Fp(\))p Fi(g)26 b(\024)2752 1902 y Fo(")p 2674 1943 199 4 v 2674 2026 a Fp(1)21 b Fi(\000)f Fo(")2899 1964 y(:)0 2245 y Fp(Next,)31 b(w)m(e)g(ha)m(v)m(e)h(that,)f Fo(\026)p Fp({a.s.)1030 2530 y(\001)1106 2544 y Fj(1)1171 2530 y Fi(\024)1464 2469 y Fp(1)21 b(+)f Fo(")p 1278 2510 571 4 v 1278 2593 a Fp(\(1)h Fi(\000)f Fo(")p Fp(\)\(1)i(+)e Fo(t)1773 2567 y Fj(2)1813 2593 y Fp(\))1880 2530 y Fi(\000)2074 2469 y Fp(1)p 1982 2510 231 4 v 1982 2593 a(1)g(+)g Fo(t)2171 2567 y Fj(2)2248 2530 y Fp(=)2411 2469 y(2)p Fo(")p 2355 2510 199 4 v 2355 2593 a Fp(1)h Fi(\000)f Fo(")2668 2469 y Fp(1)p 2576 2510 231 4 v 2576 2593 a(1)g(+)g Fo(t)2765 2567 y Fj(2)1030 2799 y Fp(\001)1106 2813 y Fj(1)1171 2799 y Fi(\025)1546 2738 y Fp(1)g Fi(\000)g Fo(")p 1278 2778 734 4 v 1278 2862 a Fp(\(1)h(+)f Fo(")p Fp(\)\(1)i(+)e Fo(t)1773 2835 y Fj(2)1813 2862 y Fp(\))h(+)f Fo(\015)2043 2799 y Fi(\000)2237 2738 y Fp(1)p 2145 2778 231 4 v 2145 2862 a(1)h(+)f Fo(t)2335 2835 y Fj(2)2411 2799 y Fi(\025)25 b(\000)2660 2738 y Fp(3)p Fo(")p 2589 2778 V 2589 2862 a Fp(1)c(+)e Fo(t)2778 2835 y Fj(2)2845 2799 y Fo(:)0 3095 y Fp(As)30 b(for)g(\001)349 3109 y Fj(2)390 3095 y Fp(,)h Fo(\026)p Fp({a.s.)41 b(w)m(e)31 b(ha)m(v)m(e)887 3391 y(\001)963 3405 y Fj(2)1028 3391 y Fi(\024)1197 3329 y Fo(t)20 b Fp(+)g Fo(")p Fp(\(1)i(+)e Fo(t)p Fp(\))p 1135 3370 571 4 v 1135 3453 a(\(1)h Fi(\000)f Fo(e)p Fp(\)\(1)i(+)e Fo(t)1630 3427 y Fj(2)1670 3453 y Fp(\))1737 3391 y Fi(\000)1937 3329 y Fo(t)p 1839 3370 231 4 v 1839 3453 a Fp(1)g(+)g Fo(t)2028 3427 y Fj(2)2105 3391 y Fp(=)2290 3329 y Fo(")p 2212 3370 199 4 v 2212 3453 a Fp(1)h Fi(\000)e Fo(")2448 3329 y Fp(1)h(+)g(2)p Fo(t)p 2448 3370 235 4 v 2450 3453 a Fp(1)h(+)f Fo(t)2640 3427 y Fj(2)887 3669 y Fp(\001)963 3683 y Fj(2)1028 3669 y Fi(\025)1279 3608 y Fo(t)g Fi(\000)g Fo(")p Fp(\(1)h(+)f Fo(t)p Fp(\))p 1135 3648 734 4 v 1135 3732 a(\(1)h(+)f Fo(e)p Fp(\)\(1)i(+)e Fo(t)1630 3705 y Fj(2)1670 3732 y Fp(\))h(+)f Fo(\015)1900 3669 y Fi(\000)2101 3608 y Fo(t)p 2002 3648 231 4 v 2002 3732 a Fp(1)h(+)e Fo(t)2191 3705 y Fj(2)2268 3669 y Fi(\025)25 b(\000)2547 3608 y Fo(")p 2446 3648 245 4 v 2446 3732 a Fp(1)20 b(+)g(2)p Fo(")2727 3608 y Fp(1)h(+)f(3)p Fo(t)p 2727 3648 235 4 v 2729 3732 a Fp(1)h(+)f Fo(t)2919 3705 y Fj(2)2988 3669 y Fo(:)0 3965 y Fp(Finally)-8 b(,)29 b(for)h(\001)542 3979 y Fj(3)583 3965 y Fp(,)g Fo(\026)p Fp({a.s.)42 b(w)m(e)31 b(ha)m(v)m(e)805 4261 y Fi(j)p Fp(\001)906 4275 y Fj(3)946 4261 y Fi(j)26 b(\024)f(j)p Fp(\001)1194 4275 y Fj(2)1235 4261 y Fi(j)1275 4156 y Fm(\014)1275 4211 y(\014)1275 4265 y(\014)1384 4199 y Fo(\026)1439 4213 y Fj(2)1479 4199 y Fo(\026)1534 4213 y Fj(1)1575 4199 y Fp(\()p Fo(g)s(f)1701 4213 y Fh(t)1733 4199 y Fp(\))p 1316 4240 521 4 v 1316 4323 a Fo(\026)1371 4337 y Fj(2)1412 4323 y Fo(\026)1467 4337 y Fj(1)1507 4323 y Fp(\()p Fo(f)1597 4292 y Fj(2)1587 4345 y Fh(t)1638 4323 y Fp(\))20 b(+)g Fo(\015)1867 4261 y Fp(+)1969 4199 y Fo(\026)p Fp(\()p Fo(g)s(f)2150 4213 y Fh(t)2183 4199 y Fp(\))p 1969 4240 249 4 v 1983 4323 a Fo(\026)p Fp(\()p Fo(f)2128 4292 y Fj(2)2118 4345 y Fh(t)2168 4323 y Fp(\))2229 4156 y Fm(\014)2229 4211 y(\014)2229 4265 y(\014)2284 4261 y Fi(\024)25 b(j)p Fp(\001)2481 4275 y Fj(2)2522 4261 y Fi(j)2650 4199 y Fp(2)p 2573 4240 199 4 v 2573 4323 a(1)c Fi(\000)f Fo(")2809 4199 y Fp(1)h(+)f(2)p Fo(t)p 2809 4240 235 4 v 2811 4323 a Fp(1)h(+)f Fo(t)3001 4297 y Fj(2)3070 4261 y Fo(:)0 4557 y Fp(Th)m(us,)30 b(collecting)g(all)f(together,)512 4858 y Fo(')571 4820 y Fe(00)571 4880 y Fh(\015)618 4858 y Fp(\()p Fo(t)p Fp(\))d Fi(\024)910 4796 y Fp(8)p Fo(")p 854 4837 199 4 v 854 4920 a Fp(1)21 b Fi(\000)f Fo(")1084 4858 y Fp(+)1275 4796 y(16)p Fo(")p 1186 4837 311 4 v 1186 4920 a Fp(\(1)h Fi(\000)f Fo(")p Fp(\))1455 4894 y Fj(2)1523 4858 y Fp(sup)1531 4934 y Fh(t)p Fe(\025)p Fj(0)1685 4796 y Fp(\(1)h(+)f(3)p Fo(t)p Fp(\))1990 4763 y Fj(2)p 1685 4837 347 4 v 1744 4920 a Fp(1)g(+)g Fo(t)1933 4894 y Fj(2)2068 4858 y Fp(=)2230 4796 y(8)p Fo(")p 2175 4837 199 4 v 2175 4920 a Fp(1)g Fi(\000)g Fo(")2405 4858 y Fp(+)g(10)2701 4796 y(16)p Fo(")p 2613 4837 311 4 v 2613 4920 a Fp(\(1)h Fi(\000)f Fo(")p Fp(\))2882 4894 y Fj(2)2959 4858 y Fi(\024)3132 4796 y Fp(168)p Fo(")p 3066 4837 V 3066 4920 a Fp(\(1)h Fi(\000)f Fo(")p Fp(\))3335 4894 y Fj(2)0 5171 y Fp(and)30 b(\(3.7\))i(follo)m(ws)90 b Fb(\003)0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([7])1390 b Fp(4.1)p eop %%Page: 8 8 8 7 bop 0 91 a Fq(4.)47 b(Log{Sob)s(olev)36 b(inequalities)f(for)g (completely)g(analytical)f(Gibbs)i(random)e(\014elds)0 249 y Fd(Gibbs)i(me)-5 b(asur)g(es)p Fp(.)53 b(W)-8 b(e)35 b(brie\015y)d(recall)i(the)g(concept)h(of)f(Gibbs)e(measures)i(and)f (refer)h(the)g(reader)g(to)h([Ge])g(for)0 376 y(a)e(comprehensiv)m(e)g (in)m(tro)s(duction)e(to)i(the)g(sub)5 b(ject.)48 b(W)-8 b(e)34 b(consider)e(the)h Fo(d)g Fp(dimensional)d(lattice)j Ff(Z)3445 343 y Fh(d)3516 376 y Fp(with)f Fd(sites)0 503 y Fo(x)25 b Fp(:=)g(\()p Fo(x)285 517 y Fj(1)327 503 y Fo(;)15 b(:)g(:)g(:)h(;)f(x)580 517 y Fh(d)623 503 y Fp(\))31 b(and)e(norm)1586 634 y Fi(j)p Fo(x)p Fi(j)d Fp(:=)106 b(max)1835 697 y Fh(i)p Fe(2f)p Fj(1)p Fh(;:::)n(;d)p Fe(g)2181 634 y Fi(j)p Fo(x)2258 648 y Fh(i)2289 634 y Fi(j)0 902 y Fp(The)37 b(asso)s(ciated)h(distance)g (function)e(is)h(denoted)g(b)m(y)h Fo(d)p Fp(\()p Fi(\001)p Fo(;)15 b Fi(\001)p Fp(\).)64 b(The)38 b(cardinalit)m(y)e(of)h(\003)h Fi(\032)f Ff(Z)3279 869 y Fh(d)3355 902 y Fp(is)f(denoted)i(b)m(y)0 1029 y Fi(j)p Fp(\003)p Fi(j)p Fp(.)82 b Ff(F)54 b Fp(is)43 b(the)h(set)h(of)f(all)e(nonempt)m(y)i(\014nite)f(subsets)g(of)h Ff(Z)2200 996 y Fh(d)2238 1029 y Fp(.)81 b(W)-8 b(e)45 b(de\014ne)e(the)h(exterior)g Fd(n{b)-5 b(oundary)55 b Fp(as)0 1157 y Fo(@)53 1124 y Fj(+)48 1179 y Fh(n)114 1157 y Fp(\003)25 b(:=)g Fi(f)p Fo(x)h Fi(2)f Fp(\003)595 1124 y Fh(c)657 1157 y Fp(:)41 b Fo(d)p Fp(\()p Fo(x;)15 b Fp(\003\))27 b Fi(\024)e Fo(n)p Fi(g)p Fp(,)30 b(where)g(\003)1599 1124 y Fh(c)1667 1157 y Fp(stands)f(for)h(the)h(complemen)m(t)g(of)f (\003)h(in)e Ff(Z)3122 1124 y Fh(d)3160 1157 y Fp(.)227 1286 y(Giv)m(en)k(an)f(arbitrary)g(probabilit)m(y)e(space)j(\()p Fo(S;)15 b Fi(E)8 b Fo(;)15 b(\027)6 b Fp(\))34 b(\(the)g Fd(single)g(spin)41 b Fp(space\),)34 b(w)m(e)f(in)m(tro)s(duce)f(the)h Fd(c)-5 b(on-)0 1413 y(\014gur)g(ation)41 b(sp)-5 b(ac)g(e)39 b Fp(\(\012)p Fo(;)15 b Fi(F)9 b Fp(\))39 b(:=)e(\()p Fo(S)1177 1380 y Fl(Z)1226 1353 y Fk(d)1263 1413 y Fo(;)15 b Fi(E)1359 1380 y Fl(Z)1408 1353 y Fk(d)1444 1413 y Fp(\).)64 b(Sometimes)37 b(w)m(e)h(consider)f(\014nite)g(v)m(olume)g (con\014guration)h(spaces)0 1541 y(\(\012)101 1555 y Fj(\003)155 1541 y Fo(;)15 b Fi(F)260 1555 y Fj(\003)315 1541 y Fp(\))26 b(:=)f(\()p Fo(S)593 1508 y Fj(\003)648 1541 y Fo(;)15 b Fi(E)744 1508 y Fj(\003)799 1541 y Fp(\),)30 b(for)f(\003)d Fi(2)e Ff(F)5 b Fp(.)46 b(Giv)m(en)30 b Fo(\033)e Fi(2)d Fp(\012)k(and)g(\003)c Fi(\032)g Ff(Z)2271 1508 y Fh(d)2339 1541 y Fp(w)m(e)30 b(denote)g(b)m(y)f Fo(\031)2941 1555 y Fj(\003)3025 1541 y Fp(the)h(natural)e(pro)5 b(jection)0 1668 y(o)m(v)m(er)38 b(\012)267 1682 y Fj(\003)357 1668 y Fp(and)e(write)g Fo(\033)831 1682 y Fj(\003)921 1668 y Fp(:=)g Fo(\031)1105 1682 y Fj(\003)1159 1668 y Fp(\()p Fo(\033)s Fp(\).)60 b(If)36 b Fo(U;)15 b(V)56 b Fi(\032)36 b Ff(Z)1849 1635 y Fh(d)1923 1668 y Fp(are)h(disjoin)m(t,) g Fo(\033)2490 1682 y Fh(U)2551 1668 y Fo(\021)2596 1682 y Fh(V)2696 1668 y Fp(is)e(the)i(con\014guration)f(on)g Fo(U)f Fi([)24 b Fo(V)0 1796 y Fp(whic)m(h)29 b(is)h(equal)f(to)j Fo(\033)h Fp(on)d Fo(U)41 b Fp(and)29 b Fo(\021)34 b Fp(on)c Fo(V)21 b Fp(.)40 b(The)30 b(action)h(of)f(the)h(translations)e (is)h(de\014ned)f(on)h(\012)g(as)1248 2035 y Fo(#)1302 2049 y Fh(x)1348 2035 y Fp(\()p Fo(\033)s Fp(\)\()p Fo(y)s Fp(\))c(:=)g Fo(\033)s Fp(\()p Fo(y)d Fi(\000)d Fo(x)p Fp(\))182 b Fo(x;)15 b(y)29 b Fi(2)24 b Ff(Z)2573 1998 y Fh(d)2627 2035 y Fo(:)0 2275 y Fp(If)30 b Fo(f)40 b Fp(is)29 b(a)i(function)e(on)h(\012,)g(\003)1009 2289 y Fh(f)1087 2275 y Fp(denotes)h(the)f(smallest)g(subset)g(of)g Ff(Z)2369 2242 y Fh(d)2438 2275 y Fp(suc)m(h)g(that)h Fo(f)10 b Fp(\()p Fo(\033)s Fp(\))30 b(dep)s(ends)f(only)g(on)h Fo(\033)3779 2289 y Fj(\003)3829 2299 y Fk(f)3875 2275 y Fp(.)0 2402 y Fo(f)40 b Fp(is)29 b(called)h Fd(lo)-5 b(c)g(al)32 b Fp(if)d(\003)782 2416 y Fh(f)860 2402 y Fp(is)g(\014nite.)40 b(The)30 b(suprem)m(um)e(norm)i(of)h Fo(f)39 b Fp(is)30 b(denoted)g(b)m(y)g Fi(k)p Fo(f)10 b Fi(k)2980 2416 y Fh(u)3053 2402 y Fp(:=)25 b(sup)3311 2424 y Fh(!)r Fe(2)p Fj(\012)3480 2402 y Fi(j)p Fo(f)10 b Fp(\()p Fo(!)s Fp(\))p Fi(j)p Fp(.)0 2588 y(In)24 b(the)h(follo)m (wing)e(w)m(e)i(consider)f(a)h(translation)f(in)m(v)-5 b(arian)m(t,)25 b(summable)e Fd(inter)-5 b(action)33 b Fo(J)9 b Fp(,)26 b(of)f(\014nite)f(range)h Fo(r)s Fp(,)h Fd(i.e.)32 b Fp(a)0 2716 y(collection)e(of)h(functions)d Fo(J)35 b Fp(=)25 b(\()p Fo(J)1165 2730 y Fh(A)1225 2716 y Fp(\))1260 2730 y Fh(A)p Fe(2)p Fl(F)1407 2716 y Fp(,)31 b(suc)m(h)f(that)h Fo(J)1915 2730 y Fh(A)2000 2716 y Fp(:)25 b(\012)g Fi(7!)g Ff(R)39 b Fp(is)30 b(measurable)f(w.r.t.)41 b Fi(F)3238 2730 y Fh(A)3298 2716 y Fp(,)30 b(and)-3 2875 y(\(H1\))47 b Fo(J)277 2889 y Fh(A)p Fj(+)p Fh(x)454 2875 y Fi(\016)21 b Fo(#)574 2889 y Fh(x)645 2875 y Fp(=)k Fo(J)791 2889 y Fh(A)880 2875 y Fp(for)31 b(all)e Fo(A)c Fi(2)g Ff(F)5 b Fp(,)36 b Fo(x)25 b Fi(2)g Ff(Z)1664 2842 y Fh(d)-3 3004 y Fp(\(H2\))47 b Fo(J)277 3018 y Fh(A)362 3004 y Fp(=)25 b(0)31 b(if)e(the)i(diameter)f(of)g Fo(A)h Fp(is)e(greater)j(than)e Fo(r)-3 3134 y Fp(\(H3\))227 3066 y Fm(P)323 3161 y Fh(A)p Fe(2)p Fl(F)p Fj(:)13 b Fh(A)p Fe(3)p Fj(0)658 3134 y Fi(k)p Fo(J)753 3148 y Fh(A)813 3134 y Fi(k)858 3148 y Fh(u)932 3134 y Fo(<)25 b Fi(1)0 3291 y Fp(The)30 b(Hamiltonian)f(\()p Fo(H)821 3305 y Fj(\003)875 3291 y Fp(\))910 3305 y Fj(\003)p Fe(2)p Fl(F)1083 3291 y Fp(asso)s(ciated)h(with)g Fo(J)39 b Fp(is)29 b(de\014ned)h(as)1279 3531 y Fo(H)1355 3545 y Fj(\003)1434 3531 y Fp(:)25 b(\012)g Fi(3)g Fo(\033)j Fi(!)2003 3445 y Fm(X)1858 3644 y Fh(A)p Fe(2)p Fl(F)p Fj(:)12 b Fh(A)p Fe(\\)p Fj(\003)p Fe(6)p Fj(=)p Fe(;)2295 3531 y Fo(J)2345 3545 y Fh(A)2430 3531 y Fi(2)24 b Ff(R)18 b Fo(:)0 3882 y Fp(Clearly)41 b Fi(k)p Fo(H)449 3896 y Fj(\003)503 3882 y Fi(k)548 3896 y Fh(u)642 3882 y Fi(\024)k(j)p Fp(\003)p Fi(jk)p Fo(J)9 b Fi(k)p Fp(.)78 b(F)-8 b(or)43 b Fo(\033)n(;)15 b(\034)56 b Fi(2)45 b Fp(\012)d(w)m(e)h(also)g(let)f Fo(H)2265 3849 y Fh(\034)2258 3907 y Fj(\003)2312 3882 y Fp(\()p Fo(\033)s Fp(\))k(:=)f Fo(H)2700 3896 y Fj(\003)2754 3882 y Fp(\()p Fo(\033)2841 3896 y Fh(V)2905 3882 y Fo(\034)2945 3896 y Fh(V)3003 3878 y Fk(c)3042 3882 y Fp(\))d(and)g Fo(\034)53 b Fp(is)41 b(called)h(the)0 4010 y Fd(b)-5 b(oundary)33 b(c)-5 b(ondition)p Fp(.)42 b(F)-8 b(or)30 b(eac)m(h)h(\003)25 b Fi(2)g Ff(F)5 b Fp(,)35 b Fo(\034)h Fi(2)24 b Fp(\012)29 b(the)h(\(\014nite)e(v)m (olume\))i(Gibbs)d(measure)i(on)h(\(\012)p Fo(;)15 b Fi(F)9 b Fp(\),)30 b(are)g(giv)m(en)0 4137 y(b)m(y)869 4267 y Fo(\026)924 4230 y Fh(\034)924 4290 y Fj(\003)978 4267 y Fp(\()p Fo(d\033)s Fp(\))c(:=)1298 4194 y Fm(\000)1339 4267 y Fo(Z)1408 4230 y Fh(\034)1401 4290 y Fj(\003)1456 4194 y Fm(\001)1497 4213 y 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(on)g Fo(X)7 b Fp(.)44 b(The)31 b(set)0 5100 y(of)g(measures)f(\(4.1\)) i(satis\014es)e(the)g(DLR)h(compatibilit)m(y)d(conditions)919 5340 y Fo(\026)974 5354 y Fj(\003)1028 5340 y Fp(\()p Fo(\026)1118 5354 y Fh(V)1181 5340 y Fp(\()p Fo(X)7 b Fp(\)\))27 b(=)e Fo(\026)1546 5354 y Fj(\003)1600 5340 y Fp(\()p Fo(X)7 b Fp(\))183 b Fi(8)15 b Fo(X)33 b Fi(2)24 b(F)192 b(8)15 b Fo(V)45 b Fi(\032)25 b Fp(\003)g Fi(2)g Ff(F)20 b Fo(:)738 b Fp(\(4)p Fo(:)p Fp(2\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([8])1390 b Fp(4.2)p eop %%Page: 9 9 9 8 bop 0 91 a Fp(Since)44 b Fo(\026)307 105 y Fh(V)370 91 y Fo(f)55 b Fp(is)45 b(measurable)f(w.r.t.)86 b Fi(F)1430 105 y Fh(V)1489 86 y Fk(c)1573 91 y Fp(and)45 b(since,)k(trivially)-8 b(,)47 b(for)e(all)g Fo(g)k Fp(measurable)44 b(w.r.t.)87 b Fi(F)3654 105 y Fh(V)3712 86 y Fk(c)3796 91 y Fp(w)m(e)0 218 y(ha)m(v)m(e)36 b Fo(\026)269 185 y Fh(\033)269 243 y(V)332 218 y Fp(\()p Fo(f)10 b(g)s Fp(\))34 b(=)f Fo(g)s Fp(\()p Fo(\033)s Fp(\))p Fo(\026)867 185 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y(p)s(o)m(w)m(erful)30 b(concept)i(of)f(complete)h(analyticit)m(y)f (of)g(an)g(in)m(teraction)g Fo(J)9 b Fp(,)32 b(within)c(the)k(framew)m (ork)f(of)g(\014nite)f(single)0 1097 y(spin)41 b(space)j Fo(S)5 b Fp(.)80 b(They)43 b(ha)m(v)m(e)h(sho)m(wn)f(ho)m(w)g(complete) h(analytical)f(in)m(teractions)g(can)h(b)s(e)f(c)m(haracterized)h(b)m (y)0 1224 y(12)c(equiv)-5 b(alen)m(t)39 b(conditions,)i(and)d(the)i (asso)s(ciated)g(Gibbs)e(\014elds)g(exhibit)f(all)i(regularit)m(y)f (prop)s(erties)g(of)i(the)0 1352 y(high{temp)s(erature)32 b(regime.)47 b(In)32 b(particular)f(complete)i(analyticit)m(y)g (implies)d(that)j(there)g(is)f(a)h(unique)e(Gibbs)0 1479 y(measure)j(for)f Fo(J)9 b Fp(.)52 b(Our)32 b(basic)i(assumption)e(on)i (the)g(in)m(teraction)g Fo(J)9 b Fp(,)35 b(is)e(condition)f(\(I)s(I)s (Id\))h(in)f([DoSh3].)52 b(While)0 1606 y(its)29 b(equiv)-5 b(alence)28 b(to)i(the)g(other)f(11)h(form)m(ulations)e(of)h(complete)h (analyticit)m(y)f(dep)s(ends)e(on)i(the)h(\014niteness)d(of)j Fo(S)5 b 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Fp(\()p Fo(f)10 b Fp(\))24 b(as)h(the)g(en)m(trop)m(y)h(of)f Fo(f)34 b Fp(w.r.t.)39 b Fo(\026)2510 5180 y Fh(\034)2510 5238 y Fj(\003)2564 5213 y Fp(.)g(When)24 b(w)m(e)i(write)e(En)m(t)3388 5227 y Fj(\003)3442 5213 y Fp(\()p Fo(f)10 b Fp(\))25 b(without)0 5340 y(the)34 b(sup)s(erscript)e Fo(\034)10 b Fp(,)35 b(w)m(e)f(mean)g(the)g(function)f Fo(\034)42 b Fi(!)31 b Fp(En)m(t)1995 5300 y Fh(\034)1995 5362 y Fj(\003)2049 5340 y Fp(\()p Fo(f)10 b Fp(\),)35 b(in)e(analogy)i(with)d Fo(\026)2949 5354 y Fj(\003)3003 5340 y Fp(\()p Fo(f)10 b Fp(\).)52 b(W)-8 b(e)35 b(consider)e(then)0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)e Fj([9])1390 b Fp(4.3)p eop %%Page: 10 10 10 9 bop 0 91 a Fp(a)35 b(\\generalized")h(Diric)m(hlet)e(form)g Fi(E)1301 58 y Fh(\034)1293 116 y Fj(\003)1347 91 y Fp(.)55 b(Again)34 b(w)m(e)i(de\014ne)e Fi(E)2150 105 y Fj(\003)2204 91 y Fp(\()p Fo(f)10 b Fp(\))33 b(:)g(\012)f Fi(7!)h Ff(R)2702 105 y Fj(+)2803 91 y Fp(as)i(the)g(function)f Fo(\034)43 b Fi(!)32 b(E)3703 58 y Fh(\034)3695 116 y Fj(\003)3750 91 y Fp(\()p Fo(f)10 b Fp(\).)0 218 y(T)m(ypically)34 b([Gr2])j Fi(E)702 185 y Fh(\034)694 243 y Fj(\003)785 218 y Fp(is)e(the)i(Diric)m(hlet)e(form)h(asso)s(ciated)g(to)h(the)g (generator)g Fo(L)2837 185 y Fh(\034)2837 243 y Fj(\003)2928 218 y Fp(of)f(a)h(symmetric,)g(p)s(ositiv)m(e)0 344 y(preserving,)31 b(con)m(traction)i(semigroup)d Fo(e)1422 311 y Fh(tL)1499 284 y Fk(\034)1499 329 y Fg(\003)1550 344 y Fp(.)45 b(W)-8 b(e)32 b(will)d(pro)s(ceed,)j(ho)m(w)m(ev)m(er,)i(in)c(a)i(more)g (abstract)h(framew)m(ork,)0 471 y(since)d(all)f(w)m(e)i(need)f(are)h (the)f(follo)m(wing)f(general)i(prop)s(erties)d(of)j Fi(E)8 b Fp(:)4 620 y(\(E1\))46 b(There)31 b(exists)h(a)g(set)g Fi(A)f Fp(of)h(measurable)e(functions)h(whic)m(h)f(is)h(a)h(domain)e (for)i(all)e Fi(fE)3167 587 y Fh(\034)3159 645 y Fj(\003)3241 620 y Fp(:)43 b(\003)27 b Fi(2)g Ff(F)5 b Fo(;)36 b(\034)i Fi(2)27 b Fp(\012)p Fi(g)p Fp(,)227 747 y(and)j Fi(E)460 714 y Fh(\034)452 772 y Fj(\003)537 747 y Fp(maps)f Fi(A)h Fp(in)m(to)h([0)p Fo(;)15 b Fi(1)p Fp(\).)4 875 y(\(E2\))46 b(F)-8 b(or)31 b(all)f Fo(V)45 b Fi(\032)25 b Fp(\003)g Fi(2)g Ff(F)5 b Fp(,)36 b Fo(\034)g Fi(2)25 b Fp(\012,)30 b Fo(f)k Fi(2)25 b(A)p Fp(,)30 b(the)h(function)e Fi(E)2133 889 y Fh(V)2196 875 y Fp(\()p Fo(f)10 b Fp(\))30 b(is)g(in)f Fo(L)2611 842 y Fj(1)2652 875 y Fp(\()p Fo(\026)2742 842 y Fh(\034)2742 900 y Fj(\003)2796 875 y Fp(\).)4 1002 y(\(E3\))46 b(If)30 b(\003)c(=)f Fo(V)556 1016 y Fj(1)617 1002 y Fi([)19 b Fo(V)750 1016 y Fj(2)791 1002 y Fp(,)31 b(then)f Fo(\026)1109 969 y Fh(\034)1109 1027 y Fj(\003)1163 1002 y Fp([)p Fi(E)1236 1016 y Fh(V)1279 1025 y Fg(1)1319 1002 y Fp(\()p Fo(f)10 b Fp(\))20 b(+)g Fi(E)1603 1016 y Fh(V)1646 1025 y Fg(2)1686 1002 y Fp(\()p Fo(f)10 b Fp(\)])26 b(=)f Fi(E)2014 969 y Fh(\034)2006 1027 y Fj(\003)2060 1002 y Fp(\()p Fo(f)10 b Fp(\))20 b(+)g Fo(\026)2351 969 y Fh(\034)2351 1027 y Fj(\003)2405 1002 y Fp(\()p Fi(E)2488 1016 y Fh(V)2531 1025 y Fg(1)2567 1016 y Fe(\\)p Fh(V)2659 1025 y Fg(2)2699 1002 y Fp(\()p Fo(f)10 b Fp(\)\).)0 1151 y(W)-8 b(e)26 b(do)e(not)h(discuss)e(densit)m (y)h(prop)s(erties)f(of)i(the)g(domain)e Fi(A)p Fp(,)j(and)e(our)g (statemen)m(t)i(will)c(b)s(e)i(giv)m(en)h(for)f(functions)0 1277 y Fo(f)42 b Fp(b)s(elonging)32 b(to)h Fi(A)p Fp(.)49 b(In)32 b(the)i(sp)s(eci\014c)d(study)i(of)g(a)g(particular)f(case,)j (one)e(can)h(in)m(v)m(estigate)g(the)f(p)s(ossibilit)m(y)d(of)0 1404 y(c)m(ho)s(osing)g Fi(A)f Fp(in)g(suc)m(h)h(a)g(w)m(a)m(y)h(that)g (all)e(statemen)m(ts)j(can)e(b)s(e)f(extended,)i(b)m(y)f(densit)m(y)-8 b(,)30 b(to)g(the)h(whole)e(domain)g(of)0 1531 y Fi(E)56 1498 y Fh(\034)48 1556 y Fj(\003)102 1531 y Fp(.)0 1679 y(F)-8 b(or)25 b(all)e(\003)i Fi(2)g Ff(F)5 b Fp(,)31 b(w)m(e)25 b(de\014ne)e(the)h Fd(lo)-5 b(garithmic)29 b(Sob)-5 b(olev)28 b(c)-5 b(onstant)35 b Fo(c)p Fp(\(\003\))26 b Fi(2)f Fp([0)p Fo(;)15 b Fi(1)p Fp(])25 b(as)g(the)f(in\014m)m(um)e (of)i(all)f(p)s(ositiv)m(e)0 1806 y(real)30 b(n)m(um)m(b)s(ers)f Fo(c)i Fp(suc)m(h)f(that)1083 1986 y(En)m(t)1229 1946 y Fh(\034)1229 2008 y Fj(\003)1283 1986 y Fp(\()p Fo(f)1373 1948 y Fj(2)1413 1986 y Fp(\))c Fi(\024)f Fp(2)p Fo(c)15 b Fi(E)1725 1948 y Fh(\034)1717 2008 y Fj(\003)1772 1986 y Fp(\()p Fo(f)10 b Fp(\))182 b Fi(8)p Fo(\034)35 b Fi(2)25 b Fp(\012)p Fo(;)106 b Fi(8)p Fo(f)34 b Fi(2)25 b(A)15 b Fo(:)0 2166 y Fp(A)m(t)25 b(this)f(lev)m(el)g(of)g(abstraction)h(it)e (is)h(clear)g(that)h(w)m(e)g(ha)m(v)m(e)h(to)f(a)m(v)m(oid)f (situations)g(where)f(a)i(LSI)f(fails,)g(for)g(instance,)0 2292 y(already)30 b(for)g(a)h(single)e(spin.)39 b(W)-8 b(e)31 b(need)f(one)h(last)f(assumption)4 2442 y(\(E4\))46 b(The)30 b(quan)m(tit)m(y)h Fo(c)p Fp(\(\003\))g(is)f(\014nite)f(for)h (all)f(\003)d Fi(2)f Ff(F)t Fp(.)0 2591 y(As)38 b(it)g(will)e(b)s(e)h (apparen)m(t)i(from)e(the)i(pro)s(of)e(b)s(elo)m(w,)j(assumption)d (\(E4\))i(can)g(b)s(e)e(replaced)h(b)m(y)g(the)g(someho)m(w)0 2717 y(milder)e(statemen)m(t)j(that)g Fo(c)p Fp(\(\003\))g(is)d (\014nite)h(for)h(all)f(\003)g Fi(2)h Ff(F)48 b Fp(whose)38 b(diameter)f(do)s(es)h(not)g(exceed)h Fo(d)3436 2731 y Fj(0)3477 2717 y Fp(,)h(where)d Fo(d)3859 2731 y Fj(0)0 2844 y Fp(dep)s(ends)29 b(on)h(the)g(complete)h(analyticit)m(y)f (constan)m(ts)i Fo(K)37 b Fp(and)30 b Fo(m)p Fp(.)0 2993 y(Ha)m(ving)h(stated)g(the)f(h)m(yp)s(otheses)h(on)f Fi(E)1386 2960 y Fh(\034)1378 3018 y Fj(\003)1432 2993 y Fp(,)h(w)m(e)f(men)m(tion)g(t)m(w)m(o)i(classical)e(examples)g(where) f(they)i(hold)66 3142 y(\(1\))46 b(The)27 b(\014rst)f(is)g(the)h(Glaub) s(er)f(dynamics)g(when)g Fo(S)k Fp(=)25 b Fi(f\000)p Fp(1)p Fo(;)15 b Fp(1)p Fi(g)29 b Fp(\(or)e(an)m(y)g(\014nite)f(set\).) 41 b(The)26 b(Diric)m(hlet)g(form)h(is)227 3269 y(giv)m(en)h(b)m(y)g Fi(E)642 3236 y Fh(\034)634 3294 y Fj(\003)688 3269 y Fp(\()p Fo(f)10 b Fp(\))25 b(=)g Fo(\026)989 3236 y Fh(\034)989 3294 y Fj(\003)1043 3269 y Fp(\()1078 3201 y Fm(P)1175 3296 y Fh(x)p Fe(2)p Fj(\003)1334 3269 y Fo(c)1373 3283 y Fh(x)1419 3269 y Fp(\()p Fo(D)1529 3283 y Fh(x)1576 3269 y Fo(f)10 b Fp(\))1666 3236 y Fj(2)1706 3269 y Fp(\),)29 b(where)f Fo(c)2095 3283 y Fh(x)2166 3269 y Fp(:)d(\012)g Fi(7!)g Ff(R)2483 3283 y Fj(+)2577 3269 y Fp(are)k(measurable,)e (usually)f(b)s(ounded,)227 3395 y(functions,)33 b(called)f(the)i (transition)e(rates,)i(and)f Fo(D)1992 3409 y Fh(x)2071 3395 y Fp(is)g(the)g(discrete)g(gradien)m(t)g(de\014ned)f(as)i(follo)m (ws:)45 b(for)227 3522 y Fo(\033)35 b Fi(2)d Fp(\012,)j Fo(x)d Fi(2)f Ff(Z)774 3489 y Fh(d)812 3522 y Fp(,)36 b(let)e Fo(\033)1063 3489 y Fh(x)1141 3522 y Fi(2)d Fp(\012)j(b)s(e)g (the)g(con\014guration)g(obtained)g(from)g Fo(\033)s Fp(,)h(b)m(y)g(\015ipping)c(the)j(spin)f(at)i Fo(x)p Fp(.)227 3649 y(Then)30 b Fo(D)540 3663 y Fh(x)586 3649 y Fo(f)10 b Fp(\()p Fo(\033)s Fp(\))25 b(:=)g Fo(f)10 b Fp(\()p Fo(\033)1057 3616 y Fh(x)1103 3649 y Fp(\))21 b Fi(\000)e Fo(f)10 b Fp(\()p Fo(\033)s Fp(\).)41 b Fi(A)30 b Fp(can)h(b)s(e)f(tak)m(en)h(as)g(the)f(set)h(of)g(all)e(lo)s(cal)h (functions.)66 3776 y(\(2\))46 b(In)37 b(the)g(second)g(example)g Fo(S)42 b Fp(is)37 b(a)g(connected,)j(compact)f(Riemannian)c(manifold,) i(and)g Fo(\027)42 b Fp(is)37 b(the)g(nor-)227 3903 y(malized)g (Riemannian)f(measure.)64 b(Consider)36 b(the)j(di\013usion)c(on)j Fo(S)2616 3870 y Fj(\003)2670 3903 y Fp(,)i(under)d(b)s(oundary)f (condition)h Fo(\034)10 b Fp(,)227 4030 y(with)30 b(generator)839 3961 y Fm(P)935 4057 y Fh(x)p Fe(2)p Fj(\003)1080 4030 y Fp(\(\001)1191 4044 y Fh(x)1257 4030 y Fi(\000)20 b(r)1424 4044 y Fh(x)1470 4030 y Fo(H)1553 3997 y Fh(\034)1546 4055 y Fj(\003)1600 4030 y Fp(\),)31 b(where)f Fi(r)2030 4044 y Fh(x)2106 4030 y Fp(and)g(\001)2359 4044 y Fh(x)2435 4030 y Fp(are)h(resp)s(ectiv)m(ely)f(the)h(gradien)m(t)g(and)f(the)227 4156 y(Laplacian)i(on)h(the)g Fo(x)985 4123 y Fh(th)1092 4156 y Fp(cop)m(y)h(of)f Fo(S)5 b Fp(.)48 b(In)32 b(this)g(case)i(w)m (e)g(ha)m(v)m(e)g Fi(E)2443 4123 y Fh(\034)2435 4181 y Fj(\003)2489 4156 y Fp(\()p Fo(f)10 b Fp(\))29 b(=)h Fo(\026)2799 4123 y Fh(\034)2799 4181 y Fj(\003)2853 4156 y Fp(\()2888 4088 y Fm(P)2984 4183 y Fh(x)p Fe(2)p Fj(\003)3144 4156 y Fi(jr)3245 4170 y Fh(x)3290 4156 y Fo(f)10 b Fi(j)3370 4123 y Fj(2)3411 4156 y Fp(\))33 b(and)f Fi(A)h Fp(can)227 4283 y(b)s(e)g(c)m(hosen)h(as)g(the)f(set)h (of)g(all)e(functions)g Fo(f)43 b Fp(whic)m(h)32 b(can)h(b)s(e)g (written)g(as)g Fo(f)40 b Fp(=)30 b Fo(g)c Fi(\016)c Fo(\031)3155 4297 y Fh(V)3252 4283 y Fp(for)33 b(some)h Fo(V)50 b Fi(2)30 b Ff(F)5 b Fp(,)227 4409 y Fo(g)29 b Fi(2)c Fo(C)457 4376 y Fe(1)534 4409 y Fp(\()p Fo(S)630 4376 y Fh(V)693 4409 y Fp(\).)0 4558 y(The)e(only)g(non)m(trivial)f (prop)s(ert)m(y)h(w)m(e)h(ha)m(v)m(e)h(to)g(c)m(hec)m(k)g(in)e(these)h (cases)h(is)d(\()p Fo(E)5 b Fp(4\):)39 b(when)23 b Fo(J)35 b Fp(=)24 b(0,)i(w)m(e)e(ha)m(v)m(e)h Fo(\026)3594 4525 y Fh(\034)3594 4583 y Fj(\003)3674 4558 y Fp(=)g Fo(\027)3821 4525 y Fj(\003)3875 4558 y Fp(,)0 4685 y(so)38 b(it)f(is)f(su\016cien)m (t)h(to)h(sho)m(w)f(that)h(a)g(LSI)e(holds)g(for)h Fo(\027)44 b Fp(with)36 b(some)h(log-Sob)s(olev)g(constan)m(t)k(^)-47 b Fo(c)p Fp(.)62 b(This)36 b(fact)i(is)0 4811 y(trivial)27 b(in)g(the)i(discrete)f(case,)i(where)e(the)h(optimal)f(constan)m(t)h (is)f(kno)m(wn)g(\(see)i([Le1],)g(Ch.5\),)f(and)f(true)h(in)e(case)0 4938 y(\(2\))39 b(\(see)h([W)-8 b(a])40 b(for)e(upp)s(er)e(b)s(ounds)g (on)k(^)-47 b Fo(c)q Fp(\).)64 b(In)38 b(order)g(to)h(deal)f(with)f(a)h (non)g(zero)h(in)m(teraction,)i(w)m(e)d(observ)m(e)0 5065 y(that)30 b Fo(\026)251 5032 y Fh(\034)251 5090 y Fj(\003)335 5065 y Fp(has)f(densit)m(y)g(exp\()p Fi(\000)p Fo(H)1137 5032 y Fh(\034)1130 5090 y Fj(\003)1184 5065 y Fp(\))p Fo(=)-5 b(Z)1328 5032 y Fh(\034)1321 5090 y Fj(\003)1406 5065 y Fp(w.r.t.)40 b Fo(\027)1709 5032 y Fj(\003)1763 5065 y Fp(.)h(It)29 b(is)g(w)m(ell)g(kno)m(wn)g(\(see)i (Lemma)e(3.5)i(in)d([SZ1]\))i(that)h(since)0 5191 y Fi(k)p Fo(H)128 5158 y Fh(\034)121 5216 y Fj(\003)175 5191 y Fi(k)220 5205 y Fh(u)294 5191 y Fi(\024)25 b(j)p Fp(\003)p Fi(jk)p Fo(J)9 b Fi(k)p Fp(,)32 b(w)m(e)f(can)f(con)m(trol)h Fo(c)p Fp(\(\003\))h(in)d(terms)h(of)i(^)-47 b Fo(c)q Fp(.)40 b(More)32 b(precisely)d(w)m(e)h(ha)m(v)m(e)i Fo(c)p Fp(\(\003\))26 b Fi(\024)h Fp(^)-47 b Fo(c)16 b(e)3428 5158 y Fj(4)p Fe(j)p Fj(\003)p Fe(jk)p Fh(J)6 b Fe(k)3683 5191 y Fp(.)0 5340 y(The)30 b(main)f(result)g(in)g(this)h (section)g(is)0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)h Fj([10])1354 b Fp(4.4)p eop %%Page: 11 11 11 10 bop 0 91 a Fq(Theorem)34 b(4.1.)53 b Fc(Let)32 b Fo(J)39 b Fc(b)s(e)31 b(a)g(translation)f(in)m(v)-5 b(arian)m(t,)30 b(summable)f(in)m(teraction)i(of)g(\014nite)e(range)j Fo(r)h Fc(suc)m(h)d(that)0 218 y(assumptions)f(\(CA\))i(holds,)e(and)g (let)i Fi(fE)1424 185 y Fh(\034)1416 243 y Fj(\003)1496 218 y Fp(:)40 b(\003)26 b Fi(2)f Ff(F)t Fo(;)37 b(\034)e Fi(2)25 b Fp(\012)p Fi(g)30 b Fc(satisfy)g(conditions)f Fp(\()p Fo(E)5 b Fp(1\))p Fo(;)15 b(:)g(:)g(:)j(;)d Fp(\()p Fo(E)5 b Fp(4\))p Fc(.)43 b(Then)1626 447 y Fp(sup)1625 523 y Fj(\003)p Fe(2)p Fl(F)1778 447 y Fo(c)p Fp(\(\003\))27 b Fo(<)e Fp(+)p Fi(1)15 b Fo(:)0 739 y Fd(Pr)-5 b(o)g(of.)32 b Fp(W)-8 b(e)31 b(start)g(with)e(the)i(follo)m(wing)e(result)0 920 y Fq(Lemma)42 b(4.2.)53 b Fc(Let)38 b Fp(\003)h Fi(2)f Ff(F)t Fc(,)47 b(and)37 b(let)h Fo(V)1463 934 y Fj(1)1504 920 y Fo(;)15 b(V)1597 934 y Fj(2)1676 920 y Fc(b)s(e)38 b(t)m(w)m(o)i(subsets)d(of)h Fp(\003)p Fc(,)j(suc)m(h)d(that)g Fp(\003)h(=)f Fo(V)3231 934 y Fj(1)3297 920 y Fi([)25 b Fo(V)3436 934 y Fj(2)3477 920 y Fc(.)64 b(Let)39 b Fo(l)h Fp(:=)0 1047 y Fo(d)p Fp(\(\003)p Fi(n)p Fo(V)243 1061 y Fj(1)285 1047 y Fo(;)15 b Fp(\003)p Fi(n)p Fo(V)486 1061 y Fj(2)528 1047 y Fp(\))p Fc(.)41 b(Assume)30 b(that)1447 1175 y Fi(j)p Fp(\()p Fo(@)1560 1137 y Fj(+)1555 1197 y Fh(r)1622 1175 y Fo(V)1675 1189 y Fj(2)1715 1175 y Fp(\))21 b Fi(\\)f Fp(\003)p Fi(j)15 b Fo(K)22 b(e)2096 1137 y Fe(\000)p Fh(ml)2271 1175 y Fi(\024)j Fp(1)15 b Fo(:)1261 b Fp(\(4)p Fo(:)p Fp(5\))0 1362 y Fc(Then)29 b(there)i(exists)f Fo(l)748 1376 y Fj(0)814 1362 y Fp(=)25 b Fo(l)937 1376 y Fj(0)978 1362 y Fp(\()p Fo(K)q(;)15 b(m)p Fp(\))32 b Fc(suc)m(h)e(that)h(for)f(all)f Fo(l)e Fi(\025)e Fo(l)2122 1376 y Fj(0)2163 1362 y Fc(,)31 b(for)f(all)f Fo(\034)36 b Fi(2)24 b Fp(\012)608 1591 y(En)m(t)753 1551 y Fh(\034)753 1613 y Fj(\003)807 1591 y Fp(\()p Fo(f)897 1553 y Fj(2)937 1591 y Fp(\))i Fi(\024)f Fp(2)1154 1517 y Fm(\000)1196 1591 y Fp(1)c(+)f Fo(K)1437 1553 y Fe(0)1462 1591 y Fo(e)1504 1553 y Fe(\000)p Fh(ml)1654 1517 y Fm(\001)1710 1591 y Fp([)p Fo(c)p Fp(\()p Fo(V)1862 1605 y Fj(1)1904 1591 y Fp(\))h Fi(_)f Fo(c)p Fp(\()p Fo(V)2168 1605 y Fj(2)2209 1591 y Fp(\)])2284 1517 y Fm(\002)2338 1591 y Fi(E)2394 1553 y Fh(\034)2386 1613 y Fj(\003)2440 1591 y Fp(\()p Fo(f)10 b Fp(\))20 b(+)g Fo(\026)2731 1553 y Fh(\034)2731 1613 y Fj(\003)2785 1591 y Fp(\()p Fi(E)2868 1605 y Fh(V)2911 1614 y Fg(1)2947 1605 y Fe(\\)p Fh(V)3039 1614 y Fg(2)3079 1591 y Fp(\()p Fo(f)10 b Fp(\)\))3254 1517 y Fm(\003)3713 1591 y Fp(\(4)p Fo(:)p Fp(6\))0 1819 y Fc(where)30 b Fo(K)347 1786 y Fe(0)397 1819 y Fp(=)25 b(2)p Fo(e\013K)39 b Fc(and)29 b Fo(\013)i Fc(is)f(giv)m(en)g(in)f(Prop)s(osition)f(2.1.)0 2000 y Fd(Pr)-5 b(o)g(of)34 b(of)g(L)-5 b(emma)34 b(4.2)p Fp(.)43 b(Let)31 b Fo(\034)36 b Fi(2)26 b Fp(\012)k(and)h(let)f Fo(F)1651 2014 y Fj(\003)p Fh(;\034)1795 2000 y Fp(:=)c Fi(f)p Fo(\033)j Fi(2)d Fp(\012)g(:)g Fo(\033)2325 2014 y Fj(\003)2375 1996 y Fk(c)2439 2000 y Fp(=)g Fo(\034)2576 2014 y Fj(\003)2626 1996 y Fk(c)2664 2000 y Fi(g)p Fp(.)43 b(By)31 b(\(4.2\))h Fo(\026)3192 1967 y Fh(\034)3192 2029 y Fj(\003)p Fh(;)p Fj(\003)p Fe(n)p Fh(V)3394 2038 y Fg(1)3464 2000 y Fp(is)e(a)h(con)m(v)m(ex)0 2127 y(com)m(bination)f (of)g Fi(f)p Fo(\026)721 2094 y Fh(\033)721 2157 y(V)764 2166 y Fg(2)800 2157 y Fh(;)p Fj(\003)p Fe(n)p Fh(V)952 2166 y Fg(1)1017 2127 y Fp(:)25 b Fo(\033)k Fi(2)24 b Fo(F)1291 2141 y Fj(\003)p Fh(;\034)1409 2127 y Fi(g)p Fp(,)31 b(th)m(us)786 2347 y Fm(\015)786 2401 y(\015)786 2456 y(\015)847 2371 y Fo(\032)894 2338 y Fh(!)894 2400 y(V)937 2409 y Fg(2)973 2400 y Fh(;)p Fj(\003)p Fe(n)p Fh(V)1125 2409 y Fg(1)p 847 2431 317 4 v 862 2514 a Fo(\032)909 2483 y Fh(\034)909 2545 y Fj(\003)p Fh(;)p Fj(\003)p Fe(n)p Fh(V)1111 2554 y Fg(1)1196 2451 y Fi(\000)19 b Fp(1)1331 2347 y Fm(\015)1331 2401 y(\015)1331 2456 y(\015)1382 2515 y Fh(u)1455 2451 y Fi(\024)107 b Fp(sup)1551 2528 y Fh(\033)n(;\021)r Fe(2)p Fh(F)1747 2537 y Fg(\003)p Fk(;\034)1851 2347 y Fm(\015)1851 2401 y(\015)1851 2456 y(\015)1913 2371 y Fo(\032)1960 2338 y Fh(\033)1960 2400 y(V)2003 2409 y Fg(2)2038 2400 y Fh(;)p Fj(\003)p Fe(n)p Fh(V)2190 2409 y Fg(1)p 1913 2431 V 1913 2514 a Fo(\032)1960 2470 y Fh(\021)1960 2545 y(V)2003 2554 y Fg(2)2038 2545 y Fh(;)p Fj(\003)p Fe(n)p Fh(V)2190 2554 y Fg(1)2261 2451 y Fi(\000)20 b Fp(1)2397 2347 y Fm(\015)2397 2401 y(\015)2397 2456 y(\015)2447 2515 y Fh(u)2677 2451 y Fi(8)p Fo(!)27 b Fi(2)e Fo(F)2956 2465 y Fj(\003)p Fh(;\034)3089 2451 y Fo(:)599 b Fp(\(4)p Fo(:)p Fp(7\))0 2767 y(Moreo)m(v)m(er,)34 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b(and)f(from)g(assumption)e(\()p Fo(E)5 b Fp(3\),)33 b(it)c(follo)m(ws)307 316 y Fo(\026)362 279 y Fh(\034)362 339 y Fj(\003)431 242 y Fm(\002)484 316 y Fp(En)m(t)629 330 y Fh(V)672 339 y Fg(1)712 316 y Fp(\()p Fo(f)802 279 y Fj(2)842 316 y Fp(\))21 b(+)f(En)m(t)1134 330 y Fh(V)1177 339 y Fg(2)1217 316 y Fp(\()p Fo(f)1307 279 y Fj(2)1347 316 y Fp(\))1397 242 y Fm(\003)1461 316 y Fi(\024)25 b Fp(2)15 b([)p Fo(c)p Fp(\()p Fo(V)1769 330 y Fj(1)1811 316 y Fp(\))21 b Fi(_)f Fo(c)p Fp(\()p Fo(V)2075 330 y Fj(2)2116 316 y Fp(\)])2191 242 y Fm(\002)2245 316 y Fi(E)2301 279 y Fh(\034)2293 339 y Fj(\003)2347 316 y Fp(\()p Fo(f)10 b Fp(\))20 b(+)g Fo(\026)2638 279 y Fh(\034)2638 339 y Fj(\003)2692 316 y Fp(\()p Fi(E)2775 330 y Fh(V)2818 339 y Fg(1)2854 330 y Fe(\\)p Fh(V)2946 339 y Fg(2)2986 316 y Fp(\()p Fo(f)10 b Fp(\)\))3161 242 y Fm(\003)3290 316 y Fb(\003)307 b Fp(\(4)p Fo(:)p Fp(11\))0 594 y Fd(Pr)-5 b(o)g(of)32 b(of)f(The)-5 b(or)g(em)33 b(4.1)p Fp(.)40 b(Previous)28 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y(the)h(extra)g(term)g Fo(\026)677 1070 y Fh(\034)677 1128 y Fj(\003)731 1103 y Fp(\()p Fi(E)814 1117 y Fh(V)857 1126 y Fg(1)893 1117 y Fe(\\)p Fh(V)985 1126 y Fg(2)1025 1103 y Fp(\()p Fo(f)10 b Fp(\)\).)54 b(A)34 b(trivial)f(upp)s(er)g(b)s (ound)f(for)j(this)f(term)g(is)g Fi(E)2889 1070 y Fh(\034)2881 1128 y Fj(\003)2935 1103 y Fp(,)i(but)e(this)g(is)f(fatal)i(to)h(the)0 1230 y(argumen)m(t)26 b(since)e(it)h(yields)f Fo(c)p Fp(\(\003\))i Fi(\024)f Fp(2\(1)10 b(+)g Fo(K)1550 1197 y Fe(0)1576 1230 y Fo(e)1618 1197 y Fe(\000)p Fh(ml)1768 1230 y Fp(\))15 b(\()p Fo(c)p Fp(\()p Fo(V)1980 1244 y Fj(1)2022 1230 y Fp(\))10 b Fi(_)g Fo(c)p Fp(\()p Fo(V)2265 1244 y Fj(2)2307 1230 y Fp(\)\).)40 b(Ho)m(w)m(ev)m(er)27 b(it)e(w)m(as)g(observ)m(ed)h(in)e([Ma])i(that)0 1357 y(one)34 b(can)g(write)f(man)m(y)-8 b(,)35 b(sa)m(y)g Fo(r)s Fp(,)f(di\013eren)m(t)f(replicas)g(of)h(inequalit)m(y)e (\(4.6\),)k(eac)m(h)f(corresp)s(onding)d(to)i(a)g(di\013eren)m(t)0 1484 y(c)m(hoice)f(of)g Fo(V)431 1498 y Fj(1)472 1484 y Fp(,)g Fo(V)583 1498 y Fj(2)656 1484 y Fp(suc)m(h)f(that)h(the)f (sets)h Fo(V)1453 1498 y Fj(1)1516 1484 y Fi(\\)21 b Fo(V)1651 1498 y Fj(2)1724 1484 y Fp(are)33 b(disjoin)m(t)d(for)j (di\013eren)m(t)e(replicas.)46 b(A)m(t)33 b(this)e(p)s(oin)m(t)h(w)m(e) h(can)0 1611 y(add)e(together)h(all)e(the)i(inequalities)c(obtained)j (and)f(the)i Fd(sum)38 b Fp(of)32 b(all)e(the)h Fo(r)j Fp(extra)e(terms)f(is)f(still)f(b)s(ounded)g(b)m(y)0 1739 y Fi(E)56 1706 y Fh(\034)48 1764 y Fj(\003)102 1739 y Fp(\()p Fo(f)10 b Fp(\).)41 b(In)30 b(this)f(w)m(a)m(y)i(w)m(e)g(get) 1033 1964 y Fo(c)p Fp(\(\003\))26 b Fi(\024)f Fp(\(1)c(+)f Fo(K)1603 1926 y Fe(0)1628 1964 y Fo(e)1670 1926 y Fe(\000)p Fh(ml)1820 1964 y Fp(\))15 b(\(1)21 b(+)f(1)p Fo(=r)s Fp(\))15 b(\()p Fo(c)p Fp(\()p Fo(V)2408 1978 y Fj(1)2451 1964 y Fp(\))21 b Fi(_)f Fo(c)p Fp(\()p Fo(V)2715 1978 y Fj(2)2756 1964 y Fp(\)\))15 b Fo(:)802 b Fp(\(4)p Fo(:)p Fp(12\))0 2189 y(and,)29 b(if)e Fo(r)k Fp(is)d(a)h(function)e(of)i(the) 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Fp(])21 b Fi(\002)e Fp([0)p Fo(;)c(l)1725 3091 y Fh(k)r Fj(+2)1863 3077 y Fp(])21 b Fi(\002)f(\001)15 b(\001)g(\001)21 b(\002)f Fp([0)p Fo(;)15 b(l)2354 3091 y Fh(k)r Fj(+)p Fh(d)2494 3077 y Fp(])2534 3003 y Fm(\001)2596 3077 y Fi(\\)20 b Ff(Z)2742 3039 y Fh(d)0 3302 y Fp(Let)31 b(also)f Fo(G)416 3316 y Fh(k)486 3302 y Fp(:=)25 b(sup)744 3324 y Fh(V)16 b Fe(2)p Fl(F)890 3334 y Fk(k)950 3302 y Fo(c)p Fp(\()p Fo(V)21 b Fp(\).)41 b(The)29 b(idea)h(b)s(ehind)d (this)i(construction)h(is)f(that)h(eac)m(h)h(rectangle)g(in)e Ff(F)3610 3316 y Fh(k)3661 3302 y Fi(n)p Ff(F)3756 3316 y Fh(k)r Fe(\000)p Fj(1)0 3429 y Fp(can)38 b(b)s(e)f(obtained)g(as)h(a) g(\\sligh)m(tly)e(o)m(v)m(erlapping)h(union")f(of)i(t)m(w)m(o)h (rectangles)f(in)e Ff(F)2944 3443 y Fh(k)r Fe(\000)p Fj(1)3088 3429 y Fp(.)62 b(More)38 b(precisely)e(w)m(e)0 3556 y(ha)m(v)m(e:)0 3736 y Fq(Prop)s(osition)28 b(4.3.)53 b Fc(F)-8 b(or)24 b(all)e Fo(k)29 b Fi(2)c Ff(Z)1274 3750 y Fj(+)1330 3736 y Fc(,)f(for)g(all)e Fp(\003)j Fi(2)g 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Fp(b)s(elong)f(to)i Ff(F)2414 109 y Fh(k)r Fe(\000)p Fj(1)2558 95 y Fp(.)46 b(In)32 b(fact)h(w)m(e)g(kno)m(w)f(that)h Fo(b)3539 109 y Fh(d)3610 95 y Fi(\024)28 b Fo(l)3736 109 y Fh(k)r Fj(+)p Fh(d)3875 95 y Fp(,)0 222 y(th)m(us,)i(for)g(all)g Fo(i)25 b Fp(=)g(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(s)933 236 y Fh(k)694 456 y Fo(b)733 470 y Fh(d)p 694 497 82 4 v 712 580 a Fp(2)807 518 y(+)908 456 y(2)p Fo(i)p 908 497 77 4 v 924 580 a Fp(8)996 434 y Fm(p)p 1087 434 72 4 v 84 x Fo(l)1114 532 y Fh(k)1184 518 y Fi(\024)1291 456 y Fo(b)1330 470 y Fh(d)p 1291 497 82 4 v 1309 580 a Fp(2)1403 518 y(+)1505 456 y(2)p Fo(s)1593 470 y Fh(k)p 1505 497 133 4 v 1549 580 a Fp(8)1649 434 y Fm(p)p 1740 434 72 4 v 84 x Fo(l)1767 532 y Fh(k)1837 518 y Fi(\024)1944 456 y Fo(l)1971 470 y Fh(k)r Fj(+)p Fh(d)p 1944 497 166 4 v 2004 580 a Fp(2)2141 518 y(+)2242 456 y(1)p 2242 497 46 4 v 2242 580 a(4)2299 518 y Fo(l)2328 471 y Fj(5)p Fh(=)p Fj(6)2326 545 y Fh(k)2467 518 y Fp(=)2574 456 y(3)p Fo(l)2646 470 y Fh(k)p 2574 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y(1)3417 1250 y Fp(and)27 b Fo(V)3664 1203 y Fj(\()p Fh(i)p Fj(\))3644 1274 y(2)3779 1250 y Fp(are)0 1378 y(nonempt)m(y)-8 b(,)34 b(since,)g(using)e(\(4.13\),)k(it)d(easy)h(to)g(see)g(that,)g (otherwise,)g(\003)f(itself)f(w)m(ould)g(b)s(elong)h(to)h Ff(F)3493 1392 y Fh(k)r Fe(\000)p Fj(1)3670 1378 y Fp(whic)m(h)0 1505 y(is)29 b(excluded)h(b)m(y)g(h)m(yp)s(othesis.)39 b(W)-8 b(e)32 b(ha)m(v)m(e)f(then)g Fo(d)p Fp(\(\003)p Fi(n)p Fo(V)1915 1458 y Fj(\()p Fh(i)p Fj(\))1894 1529 y(1)2002 1505 y Fo(;)15 b Fp(\003)p Fi(n)p Fo(V)2224 1458 y Fj(\()p Fh(i)p Fj(\))2203 1529 y(2)2311 1505 y Fp(\))26 b Fi(\025)2479 1469 y Fj(1)p 2479 1484 37 4 v 2479 1537 a(8)2526 1435 y Fi(p)p 2602 1435 72 4 v 70 x Fo(l)2629 1519 y Fh(k)2694 1505 y Fi(\000)20 b Fp(2)25 b(=)g Fo(\016)2991 1519 y Fh(k)3037 1505 y Fp(.)91 b Fb(\003)0 1696 y Fp(W)-8 b(e)32 b(can)f(conclude)f(the)h(pro)s(of)f(of) h(Theorem)g(4.1.)43 b(Cho)s(ose)30 b(a)h(p)s(ositiv)m(e)f(in)m(teger)h Fo(k)2842 1710 y Fj(0)2909 1696 y Fp(=)26 b Fo(k)3053 1710 y Fj(0)3094 1696 y Fp(\()p Fo(K)q(;)15 b(m)p Fp(\))32 b(large)f(enough)0 1823 y(suc)m(h)f(that)86 1987 y(\(i\))45 b Fo(\016)267 2001 y Fh(k)338 1987 y Fi(\025)25 b Fo(l)461 2001 y Fj(0)502 1987 y Fp(\()p Fo(K)q(;)15 b(m)p Fp(\))31 b(\()p Fo(l)863 2001 y Fj(0)935 1987 y Fp(w)m(as)g(de\014ned)e(in)g (Lemma)h(4.2\))61 2118 y(\(ii\))44 b(\()p Fo(l)289 2132 y Fh(k)328 2141 y Fg(0)364 2132 y Fj(+)p Fh(d)482 2118 y Fp(+)20 b(1\))653 2085 y Fh(d)696 2118 y Fo(K)7 b(e)822 2085 y Fe(\000)p Fh(m\016)977 2095 y Fk(k)1046 2118 y Fi(\024)25 b Fp(1)0 2279 y(Let)e(then)e Fo(k)29 b Fi(\025)c Fo(k)572 2293 y Fj(0)613 2279 y Fp(,)f(\003)h Fi(2)g Ff(F)885 2293 y Fh(k)936 2279 y Fp(,)f(and)d(let)h Fi(f)p Fo(V)1395 2232 y Fj(\()p Fh(i)p Fj(\))1374 2303 y(1)1482 2279 y Fo(;)15 b(V)1596 2232 y Fj(\()p Fh(i)p Fj(\))1575 2303 y(2)1683 2279 y Fi(g)1728 2238 y Fh(s)1762 2248 y Fk(k)1728 2304 y Fh(i)p Fj(=1)1872 2279 y Fp(b)s(e)22 b(the)g(sequence)g(giv)m(en)g(in)f(Prop)s(osition)f(4.3.)39 b(Prop)s(erties)0 2407 y(\(i\))30 b(and)g(\(ii\))g(allo)m(w)f(us)h(to)h (apply)e(Lemma)i(4.2)g(and)f(obtain,)g(for)g Fo(i)c Fp(=)f(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(s)2705 2421 y Fh(k)322 2651 y Fp(En)m(t)467 2611 y Fh(\034)467 2673 y Fj(\003)521 2651 y Fp(\()p Fo(f)611 2613 y Fj(2)652 2651 y Fp(\))25 b Fi(\024)g Fp(2)868 2577 y Fm(\000)911 2651 y Fp(1)20 b(+)g Fo(K)1151 2613 y Fe(0)1176 2651 y Fo(e)1218 2613 y Fe(\000)p Fh(m\016)1373 2623 y Fk(k)1417 2577 y Fm(\001)1474 2651 y Fp([)p Fo(c)p Fp(\()p Fo(V)1647 2603 y Fj(\()p Fh(i)p Fj(\))1626 2675 y(1)1734 2651 y Fp(\))h Fi(_)f Fo(c)p Fp(\()p Fo(V)2019 2603 y Fj(\()p Fh(i)p Fj(\))1998 2675 y(2)2106 2651 y Fp(\)])2181 2577 y Fm(\002)2235 2651 y Fi(E)2291 2613 y Fh(\034)2283 2673 y Fj(\003)2337 2651 y Fp(\()p Fo(f)10 b Fp(\))20 b(+)g Fo(\026)2628 2613 y Fh(\034)2628 2673 y Fj(\003)2682 2651 y Fp(\()p Fi(E)2765 2687 y Fh(V)2824 2653 y Fg(\()p Fk(i)p Fg(\))2808 2709 y(1)2902 2687 y Fe(\\)p Fh(V)3010 2653 y Fg(\()p Fk(i)p Fg(\))2994 2709 y(2)3092 2651 y Fp(\()p Fo(f)10 b Fp(\)\))3267 2577 y Fm(\003)3321 2651 y Fo(:)322 b Fp(\(4)p Fo(:)p Fp(14\))0 2935 y(Thanks)41 b(to)i(\(3\))g(of)g(Prop)s (osition)d(4.3,)1423 2867 y Fm(P)1519 2889 y Fh(s)1553 2899 y Fk(k)1519 2962 y Fh(i)p Fj(=1)1656 2935 y Fo(\026)1711 2902 y Fh(\034)1711 2960 y Fj(\003)1765 2935 y Fp(\()p Fi(E)1848 2971 y Fh(V)1907 2938 y Fg(\()p Fk(i)p Fg(\))1891 2993 y(1)1985 2971 y Fe(\\)p Fh(V)2093 2938 y Fg(\()p Fk(i)p Fg(\))2077 2993 y(2)2175 2935 y Fp(\()p Fo(f)10 b Fp(\)\))46 b Fi(\024)e(E)2552 2902 y Fh(\034)2544 2960 y Fj(\003)2599 2935 y Fp(\()p Fo(f)10 b Fp(\),)45 b(so)d(when)g(w)m(e)g (sum)g(\(4.14\))i(for)0 3062 y Fo(i)26 b Fp(=)f(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(s)443 3076 y Fh(k)487 3062 y Fp(,)31 b(and)f(divide)e(b)m(y)i Fo(s)1159 3076 y Fh(k)1234 3062 y Fp(w)m(e)h(get)965 3353 y(En)m(t)1110 3313 y Fh(\034)1110 3375 y Fj(\003)1164 3353 y Fp(\()p Fo(f)1254 3315 y Fj(2)1295 3353 y Fp(\))25 b Fi(\024)g Fp(2)15 b Fo(G)1582 3367 y Fh(k)r Fe(\000)p Fj(1)1736 3279 y Fm(\000)1778 3353 y Fp(1)20 b(+)g Fo(K)2018 3315 y Fe(0)2044 3353 y Fo(e)2086 3315 y Fe(\000)p Fh(m\016)2241 3325 y Fk(k)2285 3279 y Fm(\001)2341 3252 y(h)2384 3353 y Fp(1)h(+)2573 3292 y(1)p 2552 3332 88 4 v 2552 3415 a Fo(s)2595 3429 y Fh(k)2650 3252 y Fm(i)2708 3353 y Fi(E)2764 3315 y Fh(\034)2756 3375 y Fj(\003)2810 3353 y Fp(\()p Fo(f)10 b Fp(\))733 b(\(4)p Fo(:)p Fp(15\))0 3645 y(whic)m(h)29 b(yields)975 3827 y Fo(G)1046 3841 y Fh(k)1116 3827 y Fi(\024)c Fo(G)1283 3841 y Fh(k)r Fe(\000)p Fj(1)1436 3753 y Fm(\000)1478 3827 y Fp(1)c(+)f Fo(K)1719 3789 y Fe(0)1744 3827 y Fo(e)1786 3789 y Fe(\000)p Fh(m\016)1941 3799 y Fk(k)1985 3753 y Fm(\001)2042 3726 y(h)2085 3827 y Fp(1)g(+)2273 3765 y(1)p 2252 3806 V 2252 3889 a Fo(s)2295 3903 y Fh(k)2350 3726 y Fm(i)2575 3827 y Fi(8)p Fo(k)28 b Fi(\025)d Fo(k)2844 3841 y Fj(0)2900 3827 y Fo(:)743 b Fp(\(4)p Fo(:)p Fp(16\))0 4078 y(F)-8 b(rom)31 b(the)f(iteration)g(of)h(\(4.16\))i(w)m(e)d (obtain)g Fo(G)1618 4092 y Fh(k)1689 4078 y Fi(\024)25 b Fo(M)10 b(G)1954 4092 y Fh(k)1993 4101 y Fg(0)2033 4078 y Fp(,)30 b(for)h(all)e Fo(k)f Fi(\025)d Fo(k)2572 4092 y Fj(0)2613 4078 y Fp(,)31 b(where)1098 4404 y Fo(M)k Fp(:=)1391 4291 y Fe(1)1369 4318 y Fm(Y)1342 4514 y Fh(k)r Fj(=)p Fh(k)1478 4523 y Fg(0)1512 4303 y Fm(n)1573 4331 y(\000)1615 4404 y Fp(1)20 b(+)g Fo(K)1855 4367 y Fe(0)1880 4404 y Fo(e)1922 4367 y Fe(\000)p Fh(m\016)2077 4377 y Fk(k)2121 4331 y Fm(\001)2178 4303 y(h)2221 4404 y Fp(1)h(+)2409 4343 y(1)p 2389 4383 V 2389 4467 a Fo(s)2432 4481 y Fh(k)2487 4303 y Fm(io)2616 4404 y Fo(<)k Fi(1)0 4753 y Fp(thanks)36 b(to)h(the)g(explicit)e(expressions)g(of)h Fo(\016)1538 4767 y Fh(k)1620 4753 y Fp(and)g Fo(s)1846 4767 y Fh(k)1890 4753 y Fp(.)59 b(Finally)34 b(assumption)h(\(E4\))i (guaran)m(tees)h(that)f Fo(G)3723 4767 y Fh(k)3762 4776 y Fg(0)3839 4753 y Fp(is)0 4881 y(\014nite,)30 b(and)f(the)i(theorem)g (follo)m(ws)90 b Fb(\003)0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)31 b Fj([13])1354 b Fp(4.7)p eop %%Page: 14 14 14 13 bop 0 91 a Fq(References)-116 244 y Fp([BaEm])46 b(D.)28 b(Bakry)e(and)h(M.)g(Emery)-8 b(,)26 b Fd(Di\013usions)j(hyp)-5 b(er)g(c)g(ontr)g(actives)p Fp(,)31 b(S)m(\023)-43 b(eminaire)26 b(de)g(probabilit)m(\023)-43 b(es,)25 b(XIX.)i(Sprin-)227 370 y(ger,)k(Berlin,)e(1985,)k(pp.)c(177{206.)-64 498 y([BCC])45 b(L.)h(Bertini,)e(N.)i(Cancrini)d(and)i(F.)h(Cesi,)f Fd(The)h(sp)-5 b(e)g(ctr)g(al)49 b(gap)e(for)h(a)e(Glaub)-5 b(er{typ)g(e)49 b(dynamics)f(in)f(a)227 625 y(c)-5 b(ontinuous)34 b(gas)p Fp(,)d(Preprin)m(t)e(\(2000\))-1 753 y([BH])46 b(T.)28 b(Bo)s(dineau)g(and)f(B.)i(Hel\013er,)g Fd(The)h(lo)-5 b(g-Sob)g(olev)33 b(ine)-5 b(quality)31 b(for)g(unb)-5 b(ounde)g(d)32 b(spin)g(systems)p Fp(,)d(J.)f(F)-8 b(unct.)227 880 y(Anal.)30 b Fq(166)p Fp(,)i(no.)e(1,)h(168{178)j(\(1999\))-64 1008 y([DeSt])46 b(J.D.)31 b(Deusc)m(hel)g(and)f(D.W.)h(Stro)s(o)s(c)m (k,)g Fd(L)-5 b(ar)g(ge)34 b(deviations)p Fp(,)e(Academic)e(Press)g (Inc.,)h(Boston,)g(MA,)g(1989)-130 1136 y([DoSh1])46 b(R.)27 b(L.)f(Dobrushin)e(and)h(S.)h(B.)h(Shlosman,)e Fd(Constructive)k(criterion)h(for)f(the)g(uniqueness)f(of)h(Gibbs)g (\014eld)p Fp(,)227 1263 y(Statistical)35 b(ph)m(ysics)f(and)h (dynamical)f(systems)i(\(Ja\013e)g(F)-8 b(ritz)36 b(and)f(Sz\023)-45 b(asz,)36 b(eds.\),)g(Birkh\177)-45 b(auser)34 b(Boston,)227 1390 y(Mass.,)e(1985,)g(pp.)e(347{370)-130 1517 y([DoSh2])46 b(R.)21 b(L.)g(Dobrushin)e(and)h(S.)h(B.)h(Shlosman,)d Fd(Completely)26 b(analytic)-5 b(al)26 b(Gibbs)e(\014elds)p Fp(,)e(Statistical)f(ph)m(ysics)f(and)227 1644 y(dynamical)25 b(systems)i(\(Ja\013e)g(F)-8 b(ritz)27 b(and)f(Sz\023)-45 b(asz,)27 b(eds.\),)g(Birkh\177)-45 b(auser)25 b(Boston,)i(Mass.,)h (1985,)g(pp.)e(371{403)-130 1772 y([DoSh3])46 b(R.)25 b(L.)f(Dobrushin)e(and)h(S.)h(B.)h(Shlosman,)e Fd(Completely)28 b(analytic)-5 b(al)29 b(inter)-5 b(actions:)29 b(c)-5 b(onstructive)28 b(descrip-)227 1899 y(tion)p Fp(,)j(J.)g(Statist.)f (Ph)m(ys.)h Fq(46)p Fp(,)g(no.)f(5-6,)i(983{1014)i(\(1987\))-62 2027 y([DPP])46 b(P)-8 b(.)33 b(Dai)e(Pra,)h(A.)g(M.)h(P)m(aganoni)f (and)f(G.)h(P)m(osta,)i Fd(Entr)-5 b(opy)35 b(ine)-5 b(qualities)34 b(for)h(unb)-5 b(ounde)g(d)35 b(spin)g(systems)p Fp(,)227 2154 y(in)29 b(preparation)34 2282 y([F)-8 b(a])47 b(W.)34 b(G.)g(F)-8 b(aris,)34 b Fd(Pr)-5 b(o)g(duct)37 b(sp)-5 b(ac)g(es)37 b(and)f(Nelson)-9 b('s)35 b(ine)-5 b(quality)p Fp(,)35 b(Helv.)e(Ph)m(ys.)g(Acta)i Fq(48)p Fp(,)f(no.)g(5/6,)h(721{730)227 2409 y(\(1975\))20 2537 y([Ge])46 b(H.)31 b(O.)f(Georgii,)h Fd(Gibbs)h(me)-5 b(asur)g(es)34 b(and)g(phase)g(tr)-5 b(ansitions)p Fp(,)33 b(W)-8 b(alter)31 b(de)f(Gruyter,)g(1988.)-21 2664 y([Gr1])46 b(L.)31 b(Gross,)f Fd(L)-5 b(o)g(garithmic)35 b(Sob)-5 b(olev)34 b(ine)-5 b(qualities)p Fp(,)31 b(Am.)f(J.)h(Math.)g Fq(97)g Fp(1061{1083)j(\(1976\))-21 2792 y([Gr2])46 b(L.)23 b(Gross,)f Fd(L)-5 b(o)g(garithmic)28 b(Sob)-5 b(olev)26 b(ine)-5 b(qualities)26 b(and)g(c)-5 b(ontr)g(activity)27 b(pr)-5 b(op)g(erties)28 b(of)e(semigr)-5 b(oups)p Fp(,)24 b(Diric)m(hlet)227 2919 y(forms.)30 b(Lect.)i(Notes)f(in)e(Math.,)j(v)m (ol.)f(1563,)h(Springer{V)-8 b(erlag,)30 b(1993,)i(pp.)e(54{88.)-33 3047 y([HS1])46 b(R.)40 b(Holley)e(and)h(D.)h(Stro)s(o)s(c)m(k,)g Fd(L)-5 b(o)g(garithmic)43 b(Sob)-5 b(olev)41 b(ine)-5 b(qualities)42 b(and)f(sto)-5 b(chastic)43 b(Ising)e(mo)-5 b(dels)p Fp(,)41 b(J.)227 3174 y(Statist.)31 b(Ph)m(ys.)f Fq(46)p Fp(,)h(no.)g(5-6,)h(1159{1194)i(\(1987\))-33 3302 y([HS2])46 b(R.)30 b(A.)f(Holley)g(and)g(D.)h(W.)g(Stro)s(o)s(c)m (k,)g Fd(Uniform)i(and)g Fo(L)2161 3269 y Fj(2)2234 3302 y Fd(c)-5 b(onver)g(genc)g(e)32 b(in)g(one-dimensional)h(sto)-5 b(chastic)227 3429 y(Ising)33 b(mo)-5 b(dels)p Fp(,)32 b(Comm.)e(Math.)i(Ph)m(ys.)e Fq(123)p Fp(,)h(no.)g(1,)g(85{93)h (\(1989\))-11 3557 y([Le1])46 b(M.)22 b(Ledoux,)f Fd(Conc)-5 b(entr)g(ation)27 b(of)e(me)-5 b(asur)g(e)26 b(and)f(lo)-5 b(garithmic)26 b(Sob)-5 b(olev)26 b(ine)-5 b(qualities)p Fp(,)22 b(S)m(\023)-43 b(eminaire)20 b(de)h(Prob-)227 3683 y(abilit)m(\023)-43 b(es.)30 b(XXXIII.)h(Lecture)g(Notes)g(in)e (Mathematics)j(1709.)g(Springer,)c(Berlin,)h(1999,)k(pp.)c(120{216.)-11 3811 y([Le2])46 b(M.)37 b(Ledoux,)f Fd(L)-5 b(o)g(garithmic)41 b(Sob)-5 b(olev)39 b(ine)-5 b(qualities)39 b(for)f(unb)-5 b(ounde)g(d)40 b(spin)f(systems)g(r)-5 b(evisite)g(d)p Fp(,)37 b(Preprin)m(t)227 3938 y(\(1999\))4 4066 y([LO])45 b(R.)31 b(Lata )-25 b(la)31 b(and)f(K.)g(Oleszkiewicz,)f Fd(Betwe)-5 b(en)33 b(Sob)-5 b(olev)34 b(and)g(Poinc)-5 b(ar)n(\023)-44 b(e)7 b Fp(,)31 b(Preprin)m(t)e(\(2000\))14 4194 y([L)-8 b(Y])46 b(S.)33 b(L.)f(Lu)g(and)g(H.)h(T.)g(Y)-8 b(au,)33 b Fd(Sp)-5 b(e)g(ctr)g(al)38 b(Gap)d(and)h(L)-5 b(o)g(garithmic)37 b(Sob)-5 b(olev)36 b(Ine)-5 b(quality)35 b(for)h(Kawasaki)g(and)227 4321 y(Glaub)-5 b(er)34 b(Dynamics)p Fp(,)d(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fq(156)p Fp(,)i(399{433)h (\(1993\))3 4449 y([Ma])46 b(F.)35 b(Martinelli,)c Fd(L)-5 b(e)g(ctur)g(es)37 b(on)f(Glaub)-5 b(er)37 b(dynamics)g(for)g(discr)-5 b(ete)36 b(spin)g(mo)-5 b(dels)p Fp(,)36 b(Lectures)e(on)g(probabil-) 227 4576 y(it)m(y)39 b(theory)g(and)g(statistics)g(\(Sain)m(t-Flour,)f (1997\).)k(Lecture)d(Notes)h(in)e(Mathematics)i(1717)h(\(Berlin\),)227 4703 y(Springer,)29 b(Berlin,)g(1999,)j(pp.)e(93{191.)-68 4830 y([MO1])46 b(F.)32 b(Martinelli)c(and)j(E.)g(Olivieri,)c Fd(Appr)-5 b(o)g(ach)36 b(to)e(e)-5 b(quilibrium)33 b(of)g(Glaub)-5 b(er)34 b(dynamics)h(in)e(the)g(one)h(phase)227 4957 y(r)-5 b(e)g(gion)34 b(I:)e(The)h(attr)-5 b(active)34 b(c)-5 b(ase)p Fp(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)g Fq(161)p Fp(,)g(447{486)i(\(1994\))-68 5085 y([MO2])46 b(F.)32 b(Martinelli)c(and)j(E.)g(Olivieri,)c Fd(Appr)-5 b(o)g(ach)36 b(to)e(e)-5 b(quilibrium)33 b(of)g(Glaub)-5 b(er)34 b(dynamics)h(in)e(the)g(one)h(phase)227 5212 y(r)-5 b(e)g(gion)34 b(II:)e(The)h(gener)-5 b(al)34 b(c)-5 b(ase)p Fp(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fq(161)p Fp(,)h(487{514)j(\(1994\))23 5340 y([Ne])46 b(E.)31 b(Nelson,)f Fd(The)j(fr)-5 b(e)g(e)33 b(Marko\013)g(\014eld)p Fp(,)e(J.)f(F)-8 b(unctional)30 b(Analysis)f Fq(12)p Fp(,)i(211{227)i(\(1973\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)e Fj([14])1354 b Fp(4.8)p eop %%Page: 15 15 15 14 bop 19 91 a Fp([Ro])46 b(O.)27 b(Rothaus,)f Fd(A)n(nalytic)j(ine) -5 b(qualities,)30 b(isop)-5 b(erimetric)30 b(ine)-5 b(qualities,)30 b(and)g(lo)-5 b(garithmic)31 b(Sob)-5 b(olev)30 b(ine)-5 b(qual-)227 218 y(ities)p Fp(,)31 b(J.)f(F)-8 b(unct.)31 b(Anal.)f Fq(64)p Fp(,)h(296{313)j(\(1985\))-20 346 y([SZ1])45 b(D.W.)i(Stro)s(o)s(c)m(k)e(and)f(B.)i(Zegarlinski,)d Fd(The)j(lo)-5 b(garithmic)49 b(Sob)-5 b(olev)47 b(ine)-5 b(quality)46 b(for)h(c)-5 b(ontinuous)47 b(spin)227 474 y(systems)34 b(on)f(a)g(lattic)-5 b(e)p Fp(,)31 b(J.)g(F)-8 b(unct.)31 b(Anal.)f Fq(104)p Fp(,)h(no.)g(2,)g(299{326)i(\(1992\))-20 602 y([SZ1])45 b(D.)33 b(W.)f(Stro)s(o)s(c)m(k)g(and)f(B.)h (Zegarlinski,)e Fd(The)k(e)-5 b(quivalenc)g(e)33 b(of)i(the)f(lo)-5 b(garithmic)36 b(Sob)-5 b(olev)35 b(ine)-5 b(quality)34 b(and)227 729 y(the)f(Dobrushin{Shlosman)j(mixing)d(c)-5 b(ondition)p Fp(,)32 b(Comm)m(un.)e(Math.)h(Ph)m(ys.)f Fq(144)p Fp(,)i(303{323)h(\(1992\))-20 857 y([SZ3])45 b(D.)34 b(W.)g(Stro)s(o)s(c)m(k)g(and)e(B.)i(Zegarlinski,)d Fd(The)36 b(lo)-5 b(garithmic)38 b(Sob)-5 b(olev)36 b(ine)-5 b(quality)36 b(for)g(discr)-5 b(ete)36 b(spin)g(on)g(a)227 985 y(lattic)-5 b(e)p Fp(,)32 b(Comm)m(un.)d(Math.)j(Ph)m(ys.)e Fq(149)p Fp(,)h(175)h(\(1992\))0 1113 y([W)-8 b(a])47 b(F.-Y.)27 b(W)-8 b(ang,)27 b Fd(On)h(estimation)i(of)f(the)f(lo)-5 b(garithmic)31 b(Sob)-5 b(olev)29 b(c)-5 b(onstant)30 b(and)f(gr)-5 b(adient)30 b(estimates)f(of)g(he)-5 b(at)227 1240 y(semigr)g(oups)p Fp(,)32 b(Probab.)e(Theory)g(Related)h(Fields)e Fq(108)p Fp(,)i(no.)g(1,)g(87{101)i(\(1997\))-20 1368 y([Y)-8 b(o1])47 b(N.)24 b(Y)-8 b(oshida,)23 b Fd(The)k(lo)-5 b(g-Sob)g(olev)27 b(ine)-5 b(quality)27 b(for)g(we)-5 b(akly)27 b(c)-5 b(ouple)g(d)28 b(lattic)-5 b(e)27 b(\014eld)p Fp(,)e(Probab.)e(Theory)g(Related)227 1495 y(Fields)29 b Fq(115)p Fp(,)j(1{40)g(\(1999\))-20 1624 y([Y)-8 b(o2])47 b(N.)25 b(Y)-8 b(oshida,)23 b Fd(Applic)-5 b(ation)29 b(of)e(lo)-5 b(g-Sob)g(olev)28 b(ine)-5 b(quality)27 b(to)g(the)g(sto)-5 b(chastic)28 b(dynamics)g(of)f(unb)-5 b(ounde)g(d)28 b(spin)227 1751 y(systems)34 b(on)f(the)g(lattic)-5 b(e)p Fp(,)31 b(J.)g(F)-8 b(unct.)31 b(Anal.)f Fq(173)p Fp(,)h(74{102)i(\(2000\))-20 1879 y([Y)-8 b(o3])47 b(N.)21 b(Y)-8 b(oshida,)21 b Fd(The)j(e)-5 b(quivalenc)g(e)24 b(of)g(the)g(lo)-5 b(g-Sob)g(olev)25 b(ine)-5 b(quality)25 b(and)g(a)f(mixing)g(c)-5 b(ondition)25 b(for)f(unb)-5 b(ounde)g(d)227 2006 y(spin)33 b(systems)h(on)f(the)g(lattic)-5 b(e)p Fp(,)32 b(Preprin)m(t)c(\(1998\))35 2135 y([Ze])46 b(B.)28 b(Zegarlinski,)c Fd(The)29 b(str)-5 b(ong)31 b(de)-5 b(c)g(ay)30 b(to)g(e)-5 b(quilibrium)29 b(for)h(the)g(sto)-5 b(chastic)31 b(dynamics)f(of)g(unb)-5 b(ounde)g(d)30 b(spin)227 2262 y(systems)k(on)f(a)g(lattic)-5 b(e)p Fp(,)31 b(Comm)m(un.)f(Math.)h(Ph)m(ys.)g Fq(175)p Fp(,)g(401{432)i (\(1996\))0 5558 y Fg(6)p Fk(=lug)r(lio=)p Fg(2000)e Fj([15])1354 b Fp(4.9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF