A localized version of the SK model with external eld

Transcription

1 A localze verson of the SK moel wth external el Samy Tnel Freer Vens September 6, Insttut Gallee - Unverste Pars Avenue J. B. Clement 94 Vlletaneuse, France. tnel@math.unv-pars.fr Dept. Statstcs an Dept. Mathematcs Purue Unversty, 5. Unversty St. West Lafayette, I , USA. vens@purue.eu corresponng author + (765) , fax + (765) Key wors an phrases spn glasses, Sherrngton-Krpatrc, localze mean-el moel, lute moel, cavty metho, smart path metho. AMS subject classcaton () prmary 6K5 seconary 8D, 8B44. Abstract In ths note, we conser a Sherrngton-Krpatrc- (SK)-type moel on Z for, weghte by a functon allowng for any sngle spn to nteract wth a small proporton of the other ones. Both mean el an lute cases are consere, an we compute the replca symmetrc soluton at hgh temperature n those two cases. We are also able to stuy the nuence of some bounary contons on the replca symmetrc soluton uner ml assumptons. Introucton Conceve orgnally as a smple moel that coul escrbe the man behavor of some specal alloys (see [8]), the Sherrngton-Krpatrc (SK) moel has become n fact a canoncal example of sorere system, an the technques ntrouce to hanle t are also use n a we number of applcatons, rangng from neural networs (cf. [], []) to polymer measures [] or boversty [4]. On the other han, some substantal progress has been mae n the unerstanng of ths canoncal moel, through the ntroucton of Pars's ansatz [6] an the smart path metho (see [5], []). In ths paper, however, we woul le to focus on one of the man smplcatons that have been mae n orer to mae the orgnal SK moel solvable, that s the mean el approxmaton. Inee, lettng all the spns nteract wth each other allows the lmtng mxng behavor of the system to tae place n a smple way. On the other han, the results on short range moels wth ranom nteractons are scarce (see however e.g. [7] for an account on the topc). evertheless, the mean el approxmaton s often seen as an oversmplcaton, physcally unrealstc n partcular, t oes not tae nto account the geometry of the system uner conseraton. Thus, our am n ths note s to ntrouce a n of localze mean el moel, that wll partally respect the the geometrc shape of our system, but also share some features of the orgnal SK moel wth external el For, our space of conguratons wll be = = f g C, where C s the nte lattce box C = [ ] n Z. For a gven conguraton, we wll conser the Hamltonan H () = j g (j) j + h () ^ = C (j)c q

2 where stans for the nverse of the temperature of the system, ^ = +, ( j) s our (abusve) notaton for a par of stes j C (taen only once), g (j) ( j) C s a famly of IID stanar centere Gaussan ranom varables, an h represents a constant postve external el, uner whch the spns ten to tae the value +. Our localzaton s represente by the functon q, whch can be thought of as a smooth frame, an whch s only assume to be ene on [ ] such that q s of postve type, so that q s non-negatve an nvarant by symmetry about the orgn. We also assume q s contnuously erentable, nclung at ts peroc bounary. We refer to Secton for some more etale nformaton about our moel. otce that. When h =, ths n of localze moel have been consere n [], usng the martngale metho ntrouce by Comets an eveu []. In our case, however, the results wll have to be obtane thans to the smart path an cavty methos.. Havng a localzng weght of the form q( ) mples that each spn wll nteract wth at least a proporton of the ^ others, whch means that we wll stay n the mean el range. Ths paper's am s to get the replca soluton for the moel ene by the Hamltonan () an some relate ones, at hgh temperature let Z = P e H () an p () = [log(z )]. Frst, f stans for the L norm of q on [ ], an f SK( ^ h) s the usual replca soluton for the SK moel at temperature ^, we wll get that, for small enough, p () SK ( p h) C for a constant C epenng on the parameters of the moel. Observe here that the spee of convergence obtane s of orer ^ nstea of ^ n case of the SK moel. Ths s ue to the slow rate of convergence of some Remann sums, an we wll also show how to obtan some better bouns by changng the assumptons mae on q (namely by lettng q havng smoother oscllatons). It s also worth notcng that the nal result oes not epen on the actual shape of q, but only on ts L norm. The proof of ths result wll heavly rely on some Fourer analyss performe on q, combne wth the cavty an smart path methos. ote however that the usual cavty-metho ngreent of \proof by nucton on " s not use rather, some explct calculatons are performe for xe. Once the geometry of the system s rentrouce by our weght q, t wll also be natural to stuy the nuence of the bounary contons on the lmtng behavor of the system we wll show that those bounary conton wll not aect the replca symmetrc soluton at hgh temperature uner very ml assumptons, a common fact share wth the Isng moel. We wll then mae another step towars a nearest neghbor type moel, an conser on the Hamltonan H () = q ( j) (j) g (j) j + h (j)c C where (j) s an... sequence of Bernoull ranom varables wth parameter P (j) = = ^ P (j) = = ^ ^,.e. Ths enes a lute localze moel, for whch a gven spn nteracts (n mean) wth a nte number of spns n ts neghborhoo ene by q. Ths s a natural extenson of the prevous moel, also consere for the non-localze case n [9], [], an has, from our pont of vew, one atonal avantage the lmtng behavor of ths system wll nvolve the whole structure of q, through the ntroucton of a Posson pont process wth ntensty q, an not only ts L norm as n the mean el case. On the other han, the formula obtane for the replca-symmetrc soluton becomes qute nvolve, an we refer to Theorem.6 for a precse escrpton of the result we obtane. Our paper s organze as follows n Secton, we eal wth the mean el localze moel, startng wth some prelmnary results on the cavty metho an a enton of the overlap sute to our stuaton n Secton., gettng then the replca soluton n Secton., an conserng the nuence of the bounary contons n Secton.. Then, we treat the lute case n Secton.

3 The localze mean el case We conser a ranom Gbbs measure on the conguratons of spns n the nte lattce box C = [ n Z ene by ts Hamltonan H = H on = = f g C as follows. For each let H () = j g (j) j + h ^ = (j)c q C ] where,h are xe postve numbers, ^ = +, ( j) stans for a par n C counte only once, the set g (j) ( j) C s a famly of IID stanar centere Gaussan ranom varables, an q s a functon ene on [ ] such that q s of postve type, so that q s non-negatve an nvarant by symmetry about the orgn. We also assume q s contnuously erentable, nclung at ts peroc bounary. The Gbbs probablty measure s ene for each by where G () = B () =Z B () = B () = exp ( H ()) Z = Z = B () ote that whether the sum over ( j) C nclues the agonal = j or not oes not eect the Gbbs measure. The average wth respect to the Gbbs measure s enote by h, that s hf = Z P B () F (), whle the expectaton wth respect to the ranomness of the g (j) 's s enote by. When a functon F s ene on ( ) n, then F n s a ranom varable on the prouct space of several nepenent replcas of the Gbbs measure, an we stll use the notaton hf whch now enotes Z n P Q n =( n )( ) n = B ( j) F (). ote however, that the same ranom varables g (j) are common to all the replcas of the Gbbs measures. We wll commonly use the abusve notaton hf () nstea of hf. The nteracton functon q can be taen to sgnfy, for example, that nteractons ecay wth the stance between stes of C. For example, q (x) = = + = cos ( jxj). A functon q that ecays very fast near a pea at the orgn sgnes a very wea nteracton except at neghborng stes. In all cases, enotng the uclean nner prouct n R by a ot, we have the followng Fourer representaton for q q (x) = + cos ( x) () Z fg = Z e x where the gree letter enotes the magnary unt, where f g Z s a famly of nonnegatve reals, where = for all 6=, an we assume jj < whch ensures erentablty. A quantty of nterest s the expectaton of the logarthm of the so-calle partton functon = ^ = ^ (j)c q (j)c q p = p ( h) = ^ [log Z ] j j h h j Z * Proof. Follows from the entons wth some smple algebra. ^ + e = C

4 . The Cavty metho The cavty metho whch we propose s nuence by the presentaton of Talagran n []. Some of the passages below are remnscent of ths wor. In partcular, the calculatons n paragraph.. seem to be farly stanar a reaer famlar wth the subject can sp these. We have nclue these an other etals n orer to ncrease reaablty an self-contaneness... otaton We ntrouce notaton that wll be useful n mplementng the so-calle cavty metho for ths problem. The ea s to sngle out a ste m n C an create a cavty there, by progressvely ecorrelatng the spn at m from the other spns. Any calculatons to be performe on the Gbbs measure wth ecorrelate ste m wll be easer than wth the full Gbbs measure. Set ^C m = f C 6= mg. We ecompose, for all, H () = H m ( m ) + m (h + g m ( m )) where m enotes the orere spn values except for the m-th spn, that s, the ^ -tuple an we enote g m ( m ) = m q g (m) ^ = an we also use the notaton H m ( m ) = j ^C m ^C m q j g (j) j + h ^C m ^C m Ths new Hamltonan s smlar but not entcal to the Hamltonan H on. ote n partcular that H m has ^ varables, whle H has \ varables. For a functon f = f ( m ) epenng on m an other varables, let Av f ( ) enote the average of f wth respect to the two values m =. Then we can wrte, for any functon f on, hf = Z = Z f m=( ^C m )g (" m= D Av f ( m ) e (h+gm(m)) f ( m m ) e m(h+gm(m)) # exp ( H m ( m )) where h enotes the average of a functon of m only wth respect to the Gbbs measure wth Hamltonan H m, so that necessarly the normalzng functon Z above must be equal to D Z = Av e (h+gm(m)) = hcosh (h + g m ( m )) We now ntrouce the ecorrelaton scale for the spn at ste m, by progressvely llng the only term that s responsble for correlaton, namely g m ( m ), an replacng t wth an nepenent Gaussan r.v. of approprate magntue, as follows. For any t [ ] let ) g mt ( m ) = p tg m ( m ) + p^r p tg where G s Gaussan ( ) nepenent of everythng an ^r s a number that wll be chosen later. For t =, ths changes nothng. For t =, there s nee no correlaton left between m an the other spns. More precsely we can now ntrouce the averagng wth respect to a measure wth ths new partal correlaton for xe t. We o ths n a multvarable settng, where a functon f epens on an n-tuple of conguratons of spns l n, an possbly some other varables. The averagng operator Av 4

5 operates on f = f( m n m ) by calculatng the average of f wth respect to all n possbltes for l n. Let mnt = exp mt = mt n l= m l h + g mt l! m Z mt = hav mt hf mt = (Z mt ) n hav f mnt mt (f) = hhf mt = (Z mt ) n hav f mnt.. Calculaton of the varatons of mt We have an an mnt = = mnt p t * n mg l m l= l m Av mt p t mg m ( m mt * = 4 n (Z mt ) n Av f p mg l m t l= * 4 (Z mt ) n Av f p^rg n p t mnt l= " * n (Z mt ) n+ hav f mnt = A B C l! + m mnt l m + 5 p^rg p t n l= p^rg!+ p t m 5 Av mt p t mg m ( m ) l m! p^rg!+ p t m # To calculate A we use the followng entty, val f G s a Gaussan ranom varable [GF (G)] = [@F=@g (G)]. Snce we have for each 6= (m) = = mnt n l= = p t ^ = mnt (m) m m l (m) n m mq l l= l = Av p t m ^ mt m q = we get A = A na 5

6 wth an so an * A = 4 (Z mt ) n Av f p t ^C m ^C (m) n l= l m A = " mt (Z mt ) n+ Av f (m) p t mnt A = A = ^ ^C m ^ ^C m jm j q m n q l= n ll = l= + m ^ q l = l m + m ^ q l = mt f l l m l m l mt f l n+ m l m n+ where n the secon lne we combne the prouct of one average over n nepenent replcas of the Gbbs space wth one average over an atonal replca of the Gbbs space nto just one average over n + nepenent replcas. Ths yels " A = ^ ^C m m q n ll = mt f l l m l m l For B, usng agan ntegraton-by-parts, now for the Gaussan r.v. G tself, snce we obtan B = = l= l = = mnt = n n m p^r p l t l= D Av mt m p^r p n mt fm l m l To calculate C we rst wrte C = C C wth " C = n (Z mt ) n+ hav f mnt * C = n 4 (Z mt ) n+ hav f mnt n ^r n l= n l= t Av mt 5 # mt f l n+ m l m n+ # mt fm l m n+ p t mg m ( m ) Av mt p^rg p t m We rst rewrte the proucts of Gbbs averages as sngle Gbbs average, an then use the same calculatons that lea to the expresson for A to get + 5 # C = n ^ ^C m m q " n+ l= mt f l n+ m l m n+ (n + ) mt f n+ n+ m n+ m n+ # The same calculatons as for B yels smlarly that C = n ^r n+ l= mt fm l m n+ n (n + ) ^r mt fm n+ m n+ 6

7 Groupng all our calculatons, an some algebra, yel that the t mt (f) s equal to m ^ q C n n mt f l n+ m l m n+ l= l<l n + n(n + ) mt f n+ n+ m n+ m n+! mt f l l m l m l ^r mt fm l m l l<l n n n l= mt fm l m n+ + n(n + )mt fm n+ m n+! () where we use the fact that for = m, the corresponng term s zero. We now entfy overlap-type quanttes. Let R ll = ^ l l C For Z fg, set now whle for = we use nstea R ll = ^ e = l l (4) C R ll = R ll r r = ^r (5) We call R ll the basc overlap quantty for our problem as t measure the average overlap of spn values for a xe conguraton. The R ll 's are also overlaps, albet relatve to a certan Fourer moe. That all these overlaps are sgncant can alreay be seen mmeately snce, accorng to Lemma., a quantty of mportance for the behavor of the partton functon s precsely P Z fg R + R. The sum of ths last seres can thus be consere as the total overlap relevant to the problem of stuyng the partton functon. Wth these overlap notatons n mn, the expresson mt (f) =@t n () easly yels the followng, whch the reveals relevance of mt to the calculaton of expecte overlaps. Proposton. For any f n! R an t [ ], the ervatve of mt(f) s gven mt = Z e m= n l<l n n l= mt fr ll l m l m mt fr ln+ m l m n+ + n(n + )! mt fr n+n+ m n+ m n+ (6).. Bouns on mt Lemma. There exst postve constants c nq an c nq that epen only on n q an, an are unformly boune n for [ ], such that f f s a postve functon on ( ) n then for all m C an t [ ], mt (f) c nq (f) (7) an j mt (f) m (f)j c nq = jfj 4 Z = R 5 (8) 7

8 Proof. See [, Proposton.4.]. Ths lemma, whose short proof follows essentally that of the corresponng result n the cavty metho for the stanar Sherrngton-Krpatrc moel, wll be combne wth the explct expresson for mt n Proposton. an a separate calculaton of mt for t = to obtan nformaton on the actual expecte overlaps, uner = m,.e. for t =...4 Overlap calculaton at t = Lemma.4 For xe m, let f be a functon on ( ) n that oes not epen on the values m m m. n Then for any subset I of f ng we have m f Y! h m l = tanh (Y ) jij m (f) li where Y s the Gaussan ranom varable ene as follows, wth a stanar normal varable z Proof. See [, Lemma.4.]...5 Overlap expressons at t = Y = z p r + h The prevous result, whch states explctly how m separates the ste m from the other ones, wll be explote n orer to etermne the most convenent value of r when estmatng the expecte-overlap-type quantty O Z R A The rst step s to notce the followng consequence of a symmetry property among stes. Lemma.5 We have, for any xe m C, R r = R r m m r Proof. We can wrte by enton R r = R r!! ^ r C However, uner, for xe, the jont law of an R = ^ P jc j j oes not epen on nee the varables g (j) are IID an the nteractons uner the Gbbs measure, moulo the values of the xe g (j) 's, are nvarant uner translaton by enton. Therefore we have R r = R r m m r whch s the lemma's asserton. Usng Lemma.5 an expanng one of the two factors n xe arbtrary value m C, yelng the followng. R n O we can now calculate O usng a Corollary.6 O = R r m m r + ^ C Z fg R e = 8

9 In orer to explot Lemma.4, we must mofy the above expresson for O by completng the followng two tass () estmate the error mae by replacng the arguments of by functons that are of the same form as those n Lemma.4 () estmate the error mae by replacng by m (or as approprate)...6 Tas (). Separaton of cavty varable from others We eal wth tas () rst. It s sucent to replace R by the same quantty wth the m-th term omtte ene R = ^ = R ^ mm = R + ^ an smlarly let R ^C m = ^ = R ^C m e = ^ mme m= = R + ^ ote that the \bg oh" terms that are the erence of R an R, or of R an (R ), are unform n all parameters, ranom or otherwse. Speccally let O enote the expecte overlap quantty O wth replace by m or as approprate Thus, accorng to Corollary.6, an usng the quanttes wth no m-spn epenence, we must have (or ene) O = m R r m m + Z fg ^ C r R e = + m ^ m m m m r + Z fg ^ C ^ The completon of tas () can then mmeately be summarze as the followng. R Lemma.7 O = m + Z fg R ^ C r m m r R e = + ^..7 Tas (). stmatng the erence between the overlaps at t = an t = We now prepare the tools for provng an explotng tas(), begnnng wth an elementary lemma, ue to Lattala an Guerra, an whose proof can be foun.e. n [, Proposton.4.5] Lemma.8 For any choce of the parameters h >, there s a unque soluton r n [ ] to the equaton r = tanh (z p r + h) (9) Ths lemma allows us to elmnate one of the terms n our overlap calculaton O (at t = wth separate spns) by choosng r properly. 9

10 Lemma.9 Let r be the soluton of equaton (9). Then, for any m C, m R r m m r = Proof. By Lemma.4 we have m R r m m r = m R h r h tanh (z p r + h) = r whch nshes the proof. Smlarly, wth ths choce of r, regarng the other term n O we can wrte R ^ e = Z = r = r fg Z Z fg fg C ^ C A + A + A R e = () where A = A = A = ^ ^ R ^ C C h C e = R R R e = R e = Let ths s the Remann sum approxmaton of Z S = [ ] xe x = ^ e = C Y Z l= xe x l an snce ths prouct s zero because at least one of the components l of s non-zero, we can boun js j by jj =, snce jj s a boun on the ntegran's graent. However one may prefer to rewrte the Remann sum as! Y S = e l l = l= l = whch shows that t s the prouct of the Remann sum approxmatons for the ntegrals R xex l. Therefore, js j can also be boune above by (jj =) Z() where Z () s the number of components of that are non-zero. We state ths secon result formally for future use. Lemma. Conser the followng two contons (H) There exsts an nteger f g such that for every Z Z () of components of that are non-zero satses Z (). fg such that 6=, the number

11 (H') Conton (H) hols an we have jj < () Z fg Uner Conton (H) we have for all Z fg A jj For now, let us only use the rst estmaton on A, whch oes not requre any atonal assumptons on the structure of q, namely A jj () We easly obtan the followng. Lemma. There exsts a constant that epens on an q but s boune for boune such that A + A jj + ^ jj () an Proof. As note before, A = R R R ^, an 4 Z = R R 5 (4). Thus, gnorng Lemma., these facts yel (). For the estmaton (4) on A, we only nee nequalty (8) n Lemma.. Remar. In the remaner of the artcle, wll stan for a postve constant epenng on an q an that s unformly boune n the range of our parameter we wll allow to change from lne to lne. We are reay to state an prove the result whch completes tas (). Lemma. There exsts a constant that epens on an q but s boune for boune, such that jo O j O ^ A 4 R 5 (5) Z Z Proof. We can rst wrte, usng Lemma. Smlarly Z fg ^ Z nfg h R C = R R 4 Z = R e = 5 R r m m r R r m m r = R r m m r 4 R = 5 Z = R = = R r 4 R = 5 Z 4 Z = R 5

12 where we use the fact, from Lemma.8, that r [ ]. We thus have jo O j O ^ + + = R Z fg O ^ + = Z = R 4 Z = R R = O ^ + 4 R = 5 Z O ^ A Z 4 Z = R 5 4 Z = R 4 Z R whch proves the lemma...8 Self-averagng overlap lmt We are now n a poston to estmate O. We show that for small, ths expecte total overlap, whch s recentere usng the value r, converges to at the spee = as long as q has a contnuous graent. Snce r oes not epen on the ranom mea g, the appellaton \self-averagng" s use. Proposton.4 Let > an let r = r () be the soluton of r = htanh (z p r + h) where z s a stanar normal ranom varable. Let be the constant ene n Lemma.,.e. s a constant that epens on an q but s boune for boune. Assume that Z jj < (6) an that s so small that Z jj < In that case, we have, for large enough, wth R ene by (4) an R r P Z A fg jj P Z Z Proof. We have, usng (), (4), an (5) O = Z R O j + r Z fg A A + A + A

13 @ A 4 R 5 + ^ Z Z + r + r Z Z fg fg jj = R = ^ A O + r Z + r Z fg = R 4 Z = R Z fg jj 4 Z = R 5 5 (7) ^ A O+ r jj (8) Z Z where we use Jensen's nequalty to entfy the presence of O n lne (7), an we recall that O ^ s a functon that tens to zero as fast as ^ an that ths convergence hols unformly n all parameters. We now assume the C conton (6). Thus snce s boune for boune, for sucently small we can mae P Z smaller than. The result of the proposton follows. Corollary.5 Uner the same assumptons as n Proposton.4, assumng atonally that Conton (H') hols, we R r P Z A fg jj P Z Z Proof. Ths follows trvally from the proof of Proposton.4 f we use Lemma. nstea of the boun ().. Consequence for the partton functon Let s be the soluton of the equaton s = tanh z p s + h (9) where z s a stanar normal r.v. Ths s = s () s unquely ene by Lemma 9. Let an F ( h s) = ( s) =4 + log + log cosh z p s + h () SK ( h) = F ( h s ()) () The functon SK s nown to be the lmt of p n the stanar Sherrngton-Krpatrc moel. The calculatons of the prevous sectons mply the followng theorem on the behavor of the expecte logarthm of the partton functon p n our stuaton. Theorem.6 Uner the hypotheses of Proposton.4, we have p () SK ( p h) C () where the constant C epens on h, q, an, an s boune for [ ]. ()

14 Proof. Recall from = ^ (j)c q = (B + B + B ) j + Z We have that B s a Remann sum approxmaton of fg R 4 Z[ ] [ ] xyq (x y) Z = 4 Z[ y ] [ ] y x q (x ) Z = 4 Z[ y x q (x ) ] [ ] = Z[ ] x q (x ) = R + + r where we use the fact that q s peroc n all varables, wth pero. Therefore, snce by the C conton () on q, the graent n [ ] [ ] of q (x y) s boune by c q = Z jj < we have jb j c q () Usng ths an Proposton.4, we can wrte for <, r + R Z fg + R Z fg + R + r Z + rc q P Z c q = c q = c q c q To estmate the last term, f we smply say that R R = R + + r r R + + r + rr R Z R r (4) A = K hq p (5) ths oes not allow us to get the announce convergence of orer =. The followng mprovement s necessary. It s prove by showng how our choce for r allows the terms of orer = p to get lle o. Lemma.7 For <, K ( q) = where K ( q) s boune for boune. R 4

15 Proof. By symmetry among stes, we have = R r R = ^ C! = m m r = (f) for any xe value of m, where we ene the functon f = m m r. Recall also n general that from Lemma.4 we have for any functon g ene on (that epens on spn values from n nepenent replcas) but that oes not epen on any replcas of m, m g m m = m (g) [tanh n (Y )] = m (g) r (6) where Y = p rz + h wth z a stanar normal r.v. In partcular we obtan Ths s mportant because of the followng observaton we have Z Z t (f) = m (f) + m (f) + = m (f) + m (f) = (7) Z We see that our tas s to control m (f) an ms (f). From Proposton. we have ms (f) = Z e m= ms ms (f) s t Z t ms (f) s t (8) 4 Z e m= ms + Z e m= ms fr mm fr mm fr 4 4 mm 4 (9) ext, n the above formula, replace the terms R ll by ther non-m epenent terms R ll R ll = R ll ^ l m l me m= It s easy to chec that, because of fact (7), ths wll not change the values of any of the terms. I.e. ms (f) = e m= ms R f m m Z 4 ms R f m m Z + ms R 4 f m m 4 Z whch, after expanng f wth the m-epenent factors above, an usng agan the result (6) from Lemma.4, yels m (f) = c() e m= m R () Z 5

16 where r = tanh 4 Y c() = 4r + r By teratng the formula (9) for ms (f) we can calculate ms (f). Rather than wrtng own all the terms n ms (f) we wll smply note that we wll obtan a sum of terms of the form c () ms fm l mr l ll m l m l R l l where l l l an l are nteger numbers between an 6 an the constants c (), whch may be complex numbers, are boune when s boune. We then estmate each of these terms as follows c () ms fm l mr l ll m l m l R l l R ll c () jfj ms = R l l ms = R c (b) ( + r) ms R + ms = Puttng all these estmates together yels ms (f) K R ms A Z Z K A jj A Z Z = () where we use Proposton.4 wth small enough an s a constant that epens only on an q an s boune when s small enough. Agan, we wll allow below to change from lne to lne. The estmate () on the secon ervatve of mt (f), together wth (8), mply mmeately (f) m (f) () We now rewrte m (f) wth the ntenton of controllng the errors mae by replacng t = wth t =, an replacng R wth R for every. The rst replacement wll be hanle thans to Lemma. an Proposton.4, whle the secon wll only requre notcng that we trvally have m R m R () Accorngly we wrte m (f) = c() Z R + D + D (4) where an D = c () Z h m R D = c () m R Z R m R 6

17 We have by Lemma. an Jensen's nequalty jd j c Z = Z s R Z A s A R Z Z R s A R so that by Proposton.4, ow by () we mmeately get jd j (5) jd j c () (6) Z ow combnng (), (4), (5) an (7) we obtan j (f)j = (f) m (f) + m (f) + c () j (f)j+ c () (7) Z where we use agan Jensen's nequalty to apply Proposton.4. Therefore j (f)j j 4r + rj fg = R + c () j (f)j whch proves the lemma for small enough. Thans to ths lemma we now obtan r where epens only on h q an s boune for [ ]. The rst step to conclue the proof of the theorem t to perform the followng ntegraton Z p () p () r (b) b b (b) r (b) b b (8) Therefore we nee to calculate two terms. Calculatng p () s trval because we have by enton " # p () = ^ log Y e h C " = ^ Y log C = = ^ log ( cosh (h)) C e h!# = log + log (cosh (h)) (9) 7

18 For the nal calculaton we can use nown facts about the stanar Sherrngton-Krpatrc moel. Let s be the unque soluton of the equaton (9). We see that s () = q ( p ) (4) It s nown (see [, Lemma.4.5]) that wth F an SK ene by () an (), we have Z s (x) x x = SK ( h) log ( cosh (h)) By performng the trval change of varable b = p x emane by relaton (4) we obtan Z r (b) b b = SK ( p h) log ( cosh (h)) ow combnng ths (8) an (9) we obtan the theorem. The nal result we present n ths secton shows that whle the complete structure of q oes not seem to eect the lmtng behavor of the partton functon beyon the average value of q, the spee of convergence towars ths value may epen heavly on the behavor of q. We show that the spee can be ncrease to the orer as long as Conton (H') from Lemma. hols. Ths conton restrcts the Fourer ecomposton of q to contanng only terms of the form cos ( x) where x s the sum of at least terms l x l. Moreover the summablty part () of Conton (H') means that q s tmes erentable. Corollary.8 Uner the hypotheses of Corollary.5, we have p () SK ( p h) C () where the constant C epens on h, q, an, an s boune for [ ]. Proof. In the proof of Theorem.6, nclung Lemma.7, some estmates are alreay of orer =. For the others, we may use Corollary.5 nstea of Proposton (.4) n all ts occurrences. Ths mproves all estmates that were orgnally of orer to the orer, wth the excepton of the estmaton () for the term B. To eal wth ths last term, we only nee to use Conton (H'). Inee we wrte jb j = ^ = ^ = ^ (j)c q Z Z fg fg The concluson of Lemma. also means that ^ j (j)c e ( C e = C e = j)= jc e j= A whch mples mmeately that jb j Thans to Conton (), the corollary s prove. Z fg jj 8

19 . Bounary contons In ths secton, we wll show that the prevous results are not aecte f we tae nto account some bounary contons n the Hamltonan H let f! Z g be an arbtrary etermnstc famly of real numbers, such that j! j M for a gven constant M > (n fact,! coul also be allowe to epen on, but we wll not try to go eeper nto that possblty). For, conser the Hamltonan H () = H () = f = ( ) 9 j j = g otce that g stans for the bounary conton, actng on each It can also be observe that H can be wrtten as H () = j g (j) j + h + h ^ (j)c q where C = C n@c, an h = h +!. Denote by p (), hf, t the equvalent to p (), hf, t relate to the Hamltonan H. Then we have the C Theorem.9 Uner the hypotheses of Proposton.4, we have p () SK ( p h) C () where the constant C epens on h, q, an, an s boune for [ ]. Proof The result s of course easly obtane once the self-averagng property for R s establshe. Thus, we wll just try to stress the man erences between the current case an the one ealt wth at Proposton.4 rst, note that the symmetry property (among stes) s no longer true, an hence the equvalent to Corollary.6 s " O = ^ R r mm r mc Thus Lemma.7 becomes " O = ^ m mc R r + Z m m r m fg mc + Z fg R mm e # = C R m m e m= + ^ As far as Lemma.9 s concerne, t s only val for m C. However, nvong the fact that j@c j jc j = O( ), relaton becomes Z fg ^ C R e = = r Z fg A + A + A + Up to those terms of orer, the proof goes now along the same lnes as for Proposton.4. 9

20 The lute case The moel uner conseraton here wll be of the form H () = q ( j) (j) g (j) j + h (4) (j)c C where, h an q have been ene n the prevous secton, q (x) = q(x=) an (j) s an... sequence of Bernoull ranom varables wth parameter = ^,.e. P (j) = = ^ P (j) = = wth ^ stanng for ( + ), whch s the sze of C. We wll also set M q = q. Contrary to the metho we use for the mean-el case, n whch calculatons were performe for each nepenently, the stuy of our lute moel wll be base on an nucton proceure on, an ths wll force us to conser the Hamltonan H on a general subset F C, of sze K, that s the quantty H F () = H F () = q ( j) (j) g (j) j + h F (j)f Havng ths fact n mn, our basc metho wll contnue to be the cavty metho, nspre agan by the presentaton n [] t wll tae the followng form n our current stuaton for any m F, notce that H F () = H ^F m() + A where m = (), g m = g () an ^ ^F m m W m ( ) + h ^F m = f F 6= mg W m ( ) = q ( m) g () For any F C, enote by h F f F n! R, we have the Gbbs average on F wth respect to the Hamltonan H F. Then, for DAv f ^ mn where ^ mn = hf F = n l= l m ^F m D Av ^ m n^f m W m () l + haa ^F m an ^ m = ^ m. We wll enote by g m the expectaton gven the ranomness contane n f m gm ^F m g an by m (resp. g m) the expectaton when the gven the ranomness contane n f m ^F m g (resp. n fg m ^F m g).. Prelmnary results Let n an f F n! R, epenng only on a gven set of coornates ( p p ), where (p p ) F (here we suppose jf j ). We wll rst show that f f s antsymmetrc wth respect to one of ts coornates, then [hf F ] s asymptotcally small. Let us begn wth an easy lemma quantfyng the probablty of an nteracton wthn, whose proof s left to the reaer. Lemma. Dene A by Then A =! there exsts ( j) 6= j (j) = P (A ) ^

21 We also label the followng Posson law boun on the number of nteractons wth a xe ste (whose proof s postpone untl the proof of Lemma. Step ) for further use. Lemma. Let p C, J p ene by J p = [f C p = g (4) an set r = r(p) = jj p j. Then, for large enough, wth = exp((5)=) an ^ =, we have for any u. P (r = u) e ^ ^u u! Let f F n! R, wth n, an let us wrte, for j, f = f ^F pj pj ^F pj pj n^f pj n pj wth obvous notatons. Let us call T j the that transformaton that permutes the rst two occurrences of the jth spn n f T j ^F p j p j ^F p j p j n^f p j n p j = f ^F pj pj ^F pj pj n^f pj n pj The next proposton on the behavor of antsymmetrc functons, aapte from [, Theorem 4..], wll be useful n orer to get some pure state results. Proposton. Let F C such that jf j = K, an f f F n! R be two functons epenng on coornates n = (p p ) such that f T = f an jf j af (n partcular, we assume f to be a postve functon). Then, gven a constant >, there exsts strctly postve constants = ( M q ) an b = such that f, j hf F j ab hf F ^ Proof Frst, observe that, usng Lemma., we have j hf F j hf F ^ a + A c j hf F j hf F (4) We wll now wor on the last term of the rght han se, an always assume that A c s satse. Step Let J p be the set ene by (4), an note that J p \ = on A c. Snce we wll use the cavty metho wth respect to p, we wll call U j the equvalent of transformaton T j, performe on a functon f ene on ^F p. Then, t s easly seen that, on A c, h Av f ^p n Y U = Av f ^p n J p Consequently, wrtng J p = f r g (here notce that r s a ranom nteger epenng on ), we also have " # Av f ^p n = ry Av f ^p n Av f ^p n U s = r f s where f s = Av f ^p n Y U v Av f ^p n s v= sy v= s= s= U v f () s f () s

22 ote that f s satses f s U s = f s, whch means that f s s stll an antsymmetrc functon. Step We wll now get an estmate of f s n terms of Av f ^ p n, smlar to the relaton between f an f. Inee, we have n f s () = Av p "f p n ( n ) exp l p q (p v ) g p v l v + h! # qy T v wth Smlarly, wth Then, t s easly chece that = Av p n p f ( n ) ^ p n exp L () = f () s l= l p s l= q (m v= L () v )g p v l v l v = Av p p n f ( n ) ^ p n exp L () L () = jl () j p l l= v= s q (m s M q jg p v j 4M q l= v= The same estmate hols for L (), an nvong the fact that je x e y j e a jx yj v )g p v l v l v r v= jg p v j v= f x y [ a a], we get jf s j 4M q jg p s j exp 4M q r v= h jg p v j! Av f ^ p n Step We are now reay to state our nucton hypothess. We wll assume that for K, small enough, any subset F of sze K an any functons f f fulllng the hypothess of our proposton, then j hf F j ab hf F ^ (44) Ths s true f K =. Inee, then, F s a sngleton fpg an = fpg so that the event A c s empty, an the secon term on the rght-han se of (4) s null, so (44) s true wth b =. Conser then a subset F of C of sze K +, an f, f satsfyng the assumptons of our proposton. Then hf F hf F = r hf s ^F p D Av f ^ p n s= an applyng the nucton hypothess, we have, on A c, hf g F p 4 M q ab hf F ^ Thus, ntegratng wth respect to the ranomness n g yels hf F p M q abr( + M q ) hf F ^ ^F p!! r g p v s exp 4M q g p v s= s= exp 8r Mq

23 ow, usng Lemma., some elementary computatons of exponental moments for a Posson measure, an combnng wth (4), we get hf F hf F " a + b( + M q ) ^ 5 exp + 8(M q) + e # 8(Mq) Hence, f a b are such that 5 + b( + M q ) exp + 8(M q) + e 8(Mq) oes not excee b, our proposton wll be shown. Ths occurs for small an b = b () as announce. Inee, tae b =. Then t s sucent to tae 5 ( + M q ) exp + 8(M q) + e 8(Mq) whch s obvously true for small. Remar.4 otce that, applyng ths rst result to f = ( j j ) an f = for arbtrary j C, we get, for a constant >, a mean measure of how fast spns ecorrelate as the system sze ncreases, whch s ntutvely consstent wth the fact that we starte o by assumng that each correlaton probablty s = ^ [j h j h h j j] ^ Remar.5 In many of our future computatons, a rate of = coul be attane. However, some Remann sums convergence wll push ths rate own to =. Ths s why, from now, we wll often use the trval boun = = for our estmates. We wll turn now to some prelmnary results n orer to get the lmtng behavor of the magnetzaton, an let us ntrouce rst some notaton for two [ ] -value ranom varables an Y, ene the Monge-Kantorovch stance between the law of an the law of Y by (L() L(Y )) = sup [f()] [f(y )] f L () (45) = nf f[j j] ( ) M Y g (46) where L () enotes the set of Lpschtz functons boune by, wth Lpschtz constant lesser than, an M Y s the set of ranom vectors n wth margnal laws L() an L(Y ) respectvely. For a vector = ( v ) [ ] v, a functon on f g v, ene h by Z h = f (y y v ) (y ) v (y v ) g v (47) where j s a Bernoull measure on f g wth mean j. Let = (p p ) be a sequence n F, where F C. We wll also nee to ntrouce a generalze cavty metho, snglng out all the elements of at once, an changng the ntegraton over C for an ntegraton over C set ^F = f F = g an for j f g ene J pj as n (4). Dene also ^ n by n ^ n l= j= l p j W pj J pj () l + haa (48)

24 As usual, we wll esgnate ^ by ^. Then, for any f n! R, DAv f ^ n hf F = ^F DAv ^ ^F The expectaton gven the ranomness n wll be wrtten as g an, when contonng only on the values taen by, by. We wll conser the set B = A [ A () [ A(), where A () =! there l [ j=j pj [ (l) = o A () n! = there exst F j 6= j pj = pj = Then B satses the followng property, whose proof reles on results n the proof of Lemma. Step.a. Lemma.6 The probablty of the set B s boune as follows P (B ) log() (49) Proof We use the results of Step.a n the proof of Lemma. wth =. Theren, wth p a xe ste, the quantty A enotes the probablty P [r (p) = ], an s estmate as A e () =!, so that P hr (p) > log ^ = ^. Then usng the trval fact that the carnalty s of orer ^ up to a constant, we wrte h P A P 9l J pj (l) = + P (l) = j= l ^ + P 9l J pj (l) j= We estmate the last probablty above as follows P 9l J pj (l) = h P r (p j ) log ^ 9l J pj (l) = + P hr (p j ) > log ^ log ^ ^ + ^ Therefore h P A () ^ + ^ log ^ ^ + ^ whch proves the lemma, gven that the calculaton for A () s easer, an that t s trval for A. Remar.7 The lac of nteracton an [ j= J p j [, responsble for the atonal term log() n (49), s only neee n orer to get Lemma.9. Snce B s an exceptonal set, we wll always wor (unless spece) on B c. In that case, set r = F pj (5) j= 4

25 an Then, on B c, ^F = n ^F o DAv f ^ n hf F = ^F DAv ^ ^F an notce that now ^F C. In partcular, on B c, L h p F h p F DAv p ^ = ^F DAv ^ ^F DAv p ^ DAv ^ ^F A The next lemmas are then a rst step towars the lmtng behavor of h. Lemma.8 Let Y be the ranom vector ene by Y = h j ^F j ^F ^F an set, for f g, DAv p ^ DAv p ^ u = ^F DAv ^ ^F v = DAv ^ Y where the numerator an enomnator of v are ene by (47). Then, on B c, we have, for a constant = >, (L (u u ) L (v v )) r exp ((h + )) ^ Y Proof The enton of the stance easly mples that (L (u u ) L (v v )) = g [ju v j] Let fs j j Kg be an enumeraton of ^F, set f = Av ^, f = Av p ^, an for j K, f j = f s j sj sj+ sk an f j smlarly. Then, notce that u = hf ^F hf ^F v = hf K ^F hf K ^F Thus, g [ju v j] K j= g " f j ^F hf j ^F f j ^F hf j ^F The lemma s now obtane usng the antsymmetry of fj fj, the fact that all the J p j are sjont on B c, an that j P j J p j j = r where r s ene by (5). # 5

26 We wll show n the next secton that h j F converges n law to a certan ranom varable wth law by a xe pont argument. Let then = f j j ^F g be an... sequence of law on [ ], an set DAv p ^ w = DAv ^ Dene now, for, K = jf j, I () = n F C K ^ o an for,, ^ D( ) = sup L h t F h t F jf j I () t t F We also set D( ) = f > ^. Then we have the followng result, for whch we refer to [, Lemma 4.4.], an from whch there s lttle oubt that our strategy shoul nclue an nucton on the value of D ( ) wth respect to. Lemma.9 Suppose that F I (). Then outse B, we have (L(v v ) L(w w )) ( + ) exp ((h + ) () r ) D( + r) + P (B ) r! r=. Lmt law for the magnetzaton Our rst tas here wll be to gve an asymptotc expresson for the law of (w w ). Set then, for r, an = ( s ) F s,! r G Kr = G Kr ( r ") = exp q ( s )g s s " + h" where (g g s ) are... stanar Gaussan ranom varables an " represents as usual an nepenent spn. On the other han, f x = (x x r ) s a sequence n [ ], set! r G rx = exp q(x s )g s s " + h" (5) s= an ene, for a gven p [ ] r, the quantty U rx (p) = hav "G P r rx p hsnh ( s= = q(x s)g s s + h) p hav G rx p hcosh ( P r s= q(x (5) s)g s s + h) p In the sequel, we wll conser the ranom varable U rx ( (r) ), where (r) = ( r ) s an... sequence of law on [ ]. Let now Z be a Posson pont process on [ ] wth ntensty x. A gven realzaton of Z s of the form (r ), where r an = ( r ) [ ] r. Dene then ^G Z = G r an ^U Z = U r ( (r) ). If s a probablty law on [ ], one can construct the transform T as follows conser f n n g an... sequence of ranom varables of law. Then, for a realzaton (r ) of Z, set V Z as the law of the ranom varable ^U Z, an f ^ Z esgnates the expectaton wth respect to the Posson pont process Z, put T = ^ Z [V Z ] s= 6

27 whch means that f S s a ranom varable wth law T, an f s a test functon on [ h h [(S)] = ^Z ^UZ Z h = e r U r ( (r) ) r! We then have the followng. r= [ ] r Lemma. There exsts a constant > such that for any F I (), L(w w ) (T ) + + ( + + rq ) + ], then (5) Proof otce rst that, outse B, the ranom varables w w are nepenent. Hence, usng the fact that jw j j an the expresson (46) enng, we get L(w w ) (L(w )) ( + ) (54) ^ We wll now turn to some estmates on the law of w = w. For notatonal sae, we wll set p = p, w = w, g p = g. Then, f s a functon n L (), recallng enton (48), we get D Av " exp " P J [(w)] = p W p ( ) + h" D P A5 Av exp J W p p ( )" + h" p = gr r p where gr(x x r ) = U rx ( (r) ) = hsnh ( ( P r = q(x )g + h)) hcosh ( ( P r = q(x )g + h)) an where r = jj p j an r are an enumeraton of the r stes n F that nteract wth p. Then, usng some stanar calculatons for the bnomal strbuton, g [(w)] = A r B r r= wth A r = for r K, an for r K, r K A r = r ^ gr B r = ( r)a( ^F p r) K r ^ p r p (55) A r K where A( ^F p r) stans for the set of orere combnatons of r elements among those of ^F p, an A r (K )! A( K = (K r)! = ^F p r) ote n partcular that A = P [jj p j = ], an s nee the bnomal probablty of successes (nteractons from wthn F to p) among K trals wth success probablty = ^. 7

28 We wll now loo for the lmt of A r an B r separately n Steps an respectvely. Step.a From the Posson approxmaton of the bnomal strbuton, t s clear that lm A r = e r! r! but we wll loo for some sharper estmates for that convergence. Frst, remar that K r ^ r r! ry j= + j ^ Hence, usng the fact that ( x)( y) (x + y) for x y, we get ^ r r! r( + r + ) ^ C rk Invong now the fact that log( y) y for y [ ), f s large enough, we have A r r r! exp (K ) ^ ^ r r! ^ r (56) Recall that K ^ + ^. Hence A r e r r! exp ( + ) ^ ^ r (57) We wll rst use a crue boun on A r f s large enough, then ^ an f (+) ^ <, whch of course occurs for large enough, we get A r e wth = exp(5=) an ^ =. ow, f V s a Posson ranom varable wth parameter ^, the followng exponental boun hols true, for v > ^ ^r r! )^ P (V > v) e(e e v In partcular, f v log( ^ ) + (e )^, whch s true for v log( ^ ) when s large enough, we have P (V > v) ^ Thus, up to an exceptonal set of probablty less than = ^, we can assume that r log( ^ ). notatonal convenence, we wll set G = f! r log( ^ )g. For Step.b Recall that (57) hols true for any r. Hence, f r log( ^ ), A r e r r! e exp (+) ^ r! r + r A ^ ^ 8

29 Furthermore, t s easly chece that, for large enough, an r log( ^ ), ^ r r ^ r ^ an snce e u ( + ) exp ^ ( + ) ^ u for u. Hence, for large enough, on G c, we have A r e r r ( + ) + r e (58) r! r! ^ On the other han, t s clear from (56) that A r r r! r r! r r! e K ^ K ^ (r+) ^ r( + r + ) ^ Moreover, f x >, t hols that x e x ( x ), an hence A r r r! K^ exp ^ + r( + r + ) whch means that A r e r r! e r r! ^ r( + r + ) ^ r( + r + ) ^ + r( + r + ) ow, puttng together (58) an (59), we get, on G c, A r e r r! e r 4( + + ) + r(= + + ) + r = r! ^ ^ (59) Pluggng the values of the rst two moments of a Posson law nto ths last equaton, an usng the fact that G s an exceptonal set, we get A r e r 4 + ( + ) + 5 r! ^ r= Step.a Our next tas wll be to obtan some sharp estmates on B r, ene by (55). Set rst B () r = ^ r ( r)a( ^F p r) gr p r p Then However, by nequalty (56), we have B r B () gr r ( r)a( ^F p r) ^ r A r K ^ r A r K ^ r A r K r( + r + ) ^ r ^ (6) 9

30 Hence Br B r () r( + r + ) gr ^ (6) Furthermore, nequalty (6) also yels where ^B r = ^ r ^B r B r () r( + r + ) gr ^ (6) ( r)(c ) r gr p r p Because of the perocty of q, we can entfyng B r as the Remann sum corresponng to the ntegral of gr over [ ] r. Thus, settng Z I gr = gr(x x r ) x x r [ ] r we get that j ^B r I gr j gr L () (6) Puttng together (6), (6) an (6), we obtan jb r I gr j [ gr L () + gr r( + r + )] (64) Step.b Let us compute now gr an gr L (). The rst one s easly obtane, snce gr To estmate the Lpschtz constant of gr, for r an x = (x x r ) [ ] r, set `r(x) = hsnh(v r(x) hcosh(v r (x) where! r v r (x) = q(x )g + h For each f rg, notce that x s of the form fx (j) j g, wth x (j) [ notatons, = ]. Thus, wth x(j)`r(x) = x(j)q(x ) " h cosh(v r (x) hcosh(v r (x) # hsnh(v r (x) h snh(v r (x) hcosh(v r (x) an rq jg j whch yels r r gr L () L ()rq jg j rq jg j = = an hence, tang the expectaton wth respect to the g's only, r [ gr L ()] () = rq r (65)

31 Pluggng ths estmate nto (64), we get, for a constant >, r [jb r I gr j] [rq + + r + ] r On the other han, we wll also nee the followng trval boun on B r jb r j gr (66) Step (concluson) We are now reay to estmate the stance (L(w w ) (T ) ). ote rst that (L(w)) (T ) (L(w) T ) Let now be a functon n L () an S be a ranom varable of law T. Invong the enton (5), relatons (5), (65), an (66), we have prove that j[(w)] [(S)]j K = [A r B r ] e r r! [I gr] r= r= A r e r r! + e r r! r [jb r I gr j] r= r= ^ + ( + ) + + [rq ] + ( + + rq ) + Ths, together wth (54), ens the proof. Recallng the enton of D( ), Lemma.9 an., t s obvous that the natural canate for the lmt law of h s the soluton to the equaton = T. We wll now show that ths equaton has a unque soluton. Proposton. Suppose that an M q satsfy M q ( + M q ) exp (M q ) ( + ) (67) for a constant large enough. Then there exsts a unque soluton to the equaton = T. Proof We wll rst state some elementary results, an then apply them to prove that T s a contractng map for the stance. Step It can be shown, by a slght varaton of [, Lemma 4..], that, fa r r g beng a sequence of postve numbers such that P r a r =, we have! a r r a r r a r ( r r ) (68) Z[ ] Z[ r ] Z[ r ] r r= r= Furthermore, f f f g r 7! R, an p [ ] r, hf p beng ene by (47), we have (cf. [, Lemma 4..]), for f p hf p = h f p where, for f g r, r= f() = f(+ ) f( )

32 wth = ( + r ) Step From the prevous step, for any p = (p p r ) [ ] r, an U rx (p) ene by (5), we p [U rx (p)] = h (Av "G rx ) p hav G rx p hav "G rx p h (Av G rx ) p hav G rx p Moreover j (Av "G rx )j = jav (" G rx )j Av j G rx j an havng n mn equaton (5) enng G rx, t s easly seen that j G rx j M q jg j exp (M q jg j) from whch the upper boun follows. j@ U rx (p)j M q jg j exp (M q jg j) (69) Step Let p p be two elements of [ ] r. Then, from (69), we have U rx (p ) U rx (p ) M q jg j exp (M q jg j) p p Conser now two probablty measures on [ ], an f () () rg some nepenent copes of ranom varables of law an respectvely. Set, for j f g, T rx ( j ) = L U rx (j) Then we have (T rx ( ) T rx ( )) M q r [jgj exp (M q jgj)] hj () () j M q ( + M q )e (Mq) r hj () () j where jzj s ene on r by jzj = P r jz j. Hence, tang the nmum of ths last quantty over M () (), we get (T rx ( ) T rx ( )) M q ( + M q )e (Mq) r ( ) ow, usng (68), we have (T ( ) T ( )) M q ( + M q )e (Mq) r= =M q ( + M q )e (Mq) ( + )( ) Z r e r ( ) r! [ ] r Hence, uner the hypothess (67) of our proposton, T s a contractng map ene on the probablty measures on [ ], whch shows our clam. We can now turn to the man result of ths secton, that s the lmt law for the magnetzaton. Theorem. Let. Then, for a strctly postve constant = q, we have L (h h ) + log()

33 prove exp ((h + )) ( + + 4( + )) < (7) Proof We assume the followng nucton hypothess for some constant = q an for all, fg, an F I (), D ( ) ( + ) + log (7) We can begn the nucton at = by choosng the constant large enough, snce the above nequalty nees only to be chece for ^, an ts left-han se s some xe nte value that can be absorbe nto. ote rst that usng = f g an F = C tself, then we have that the sequence h h s entcal to h p F h p F. From Lemma.8 an.9, we then have, for any an F I (), (L (h h ) L (w w )) ( + )( + )( + log()) exp ((h + )) +( + ) e r=! ()r D( r! + r) an nvong now Lemma., we get, f s the soluton to the equaton T =, L (h h )! ( + )( + ) ( + log()) q exp ((h + )) ()r +( + ) e D( ( + ) + r) q exp ((h + )) ( + log()) + + 4( r! r= + ) + log In the last step we use the nucton hypothess (7) an the fact that + + ( + ). If conton (7) s satse, ths allows to propagate the nucton hypothess to the next step, whch ens the proof.. The replca symmetrc soluton As usual, set Z = P exp( H ()). In our context, t wll be easer to conser ths quantty as a functon of the parameter, nstea of. Set then p () = ^ [log(z )] an call p () the rght ervatve of p wth respect to. It can be shown, as n [, Lemma 4.4.5], that for a constant >, jp () A j where A = ^ (j)c h log De q ( j)g (j) j ote that, settng C = C nfg, g j = g (j), ths last expresson can be symmetrze to obtan A = ^ jc h log De q (j)g j j Denote by g C the expectaton on C contone on the values of g (j), (j). The followng Lemma wll be essental n the computaton of p ().

34 Lemma. Let (j) = (h h j ). Then, for any j C, f an s small enough, we have for a strctly postve constant h g log C De q (j)g j j e (Mqjgjj+h) log() log De q (j)g (j) j j Proof Ths result s obtane usng the same technques as n Lemma.8. One can also go one step further, an replace (j) by Lemma.4 Let j C an Y (j) = (Y Y j ) be two nepenent ranom varables of law, such that Then [jh Y j] + [jh j Y j j] log() C h log D e q (j)g j j log De q (j)g j j Y (j) e (Mq+h) log() Proof By stanar methos, usng Theorem. an the boun j log(z ) log(z )j jz z j a that hols for z z [a ) wth a >, we have h g log C De q (j)g j j log De q (j)g j j Y (j) e (Mqjgjj+h) log() Integratng ths result wth respect to the remanng sorer n g, we get the esre result. We can now state our man result concernng the ervatve of p (). Proposton.5 For any >, there exsts a constant such that for an beta small enough, Z Z D p () log e q(x)g"" (y ) (y ) y e Mq(Mq+h) log() [ ] x [ ] Proof Lemma.4 mples rectly that Z D p () log e ^ q (j)g" " (y ) (y ) [ ] y e Mq(Mq+h) log() jc The result s then obtane by Remann sums approxmaton, once the graent of D x 7! log e q(x)g"" y s controlle, whch can be one as n the proof of Lemma., Step.b. There s now lttle to change to the proof of [, Theorem 4.4.] to obtan the followng 4

35 Theorem.6 Uner the contons of Proposton.5, we have wth F () = log() where jp () Z Z x L (x y) (y y )+ [ ] [ ] F ()j K log r e Z r= r! [ ] x D L (x y) = log e q(x)g"" y D H (x y) = log e P r q(x)gss" s= y Z [ ] r H (x y) r (y y r ) References [] Bover, A Gayrar, V Pcco, P. Gbbs states of the Hopel moel n the regme of perfect memory, Probab. Theory. Relate Fels,, no (994), pp [] Carmona, P Hu, Y. On the partton functon of a recte polymer n a Gaussan ranom envronment, Probab. Theory. Relate Fels 4 (), pp [] Comets, F eveu, J. The Sherrngton-Krpatrc moel of spn glasses an stochastc calculus the hgh temperature case. Comm. Math. Phys. 66, n. (995), pp. 549{564. [4] Derra, B Jung-Muller, B. The genealogcal tree of a chromosome. J. Statst. Phys. 94 (999), no. -4, pp. 77{98. [5] Guerra, F Tonnell F Quaratc replca couplng n the Sherrngton-Krpatrc mean el spn glass moel, J. Math. Phys. 4, no. 7 (), pp [6] Mezar, M Pars, G Vrasoro, M.A. Spn glass theory an beyon. Worl Scentc Publshng Co. (987). [7] ewman, C Sten, D. on mean-el behavor of realstc spn glasses, Phys. Rev. Lett. 76 (997), pp [8] Sherrngton, D Krpatrc, S. Solvable moel of a spn-glass, Phys. Rev. Lett. 5 (975), pp [9] Talagran, M. The hgh temperature phase of the ranom K-sat problem, Probab. Theory. Relate Fels 9 (), pp [] Talagran, M. Spn Glasses a Challenge for Mathematcans, to appear n Sprnger V. [] Toubol, A. tue e quelques moeles e systemes esoronnes, Ph. D. Dssertaton at the Unversty Pars 7 (997). 5