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Fc(1)1271 5235 y Fk(;i)1328 5220 y Fl(;)g(:)g(:)g(:)d(;)j(\021)1599 5235 y Fk(i;i)p Fh(+)p Fk(e)1778 5246 y Ff(d)1825 5220 y Fl(;)g(\021)1919 5235 y Fk(i)p Fe(\000)p Fk(e)2047 5246 y Ff(d)2089 5235 y Fk(;i)2145 5220 y Fm(\),)43 b(whic)m(h)f(can)f (b)s(e)h(in)m(terpreted)g(as)f(iden)m(tical)0 5340 y(list)33 b(of)g(particles)g(presen)m(t)i(around)f Fl(i)p Fm(.)1925 5579 y(2)p eop %%Page: 3 3 3 2 bop 199 100 a Fm(The)38 b(lattice)e Fi(Z)785 63 y Fk(d)866 100 y Fm(is)h(bipartite:)51 b(sites)37 b(are)g(ev)m(en)h(or)f (o)s(dd)g(according)g(to)g(the)g(parit)m(y)g(of)g(the)g(sum)0 219 y(of)g(their)g(co)s(ordinates.)56 b(This)37 b(allo)m(ws)g(to)f (de\014ne)j(t)m(w)m(o)e(equiv)-6 b(alence)38 b(relations)3047 212 y Fe(\030)3058 253 y Fk(o)3185 219 y Fm(and)3411 212 y Fe(\030)3423 253 y Fk(e)3549 219 y Fm(b)s(et)m(w)m(een)0 339 y(con\014gurations:)1000 466 y Fl(\021)1082 459 y Fe(\030)1093 501 y Fk(o)1183 466 y Fl(\021)1236 425 y Fe(0)1396 466 y Fm(if)c(and)f(only)g(if)100 b Fl(\021)32 b Fj(\030)c Fl(\021)2297 425 y Fe(0)2358 466 y Fm(at)33 b Fl(i)100 b Fj(8)p Fl(i)34 b Fm(o)s(dd)0 664 y(and)985 792 y Fl(\021)1067 784 y Fe(\030)1079 826 y Fk(e)1167 792 y Fl(\021)1220 751 y Fe(0)1381 792 y Fm(if)g(and)f(only)g(if)100 b Fl(\021)31 b Fj(\030)d Fl(\021)2281 751 y Fe(0)2343 792 y Fm(at)k Fl(i)100 b Fj(8)p Fl(i)34 b Fm(ev)m(en)0 989 y(Next)e(w)m(e)i(in)m(tro)s(duce)g(2)p Fl(d)f Fm(probabilit)m(y)f (measures)h(on)h Fi(Z)c Fm(denoted)k Fl(Q)2597 1004 y Fh(1)2642 989 y Fl(;)17 b(R)2763 1004 y Fh(1)2806 989 y Fl(;)g(:)g(:)g(:)d(;)j(Q)3107 1004 y Fk(d)3153 989 y Fl(;)g(R)3274 1004 y Fk(d)3319 989 y Fm(,)33 b(and)h(the)f(t)m(w)m(o) 0 1109 y(candidates)h(for)g(stationary)e(measures)h(under)i(the)e (dynamics)g(to)g(b)s(e)g(de\014ned:)102 1449 y Fl(\026)162 1408 y Fk(e)205 1449 y Fm(\()p Fl(\021)t Fm(\))27 b(=)503 1354 y Fg(Y)468 1566 y Fk(i)14 b Fh(ev)n(en)736 1324 y Fk(d)693 1354 y Fg(Y)681 1565 y Fk(n)p Fh(=1)848 1449 y Fl(Q)927 1464 y Fk(n)981 1449 y Fm(\()p Fl(\021)1069 1464 y Fk(i)p Fe(\000)p Fk(e)1197 1474 y Ff(n)1246 1464 y Fk(;i)1303 1449 y Fm(\))p Fl(R)1418 1464 y Fk(n)1472 1449 y Fm(\()p Fl(\021)1560 1464 y Fk(i;i)p Fh(+)p Fk(e)1739 1474 y Ff(n)1793 1449 y Fm(\))j Fl(;)215 b(\026)2152 1408 y Fk(o)2196 1449 y Fm(\()p Fl(\021)t Fm(\))27 b(=)2482 1354 y Fg(Y)2459 1568 y Fk(i)14 b Fh(o)r(dd)2702 1324 y Fk(d)2659 1354 y Fg(Y)2648 1565 y Fk(n)p Fh(=1)2814 1449 y Fl(Q)2893 1464 y Fk(n)2947 1449 y Fm(\()p Fl(\021)3035 1464 y Fk(i)p Fe(\000)p Fk(e)3163 1474 y Ff(n)3213 1464 y Fk(;i)3269 1449 y Fm(\))p Fl(R)3384 1464 y Fk(n)3438 1449 y Fm(\()p Fl(\021)3526 1464 y Fk(i;i)p Fh(+)p Fk(e)3705 1474 y Ff(n)3759 1449 y Fm(\))3695 1663 y(\(2)p Fl(:)p Fm(3\))0 1783 y(Time)32 b(is)i(discrete,)g Fl(t)28 b Fj(2)g Fi(Z)982 1747 y Fh(+)1046 1783 y Fm(.)45 b(Giv)m(en)33 b(a)h(con\014guration)f Fl(\021)2147 1747 y Fh(2)p Fk(t)2250 1783 y Fm(=)c Fl(\021)37 b Fm(at)c(some)g(ev)m(en)h(time,)e(the)i (con\014gura-)0 1902 y(tion)f(at)g(time)e(2)p Fl(t)22 b Fm(+)h(1)33 b(is)g(distributed)h(according)g(to)662 2306 y Fi(P)721 2226 y Fg(\000)768 2306 y Fl(\021)821 2265 y Fh(2)p Fk(t)p Fh(+1)1025 2306 y Fm(=)28 b Fl(\021)1183 2265 y Fe(0)1227 2306 y Fj(j)17 b Fl(\021)1325 2265 y Fh(2)p Fk(t)1427 2306 y Fm(=)28 b Fl(\021)1585 2226 y Fg(\001)1659 2306 y Fm(=)1764 2072 y Fg(8)1764 2162 y(>)1764 2192 y(<)1764 2371 y(>)1764 2401 y(:)2094 2124 y Fl(\026)2154 2088 y Fk(o)2198 2124 y Fm(\()p Fl(\021)2290 2088 y Fe(0)2318 2124 y Fm(\))p 1881 2169 689 4 v 1881 2185 a Fg(P)1986 2290 y Fk(\021)2030 2270 y Fd(00)2104 2293 y Fe(\030)2116 2334 y Fk(o)2202 2290 y(\021)2284 2260 y Fl(\026)2344 2231 y Fk(o)2388 2260 y Fm(\()p Fl(\021)2480 2231 y Fe(00)2531 2260 y Fm(\))2781 2192 y(if)107 b Fl(\021)2999 2155 y Fe(0)3056 2184 y(\030)3067 2226 y Fk(o)3157 2192 y Fl(\021)2200 2479 y Fm(0)540 b(otherwise)3695 2306 y(\(2)p Fl(:)p Fm(4\))0 2711 y(and)34 b(similarly)c(at)j(the)h(follo)m(wing)e(time)599 3115 y Fi(P)658 3034 y Fg(\000)705 3115 y Fl(\021)758 3073 y Fh(2)p Fk(t)p Fh(+2)962 3115 y Fm(=)c Fl(\021)20 b Fj(j)d Fl(\021)1234 3073 y Fh(2)p Fk(t)p Fh(+1)1437 3115 y Fm(=)28 b Fl(\021)1595 3073 y Fe(0)1623 3034 y Fg(\001)1697 3115 y Fm(=)1802 2881 y Fg(8)1802 2970 y(>)1802 3000 y(<)1802 3179 y(>)1802 3209 y(:)2159 2932 y Fl(\026)2219 2896 y Fk(e)2262 2932 y Fm(\()p Fl(\021)t Fm(\))p 1919 2977 714 4 v 1919 2993 a Fg(P)2024 3098 y Fk(\021)2068 3078 y Fd(00)2142 3101 y Fe(\030)2154 3142 y Fk(e)2240 3098 y(\021)2284 3078 y Fd(0)2348 3068 y Fl(\026)2408 3039 y Fk(e)2451 3068 y Fm(\()p Fl(\021)2543 3039 y Fe(00)2594 3068 y Fm(\))2844 3000 y(if)107 b Fl(\021)3091 2992 y Fe(\030)3103 3034 y Fk(e)3192 3000 y Fl(\021)3245 2964 y Fe(0)2251 3287 y Fm(0)552 b(otherwise)3695 3115 y(\(2)p Fl(:)p Fm(5\))0 3518 y(Let)33 b(us)h(compute)139 3722 y Fg(X)189 3932 y Fk(\021)299 3817 y Fi(P)358 3736 y Fg(\000)406 3817 y Fl(\021)459 3776 y Fh(2)p Fk(t)p Fh(+1)662 3817 y Fm(=)29 b Fl(\021)821 3776 y Fe(0)865 3817 y Fj(j)17 b Fl(\021)963 3776 y Fh(2)p Fk(t)1065 3817 y Fm(=)28 b Fl(\021)1223 3736 y Fg(\001)1285 3817 y Fl(\026)1345 3776 y Fk(e)1388 3817 y Fm(\()p Fl(\021)t Fm(\))f(=)1698 3722 y Fg(X)1652 3936 y Fk(\021)1721 3939 y Fe(\030)1732 3980 y Fk(o)1818 3936 y(\021)1862 3916 y Fd(0)2130 3750 y Fl(\026)2190 3713 y Fk(o)2235 3750 y Fm(\()p Fl(\021)2327 3713 y Fe(0)2354 3750 y Fm(\))p 1918 3794 689 4 v 1918 3811 a Fg(P)2023 3915 y Fk(\021)2067 3895 y Fd(00)2141 3918 y Fe(\030)2152 3960 y Fk(o)2238 3915 y(\021)2320 3885 y Fl(\026)2380 3857 y Fk(o)2425 3885 y Fm(\()p Fl(\021)2517 3857 y Fe(00)2567 3885 y Fm(\))2635 3817 y Fl(\026)2695 3776 y Fk(e)2737 3817 y Fm(\()p Fl(\021)t Fm(\))1546 4248 y(=)i Fl(\026)1712 4207 y Fk(o)1756 4248 y Fm(\()p Fl(\021)1848 4207 y Fe(0)1876 4248 y Fm(\))1994 4045 y Fg(P)2099 4150 y Fk(\021)2168 4153 y Fe(\030)2179 4194 y Fk(o)2266 4150 y(\021)2310 4130 y Fd(0)2358 4120 y Fl(\026)2418 4084 y Fk(e)2461 4120 y Fm(\()p Fl(\021)t Fm(\))p 1943 4225 699 4 v 1943 4242 a Fg(P)2048 4346 y Fk(\021)2092 4326 y Fd(00)2166 4349 y Fe(\030)2178 4391 y Fk(o)2264 4346 y(\021)2308 4326 y Fd(0)2356 4316 y Fl(\026)2416 4288 y Fk(o)2460 4316 y Fm(\()p Fl(\021)2552 4288 y Fe(00)2603 4316 y Fm(\))1546 4626 y(=)g Fl(\026)1712 4585 y Fk(o)1756 4626 y Fm(\()p Fl(\021)1848 4585 y Fe(0)1876 4626 y Fm(\))1970 4532 y Fg(Y)1948 4746 y Fk(i)13 b Fh(o)r(dd)2172 4469 y Fg(P)2277 4494 y Fk(i)2277 4574 y(\021)s Fe(\030)p Fk(\021)2427 4554 y Fd(0)2475 4469 y Fg(Q)2569 4494 y Fk(d)2569 4574 y(n)p Fh(=1)2741 4544 y Fl(Q)2820 4559 y Fk(n)2874 4544 y Fm(\()p Fl(\021)2962 4559 y Fk(i)p Fe(\000)p Fk(e)3090 4569 y Ff(n)3139 4559 y Fk(;i)3196 4544 y Fm(\))p Fl(R)3311 4559 y Fk(n)3365 4544 y Fm(\()p Fl(\021)3453 4559 y Fk(i;i)p Fh(+)p Fk(e)3632 4569 y Ff(n)3686 4544 y Fm(\))p 2148 4603 1602 4 v 2148 4639 a Fg(P)2253 4664 y Fk(i)2253 4744 y(\021)2297 4724 y Fd(00)2345 4744 y Fe(\030)p Fk(\021)2451 4724 y Fd(0)2500 4639 y Fg(Q)2594 4664 y Fk(d)2594 4744 y(n)p Fh(=1)2766 4714 y Fl(R)2842 4729 y Fk(n)2895 4714 y Fm(\()p Fl(\021)2987 4680 y Fe(00)2983 4742 y Fk(i)p Fe(\000)p Fk(e)3111 4752 y Ff(n)3161 4742 y Fk(;i)3218 4714 y Fm(\))p Fl(Q)3336 4729 y Fk(n)3389 4714 y Fm(\()p Fl(\021)3481 4680 y Fe(00)3477 4742 y Fk(i;i)p Fh(+)p Fk(e)3656 4752 y Ff(n)3710 4714 y Fm(\))3695 4842 y(\(2)p Fl(:)p Fm(6\))0 4973 y(where)286 4898 y Fg(P)391 4923 y Fk(i)456 4973 y Fm(means)31 b(a)g(sum)g(o)m(v)m (er)h(b)s(ond)g(v)-6 b(ariables)31 b(inciden)m(t)h(up)s(on)g Fl(i)p Fm(.)44 b(No)m(w)31 b(\(2.1\))f(allo)m(ws)h(a)g(c)m(hange)i(of)0 5093 y(summation)e(v)-6 b(ariable)33 b(in)g(the)h(denominator:)1171 5340 y Fl(\021)1224 5299 y Fe(00)1220 5365 y Fk(i)p Fe(\000)p Fk(e)1348 5375 y Ff(n)1398 5365 y Fk(;i)1482 5340 y Fm(=)28 b Fl(\021)1636 5355 y Fk(i;i)p Fh(+)p Fk(e)1815 5365 y Ff(n)1886 5340 y Fl(;)116 b(\021)2083 5299 y Fe(00)2079 5365 y Fk(i;i)p Fh(+)p Fk(e)2258 5375 y Ff(n)2340 5340 y Fm(=)28 b Fl(\021)2494 5355 y Fk(i)p Fe(\000)p Fk(e)2622 5365 y Ff(n)2672 5355 y Fk(;i)1925 5579 y Fm(3)p eop %%Page: 4 4 4 3 bop 0 100 a Fm(so)33 b(that)g(n)m(umerator)g(and)h(denominator)e (cancel)i(out)f(and)1062 228 y Fg(X)1112 438 y Fk(\021)1223 323 y Fi(P)1282 242 y Fg(\000)1329 323 y Fl(\021)1382 282 y Fh(2)p Fk(t)p Fh(+1)1586 323 y Fm(=)28 b Fl(\021)1744 282 y Fe(0)1788 323 y Fj(j)17 b Fl(\021)1886 282 y Fh(2)p Fk(t)1988 323 y Fm(=)28 b Fl(\021)2146 242 y Fg(\001)2209 323 y Fl(\026)2269 282 y Fk(e)2312 323 y Fm(\()p Fl(\021)t Fm(\))f(=)h Fl(\026)2635 282 y Fk(o)2679 323 y Fm(\()p Fl(\021)2771 282 y Fe(0)2799 323 y Fm(\))857 b(\(2)p Fl(:)p Fm(7\))0 644 y(Similarly)1012 684 y Fg(X)1049 897 y Fk(\021)1093 877 y Fd(0)1172 778 y Fi(P)1231 697 y Fg(\000)1279 778 y Fl(\021)1332 737 y Fh(2)p Fk(t)p Fh(+2)1535 778 y Fm(=)28 b Fl(\021)21 b Fj(j)c Fl(\021)1808 737 y Fh(2)p Fk(t)p Fh(+1)2011 778 y Fm(=)28 b Fl(\021)2169 737 y Fe(0)2197 697 y Fg(\001)2259 778 y Fl(\026)2319 737 y Fk(o)2363 778 y Fm(\()p Fl(\021)2455 737 y Fe(0)2483 778 y Fm(\))g(=)g Fl(\026)2715 737 y Fk(e)2758 778 y Fm(\()p Fl(\021)t Fm(\))806 b(\(2)p Fl(:)p Fm(8\))0 1063 y(Therefore)35 b Fl(\026)512 1026 y Fk(e)588 1063 y Fm(and)f Fl(\026)842 1026 y Fk(o)920 1063 y Fm(are)g(exc)m(hanged)g(under)h(the) f(dynamics,)f(and)h(b)s(oth)f(are)h(in)m(v)-6 b(arian)m(t)33 b(from)g Fl(t)g Fm(to)0 1182 y Fl(t)22 b Fm(+)h(2.)0 1464 y Fo(3.)51 b(Dynamics)38 b(for)h Fl(d)27 b Fm(=)i(1)38 b Fo(SOS)f(mo)s(dels)0 1662 y Fm(In)47 b Fl(d)i Fm(=)h(1)c(the)h (notation)f(ma)m(y)f(b)s(e)h(simpli\014ed)g(with)g Fl(\021)2187 1677 y Fk(i)2270 1662 y Fm(=)k Fl(\021)2446 1677 y Fk(i;i)p Fh(+)p Fk(e)2625 1687 y Fc(1)2716 1662 y Fm(and)d Fl(\021)2972 1677 y Fk(i)p Fe(\000)p Fh(1)3157 1662 y Fm(=)j Fl(\021)3333 1677 y Fk(i)p Fe(\000)p Fk(e)3461 1687 y Fc(1)3501 1677 y Fk(;i)3558 1662 y Fm(.)83 b(Bond)0 1781 y(v)-6 b(ariables)46 b(ma)m(y)e(b)s(e)j(iden)m(ti\014ed)g(with)e(the)h(gradien)m(t)h(of)f (heigh)m(t)g(v)-6 b(ariables,)49 b Fl(\021)3023 1796 y Fk(i)3105 1781 y Fm(=)g Fl(h)3288 1796 y Fk(i)p Fh(+1)3453 1781 y Fj(\000)31 b Fl(h)3618 1796 y Fk(i)3652 1781 y Fm(.)82 b(W)-8 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(the)h(Leb)s(esgue)h(measure,)1354 4259 y Fl(Z)7 b Fm(\()p Fl(\015)f Fm(\))26 b(=)1697 4123 y Fg(Z)1813 4259 y Fl(Q)p Fm(\()p Fl(\013)p Fm(\))p Fl(R)q Fm(\()p Fl(\015)g Fj(\000)c Fl(\013)p Fm(\))p Fl(d\013)0 4508 y Fm(and)34 b(\(3.1\)\(3.7\))d(b)s (ecome)200 4725 y Fl(\026)260 4684 y Fk(e)302 4725 y Fm(\()p Fl(d\021)t Fm(\))c(=)652 4631 y Fg(Y)618 4843 y Fk(i)13 b Fh(ev)n(en)831 4725 y Fl(Q)p Fm(\()p Fl(\021)998 4740 y Fk(i)p Fe(\000)p Fh(1)1133 4725 y Fm(\))p Fl(R)q Fm(\()p Fl(\021)1337 4740 y Fk(i)1369 4725 y Fm(\))p Fl(d\021)1509 4740 y Fk(i)p Fe(\000)p Fh(1)1644 4725 y Fl(d\021)1745 4740 y Fk(i)1796 4725 y Fl(;)115 b(\026)1999 4684 y Fk(o)2044 4725 y Fm(\()p Fl(d\021)t Fm(\))27 b(=)2381 4631 y Fg(Y)2359 4845 y Fk(i)14 b Fh(o)r(dd)2547 4725 y Fl(Q)p Fm(\()p Fl(\021)2714 4740 y Fk(i)p Fe(\000)p Fh(1)2849 4725 y Fm(\))p Fl(R)q Fm(\()p Fl(\021)3053 4740 y Fk(i)3086 4725 y Fm(\))p Fl(d\021)3226 4740 y Fk(i)p Fe(\000)p Fh(1)3361 4725 y Fl(d\021)3462 4740 y Fk(i)3695 4725 y Fm(\(3)p Fl(:)p Fm(8\))533 5105 y Fi(P)592 5024 y Fg(\000)656 5105 y Fl(dh)765 5064 y Fh(0)p Fe(\024)p Fk(t)p Fe(\024)p Fk(T)1039 5105 y Fj(j)j Fl(h)1141 5064 y Fh(0)1202 5024 y Fg(\001)1276 5105 y Fm(=)1483 5010 y Fg(Y)1432 5211 y Ff(i)p Fc(+)p Ff(t)12 b Fc(o)r(dd)1393 5268 y(0)p Fd(\024)p Ff(t)p Fd(\024)p Ff(T)d Fd(\000)p Fc(1)1741 5031 y Fl(Q)1820 4950 y Fg(\000)1865 5031 y Fl(h)1922 4988 y Fk(t)p Fh(+1)1922 5059 y Fk(i)2081 5031 y Fj(\000)23 b Fl(h)2238 4995 y Fk(t)2238 5057 y(i)p Fe(\000)p Fh(1)2373 4950 y Fg(\001)2435 5031 y Fl(R)2512 4950 y Fg(\000)2557 5031 y Fl(h)2614 4995 y Fk(t)2614 5057 y(i)p Fh(+1)2771 5031 y Fj(\000)g Fl(h)2928 4988 y Fk(t)p Fh(+1)2928 5059 y Fk(i)3064 4950 y Fg(\001)p 1741 5082 V 2089 5179 a Fl(Z)2164 5098 y Fg(\000)2210 5179 y Fl(h)2267 5144 y Fk(t)2267 5206 y(i)p Fh(+1)2424 5179 y Fj(\000)f Fl(h)2580 5144 y Fk(t)2580 5206 y(i)p Fe(\000)p Fh(1)2716 5098 y Fg(\001)3122 5105 y Fl(dh)3231 5062 y Fk(t)p Fh(+1)3231 5133 y Fk(i)3695 5105 y Fm(\(3)p Fl(:)p Fm(9\))1925 5579 y(5)p eop %%Page: 6 6 6 5 bop 0 100 a Fo(4.)51 b(Examples)0 296 y Fm(The)29 b(case)g Fl(\021)455 311 y Fk(i)516 296 y Fj(2)f(f)p Fm(+1)p Fl(;)17 b Fm(0)p Fl(;)g Fj(\000)p Fm(1)p Fj(g)28 b Fm(has)g(b)s(een)i(thoroughly)e(studied)h(in)g([CDF)m(G].)f(Here)g(w) m(e)h(concen)m(trate)h(on)0 416 y(cases)g(where)g(the)g(probabilit)m(y) e(measures)i Fl(Q)f Fm(and)g Fl(R)h Fm(ha)m(v)m(e)g(densities)g(with)f (resp)s(ect)h(to)f(the)g(Leb)s(esgue)0 536 y(measure)k(on)h Fi(R)44 b Fm(or)33 b(the)g(uniform)g(measure)h(on)f Fi(Z)e Fm(of)i(the)h(form)497 776 y Fl(Q)576 791 y Fk(\025)628 776 y Fm(\()p Fl(\021)t Fm(\))27 b(=)h Fl(e)937 704 y Fe(\000)p Fk(J)7 b Fe(j)p Fk(\021)s Fe(j)1142 673 y Ff(n)1190 704 y Fh(+\()p Fk(\025)p Fh(+)1403 677 y Ff(E)p 1403 688 53 4 v 1413 728 a Fc(2)1468 704 y Fh(\))14 b Fk(\021)1566 776 y Fl(=)p Fm(norm)p Fl(:)h(;)116 b(R)2107 791 y Fk(\025)2159 776 y Fm(\()p Fl(\021)t Fm(\))27 b(=)h Fl(e)2468 704 y Fe(\000)p Fk(J)7 b Fe(j)p Fk(\021)s Fe(j)2673 673 y Ff(n)2722 704 y Fh(+\()p Fk(\025)p Fe(\000)2935 677 y Ff(E)p 2935 688 V 2945 728 a Fc(2)3000 704 y Fh(\))14 b Fk(\021)3098 776 y Fl(=)p Fm(norm)p Fl(:)291 b Fm(\(4)p Fl(:)p Fm(1\))0 993 y(with)27 b Fl(n)h Fm(=)g(1)p Fl(;)17 b Fm(2)26 b(or)i(4,)g Fl(J)37 b(>)28 b Fm(0,)g Fl(E)33 b(>)c Fm(0.)41 b(The)29 b(parameter)d Fl(\025)i Fm(is)f(in)h(the)g (range)g Fl(\025)f Fj(2)h Fm(\()p Fj(\000)p Fl(J)21 b Fm(+)11 b Fl(E)6 b(=)p Fm(2)p Fl(;)17 b(J)i Fj(\000)11 b Fl(E)6 b(=)p Fm(2\))0 1113 y(for)33 b Fl(n)28 b Fm(=)g(1)33 b(and)h Fl(\025)28 b Fj(2)g Fm(\()p Fj(\0001)p Fl(;)17 b Fm(+)p Fj(1)p Fm(\))32 b(for)h Fl(n)28 b(>)g Fm(1.)199 1282 y(The)34 b(transition)f(rates)g(\(3.2\))f(tak)m(e)h(the)h(form)50 1614 y Fi(P)109 1533 y Fg(\000)156 1614 y Fl(\021)209 1573 y Fh(2)p Fk(t)p Fh(+1)413 1614 y Fm(=)28 b Fl(\021)571 1573 y Fe(0)615 1614 y Fj(j)17 b Fl(\021)713 1573 y Fh(2)p Fk(t)815 1614 y Fm(=)28 b Fl(\021)973 1533 y Fg(\001)1047 1614 y Fm(=)1174 1520 y Fg(Y)1152 1734 y Fk(i)14 b Fh(o)r(dd)1352 1547 y Fl(e)1398 1475 y Fe(\000)p Fk(J)7 b Fh(\()p Fe(j)p Fk(\021)1610 1445 y Fd(0)1607 1495 y Ff(i)p Fd(\000)p Fc(1)1727 1475 y Fe(j)1751 1445 y Ff(n)1799 1475 y Fh(+)p Fe(j)p Fk(\021)1928 1445 y Fd(0)1925 1495 y Ff(i)1957 1475 y Fe(j)1981 1445 y Ff(n)2029 1475 y Fh(\)+)2133 1448 y Ff(E)p 2133 1459 V 2143 1499 a Fc(2)2198 1475 y Fh(\()14 b Fk(\021)2287 1445 y Fd(0)2284 1495 y Ff(i)p Fd(\000)p Fc(1)2403 1475 y Fe(\000)p Fk(\021)2509 1445 y Fd(0)2506 1495 y Ff(i)2538 1475 y Fh(\))p 1352 1591 1228 4 v 1709 1670 a Fm(~)1688 1695 y Fl(Z)1763 1614 y Fg(\000)1809 1695 y Fl(\021)1858 1710 y Fk(i)p Fe(\000)p Fh(1)2016 1695 y Fm(+)22 b Fl(\021)2164 1710 y Fk(i)2198 1614 y Fg(\001)364 b Fl(\016)2656 1533 y Fg(\000)2702 1614 y Fl(\021)2755 1573 y Fe(0)2751 1639 y Fk(i)p Fe(\000)p Fh(1)2908 1614 y Fm(+)22 b Fl(\021)3060 1573 y Fe(0)3056 1639 y Fk(i)3112 1614 y Fj(\000)g Fl(\021)3260 1629 y Fk(i)p Fe(\000)p Fh(1)3418 1614 y Fj(\000)g Fl(\021)3566 1629 y Fk(i)3600 1533 y Fg(\001)3695 1614 y Fm(\(4)p Fl(:)p Fm(2\))0 1920 y(with)1157 2039 y(~)1136 2064 y Fl(Z)7 b Fm(\()p Fl(\015)f Fm(\))26 b(=)1486 1969 y Fg(X)1478 2190 y Fk(\013)p Fe(2)p Fa(Z)1654 2064 y Fl(e)1700 1991 y Fe(\000)p Fk(J)7 b Fh(\()p Fe(j)p Fk(\013)p Fe(j)1944 1961 y Ff(n)1992 1991 y Fh(+)p Fe(j)p Fk(\015)t Fe(\000)p Fk(\013)p Fe(j)2261 1961 y Ff(n)2310 1991 y Fh(\)+)2414 1964 y Ff(E)p 2414 1975 53 4 v 2423 2015 a Fc(2)2478 1991 y Fh(\()14 b Fk(\015)t Fe(\000)p Fh(2)p Fk(\013)p Fh(\))3695 2064 y Fm(\(4)p Fl(:)p Fm(3\))0 2344 y(The)39 b(dynamics)f(is)h(th)m(us)h(indep)s(enden)m(t)h(of)e Fl(\025)p Fm(,)g(and)h(admits)d(a)i(one-parameter)f(family)f(of)i(in)m (v)-6 b(arian)m(t)0 2464 y(measures)34 b(indexed)g(b)m(y)f Fl(\025)p Fm(.)199 2633 y(The)h(transition)f(rates)g(for)h(heigh)m(ts)g (\(3.4\)\(3.5\)\(3.6\))29 b(tak)m(e)k(the)h(form)209 2996 y Fi(P)268 2915 y Fg(\000)316 2996 y Fl(h)373 2955 y Fh(2)p Fk(t)p Fh(+1)576 2996 y Fm(=)29 b Fl(h)739 2955 y Fe(0)784 2996 y Fj(j)17 b Fl(h)886 2955 y Fh(2)p Fk(t)988 2996 y Fm(=)28 b Fl(h)1150 2915 y Fg(\001)1224 2996 y Fm(=)1351 2901 y Fg(Y)1329 3116 y Fk(i)14 b Fh(o)r(dd)1529 2929 y Fl(e)1575 2850 y Fe(\000)p Fk(J)1688 2769 y Fg(\000)1748 2850 y Fe(j)p Fk(h)1819 2820 y Fd(0)1819 2870 y Ff(i)1849 2850 y Fe(\000)p Fk(h)1958 2860 y Ff(i)p Fd(\000)p Fc(1)2078 2850 y Fe(j)2102 2820 y Ff(n)2150 2850 y Fh(+)p Fe(j)p Fk(h)2282 2860 y Ff(i)p Fc(+1)2398 2850 y Fe(\000)p Fk(h)2507 2820 y Fd(0)2507 2870 y Ff(i)2539 2850 y Fe(j)2563 2820 y Ff(n)2625 2769 y Fg(\001)2671 2850 y Fh(+)p Fk(E)t(h)2841 2820 y Fd(0)2841 2870 y Ff(i)2872 2850 y Fe(\000)2947 2823 y Ff(E)p 2947 2834 V 2956 2874 a Fc(2)3011 2850 y Fh(\()p Fk(h)3089 2860 y Ff(i)p Fd(\000)p Fc(1)3208 2850 y Fh(+)p Fk(h)3316 2860 y Ff(i)p Fc(+1)3433 2850 y Fh(\))p 1529 2973 1946 4 v 2186 3052 a Fm(~)2165 3077 y Fl(Z)2241 2996 y Fg(\000)2286 3077 y Fl(h)2343 3092 y Fk(i)p Fh(+1)2500 3077 y Fj(\000)22 b Fl(h)2656 3092 y Fk(i)p Fe(\000)p Fh(1)2792 2996 y Fg(\001)857 b Fm(\(4)p Fl(:)p Fm(4\))0 3325 y(where)312 3300 y(~)291 3325 y Fl(Z)43 b Fm(is)35 b(the)i(same)e(as)h(\(4.3\).)50 b(If)36 b(w)m(e)h(de\014ne)g Fl(w)1953 3342 y Fk(h)2000 3352 y Ff(i)2032 3342 y Fe(!)p Fk(h)2158 3318 y Fd(0)2158 3367 y Ff(i)2230 3325 y Fm(as)f(the)g(factor)g(indexed)h(b)m(y)f Fl(i)g Fm(in)g(\(4.4\),)f(w)m(e)0 3444 y(\014nd)f(that)f(it)g(ob)s(eys) g(\(1.2\).)199 3614 y(F)-8 b(or)24 b(the)h(Gaussian)g(case,)i Fl(n)g Fm(=)i(2)24 b(with)g(con)m(tin)m(uous)i(v)-6 b(ariables,)26 b(w)m(e)f(\014nd)g(that)3149 3588 y(~)3128 3614 y Fl(Z)7 b Fm(\()p Fl(\015)f Fm(\),)24 b(no)m(w)h(de\014ned)0 3733 y(with)39 b(an)h(in)m(tegral)e(o)m(v)m(er)i Fl(\013)p Fm(,)h(is)e(prop)s(ortional)f(to)h(exp\()p Fj(\000)p Fl(J)10 b(\015)2303 3697 y Fh(2)2347 3733 y Fl(=)p Fm(2\),)40 b(hence)h(a)e(more)g(explicit)f(form)h(for)0 3853 y(\(4.4\))32 b(and)i(for)f(the)h(space-time)f(measure:)248 4153 y Fi(P)307 4072 y Fg(\000)371 4153 y Fl(dh)480 4112 y Fh(0)p Fe(\024)p Fk(t)p Fe(\024)p Fk(T)754 4153 y Fj(j)17 b Fl(h)856 4112 y Fh(0)917 4072 y Fg(\001)990 4153 y Fm(=)1198 4058 y Fg(Y)1147 4259 y Ff(i)p Fc(+)p Ff(t)12 b Fc(o)r(dd)1107 4316 y(0)p Fd(\024)p Ff(t)p Fd(\024)p Ff(T)d Fd(\000)p Fc(1)1427 4013 y Fg(\022)1500 4153 y Fl(e)1546 4070 y Fe(\000)p Fk(J)1659 3989 y Fg(\000)1719 4070 y Fe(j)p Fk(h)1790 4037 y Ff(t)p Fc(+1)1790 4095 y Ff(i)1907 4070 y Fe(\000)p Fk(h)2016 4040 y Ff(t)2016 4090 y(i)p Fd(\000)p Fc(1)2136 4070 y Fe(j)2160 4040 y Fc(2)2198 4070 y Fh(+)p Fe(j)p Fk(h)2330 4040 y Ff(t)2330 4090 y(i)p Fc(+1)2446 4070 y Fe(\000)p Fk(h)2555 4037 y Ff(t)p Fc(+1)2555 4095 y Ff(i)2674 4070 y Fe(j)2698 4040 y Fc(2)2736 4070 y Fe(\000)2810 4043 y Fc(1)p 2810 4054 34 4 v 2810 4094 a(2)2856 4070 y Fe(j)p Fk(h)2927 4040 y Ff(t)2927 4090 y(i)p Fc(+1)3043 4070 y Fe(\000)p Fk(h)3152 4040 y Ff(t)3152 4090 y(i)p Fd(\000)p Fc(1)3271 4070 y Fe(j)3295 4040 y Fc(2)3348 3989 y Fg(\001)3420 4153 y Fl(:)1554 4521 y(:)17 b(e)1657 4418 y Ff(E)p 1656 4429 53 4 v 1666 4468 a Fc(2)1721 4444 y Fh(\()p Fk(h)1799 4412 y Ff(t)p Fc(+1)1799 4469 y Ff(i)1917 4444 y Fe(\000)p Fk(h)2026 4414 y Ff(t)2026 4464 y(i)p Fd(\000)p Fc(1)2145 4444 y Fh(\)+)2249 4418 y Ff(E)p 2249 4429 V 2259 4468 a Fc(2)2314 4444 y Fh(\()p Fk(h)2392 4412 y Ff(t)p Fc(+1)2392 4469 y Ff(i)2510 4444 y Fe(\000)p Fk(h)2619 4414 y Ff(t)2619 4464 y(i)p Fc(+1)2736 4444 y Fh(\))2777 4521 y Fl(dh)2886 4479 y Fk(t)p Fh(+1)2886 4549 y Fk(i)3038 4381 y Fg(\023)3112 4411 y(.)3193 4521 y Fm(norm)p Fl(:)3695 4325 y Fm(\(4)p Fl(:)p Fm(5\))0 4952 y Fo(5.)51 b(In)m(terface)37 b(mean)g(slop)s(e)h(and)f(sp)s(eed)0 5148 y Fm(Let)958 5294 y Fj(h)17 b Fl(\021)j Fj(i)1122 5309 y Fk(Q)1217 5294 y Fm(=)1322 5158 y Fg(Z)1438 5294 y Fl(\021)h(Q)p Fm(\()p Fl(\021)t Fm(\))p Fl(d\021)f(;)215 b Fj(h)17 b Fl(\021)j Fj(i)2246 5309 y Fk(R)2339 5294 y Fm(=)2444 5158 y Fg(Z)2561 5294 y Fl(\021)g(R)q Fm(\()p Fl(\021)t Fm(\))p Fl(d\021)1925 5579 y Fm(6)p eop %%Page: 7 7 7 6 bop 0 100 a Fm(and)38 b(similarly)d(with)i(sums)h(instead)g(of)g (in)m(tegrals)g(in)f(the)h(case)g(of)g(discrete)h(v)-6 b(ariables.)57 b(In)38 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4693 l 1744 4701 l 1716 4716 l 1688 4736 l 1661 4760 l 1635 4790 l 1609 4823 l 1584 4860 l 1559 4901 l 1536 4943 l 1513 4988 l 1492 5034 l 1471 5080 l 1451 5127 l 1433 5174 l 1415 5220 l 1399 5266 l 1383 5310 l 1369 5353 l 1355 5394 l 1342 5433 l 1330 5470 l 1320 5506 l 1309 5539 l 1300 5571 l 1291 5600 l 1283 5628 l 1276 5653 l 1269 5677 l 1263 5699 l 1257 5720 l 1252 5738 l 1248 5756 l 1243 5772 l 1239 5786 l 1236 5800 l 1232 5812 l 1229 5823 l 1227 5833 l 1224 5843 l 1222 5851 l 1220 5859 l 1218 5866 l 1216 5873 l 1215 5879 l 1213 5884 l 1212 5889 l 1211 5893 l 1210 5897 l 1209 5901 l 1208 5904 l 1207 5907 l 1207 5910 l 1206 5912 l 1205 5915 l 1205 5917 l 1205 5918 l 1204 5920 l 1204 5922 l 1203 5923 l 1203 5924 l 1203 5925 l 1202 5926 l 1202 5927 l 1202 5928 l 1202 5929 l 1202 5930 l 1201 5931 l 1201 5932 l 1201 5933 l 1201 5934 l 1200 5934 l 1200 5935 l 1200 5936 l gs col0 s gr $F2psEnd rs %%EndDocument @endspecial 1064 2755 a Fn(Fig.2:)40 b(Quartic)30 b(RSOS)h(mo)s(del,)d Fl(N)39 b Fm(=)28 b(3)p Fn(,)i Fl(E)j Fm(=)28 b(1)0 3044 y Fo(References)-228 3244 y Fm([CDF)m(G])83 b(P)-8 b(.)41 b(Collet,)i(F.)e(Dunlop,)j(D.)e(F)-8 b(oster,)43 b(T.)e(Gobron)i(:)61 b Fb(Pro)s(duct)41 b(Measures)j(and)e(Dynamics)f(for)199 3363 y(Solid-On-Solid)34 b(In)m(terfaces)p Fm(,)h(J.)e(Stat.)44 b(Ph)m(ys.)g Fo(89)p Fm(,)33 b(509{536)f(\(1997\))29 3532 y([F])49 b(F)-8 b(ranks)33 b(J.)g(Crystal)g(Gro)m(wth)g Fo(22)p Fm(,)h(233)e(\(1974\))-39 3702 y([KS])50 b(J.)35 b(Krug)h(and)g(H.)f(Sp)s(ohn,)h Fb(Kinetic)g(Roughening)g(of)g(Gro)m (wing)f(Surfaces)p Fm(,)j(in)d(\\Solids)g(far)h(from)199 3821 y(equilibrium",)c(C.)g(Go)s(dr)m(\022)-47 b(ec)m(he)35 b(ed.)45 b(\(Cam)m(bridge)32 b(Univ)m(ersit)m(y)h(Press,)g(Cam)m (bridge,)g(1991\))1925 5579 y(9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF