On the multiple overlap function of the SK model

Publ. Mat. 51 (2007), 163–199 ON THE MULTIPLE OVERLAP FUNCTION OF THE SK MODEL S´ ergio de Carvalho Bezerra1and Samy Tindel Abstract In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap R1,...,s of the SK model, defined for given N, s ≥1 by R1,...,s =N−1Pi≤Nσ1 i. . . σs i. These results are obtained by a careful analysis of the terms appearing in the cavity derivation formula, as well as some graph induction procedures. Our method could hopefully be applied to other spin glasses models. 1. Introduction The celebrated SK model, which can be seen as a generic spin model with random interactions, also happened to model (together with some of its generalizations) different situations, such as disordered particle systems or neural capacity (see [5], [7]). Briefly speaking, the canonical space of the model is the set ΣN={−1,1}N, called space of configurations, where Nis a positive integer which represents the number of spins. A configuration σ= (σ1,...,σN)∈ΣNspecifies the values of all spins and the probabilistic feature of the model emerges when we suppose that the spin interactions occur randomly and the energy of each configuration, sum of all the interactions, can be written as (1) −HN(σ) = 1 N1 2X 1≤i<j≤N gi,jσiσj, where gi,j is a family of independent standard Gaussian random variables defined on a probability space (Ω,F,P) and 1/N 1 2is a normalization factor. 2000 Mathematics Subject Classification. 82D30, 60G15. Key words. SK model, overlap function, cavity method. 1Research supported by CAPES. 164 S. de Carvalho Bezerra, S. Tindel As usual in statistical mechanics, we associate a Gibbs measure GN on ΣNto the Hamiltonian HN, and this Gibbs measure depends on a parameter βwhose meaning is the inverse of the system temperature. The model constructed then starting from (1) has been introduced first [7] in order to describe spin glass systems, i.e. magnetic systems in which the interaction between the magnetic moments are ‘in conflict’ with each other. Since then, the Physicists have been mostly interested in the behavior of the SK model for large values of β, but let us mention at this point that during all this work, we assume to be in the region of high temperature (i.e. β < 1) for which a huge amount of information is available (see [8], [2], [6], and the path-breaking papers [3], [9] for the SK model with external field). Let us introduce also some classical notation, which will allow us to state our main results: given a positive integer number n(number of system replicas) and fa function on Σn N, we define hfias the expected value of fwith respect to the product measure dG⊗n Nand ν(f) as the expected value of hfiwith respect to the randomness contained in the coefficients gi,j, that is ν(f) = E[hfi]. The problem we will deal with starts from the following observation: a large proportion of the structural information about the behavior of ΣN under GNis usually obtained by studying the so-called overlap between two configurations σ1and σ2, defined by R1,2,1 N N X i=1 σ1 iσ2 i, which can be also related to the Hamming distance between σ1and σ2 (understood as two independent configurations under G⊗2 N). And a natural extension of R1,2would be a quantity that measures the correlation among sconfigurations, for example: (2) R1,2,...,s ,1 N N X i=1 σ1 iσ2 i...σs i. Clearly, the asymptotic behavior of such a quantity would give us some additional information about the limiting spin system when Ngoes to infinity. However, in spite of the sharp asymptotic estimates available for R1,2, the study of R1,2,...,s for s > 2 is still poorly developed, and this paper proposes to make a step in that direction: we will prove the following CLT (central limit theorem) for a family (Rℓ1,ℓ2,...,ℓs)1≤ℓ1<···<ℓs≤n for any s > 2. Multiple Overlap Function 165 Theorem 1.1. Consider two integers 3≤s≤n, and for 1≤ℓ1< ···< ℓs≤nsome non-negative integers k(ℓ1,...,ℓs). Set k=Pℓ1,...,ℓs k(ℓ1,...,ℓs). Then (3) ν Y ℓ1<···<ℓs Rk(ℓ1,...,ℓs) ℓ1,...,ℓs!−Y ℓ1<···<ℓs a(k(ℓ1,...,ℓs)) Nk(ℓ1,...,ℓs) 2 =O(k+ 1), where we denote by a(k)the kth moment of a standard Gaussian random variable, and where the relation g=O(k)means the existence of a constant csuch that |g|<c Nk/2. This theorem implies that for a typical disorder, a finite family of functions, ˆ Rℓ1,...,ℓs=√NRℓ1,...,ℓs defined on (Σn N, G⊗n N), with s≥3, asymptotically looks like an independent family of standard centered Gaussian random variables. It is worth observing that, contrarily to the case s= 2 treated in [8], for s > 2, the dependence on βin the normalization of Rℓ1,...,ℓsdisappears. This has been a surprise for us. Let us also mention at this point that, from our point of view, the study of multiple overlaps is a natural question, which illustrates the fact that the understanding of the SK model is still far from being complete. On our way to the proof of Theorem 1.1, we will have to compute the first two terms in the expansion of ν(R2 1,2,...,s), and we will obtain a result which generalizes a result obtained by Talagrand [8, Proposition 2.3.5] for s= 2: Theorem 1.2. Given s∈Nand β < 1, the following relations hold true: i) If sis odd (s≥3), then (4) νR2 1,2,...,s=1 N+O(2p),for all p≥2. ii) If sis even (s= 2k), we have (5) νR2 1,2,...,s=1 N+c(β, s) Nk+O(2k+ 1), where c(β, k) = (2k)! k!(β2 2(1−β2))k. Theorem 1.2 can be seen in fact as the main contribution of this paper. Indeed, on one hand, once these relations are proven, the announced CLT can be deduced by means of the standard methods introduced e.g. in [8], and one could also argue that it is implicitly contained in [4] (or at 166 S. de Carvalho Bezerra, S. Tindel least, that the techniques involved in [4] could yield the proof of our Theorem 1.1); notice however that this latter reference relies heavily on the fact that the SK model without external field is considered. On the other hand, our expansion of ν(R2 1,2,...,s) is new; it will be achieved thanks to some graph-type methods, which have their own interest in the SK context, and are introduced here for the first time (as far as we know). Furthermore, it seems that our computations don’t depend too much on the specific model we have considered, and thus we hope to extend this kind of methodology to other situations, like the pspins models with external field or the perceptron model. Our paper is organized as follows: In the next section, we introduce some notations and definitions. In the third section, performing a Taylor expansion, we obtain a general expression of νt(f) where fis a function defined on Σn N, and we evaluate the leading term of ν(Qm i=1 ǫℓiǫjiRℓi,ji) for some specific ℓi’s and ji’s (where ǫl=σl N). The fourth section will be devoted to the computation of ν0(f) for a certain class of functions f. Eventually, in the last two sections, we conclude with the proof of Theorems 1.2 and 1.1. 2. Preliminaries In this section, we will first introduce some notations, and then give briefly some definitions which will be used in the sequel of the paper. Eventually, we will explain the strategy of the proof of Theorem 1.2. 2.1. Smart path and overlap products. In order to expand ν(R2 1,2,...,s) in terms of N, the use of Taylor series is certainly a natural idea. So, for a given configuration σ∈ΣNand a parameter t∈[0,1], define a new energy function HN,t(σ) = 1 N1/2X 1≤i<j≤N−1 gi,jσiσj+t N1/2 σNX 1≤i≤N−1 σigi,N , where the coefficients gi,j are, as before, independent Gaussian standard random variables. Set now GN,t({σ}) = exp(−βHN,t(σ)) ZN,t ,where ZN,t =X σ∈ΣN exp(−βHN,t(σ)). Multiple Overlap Function 167 These random measures induce some averages hfitand νt(f) defined, for a function f: Σn N→R, by hfit=Pσ1,...,σn∈ΣNf(σ1,...,σn) exp Pn i=1 −βHN,t(σi) Zn N,t and νt(f) = E[hfit]. Define also the overlap functions Rℓi,jiand R− ℓi,ji by Rℓi,ji,1 NX k≤N σℓi kσji k,and R− ℓi,ji ,1 NX k≤N−1 σℓi kσji k. Then the function t7→ νt(f) can be differentiated in the following way (see [8]): Proposition 2.1. Given a function fon Σn Nand t≥0, we have ν′ t(f) = β2X 1≤l<l′≤n νt(fǫlǫl′R− l,l′) −β2nX l≤n νt(fǫlǫn+1R− l,n+1) +β2n(n+ 1) 2νt(fǫn+1ǫn+2R− n+1,n+2). This proposition will the basis of our future expansions. Apart from the usual overlap function R1,2, we will have to introduce a specific notation for some products of overlaps which will appear throughout our computations: given some arbitrary positive integer numbers ℓ1, j1,...,ℓm, jmsuch that ℓi≤jifor all i≤m, we set (6) Sℓ1,j1,...,ℓm,jm, m Y i=1 ǫℓiǫjiRℓi,ji, S− ℓ1,j1,...,ℓm,jm , m Y i=1 ǫℓiǫjiR− ℓi,ji. Remark 2.2.The importance of the products ǫℓiǫjiRℓi,jistems basically from Proposition 2.1, in which they appear naturally. 2.2. Sets and graphs. Our proofs will also make use of two subsets of tuples of positive integers: given a positive integer k, set (7) Ω2k,{(r1,...,r2k)∈N2k|ri≤N, ri6=rjif 1 ≤i < j ≤2k and r2u−1< r2ufor all u≤k} 168 S. de Carvalho Bezerra, S. Tindel and (8) Ck,{α= (ℓ1, j1, . . . , ℓm, jm)|(H) holds true}, where (H) is the following assumption: •ℓiis smaller than jifor any i≤m; •If αalso designates the set {ℓ1, j1,...,ℓm, jm}, then {1,2,...,2k}⊂ α; •The only elements of αwhich appear an odd number of times are 1,2,...,2k. Obviously, the definition of the quantity Sℓ1,j1,...,ℓm,jmdepends on the sequence (ℓ1, j1,...,ℓm, jm). For sake of clarity, we will associate a graph to such kind of sequence, where a graph is understood for us in the following sense: Definition 2.3. Let Ibe a set of positive integers and Ebe a subset of I×I. We refer to Ias the vertex set and to Eas the edge set. In addition, if (i, j)∈E, assume that i < j and let Υ: E→N∗be a function which counts the number of edges of type (i, j). Then, the triple (I, E, Υ) is called a graph. Given a graph (I, E, Υ), for each J⊆I,F⊆J×J with F⊆Eand V:F→N∗such that for all e∈F,V(e)≤Υ(e), we call (J, F, V ) a subgraph of (I, E, Υ). Obviously, a subgraph is also a graph. Here is now the procedure we will use for our graph construction: pick a sequence (ℓ1, j1,...,ℓm, jm) of 2mnumbers, and assume, for sake of simplicity, that ℓi< jifor all 1 ≤i≤m. Define then •I={ℓ1, j1,...,ℓm, jm}; •E={(ℓi, ji)|i≤m}; •Υ((ℓi, ji)) = #{r≤m|(ℓi, ji) = (ℓr, jr)}. We denote this graph by G((ℓ1, j1,...,ℓm, jm)). In particular, given our set Ckwe can associate the family of graphs Gk={G(c)|c∈ Ck}. Let us define some local and global objects on a graph g= (I, E, Υ). Set first Ng(i),X e∈E:i∈e Υ(e) and N(g) = X e∈E Υ(e). Obviously, Ng(i) represents the number of edges having ias an endpoint, and N(g) the total number of edges of g. Furthermore, it is easily checked that N(g) = 1 2Pi∈INg(i). Let us also define a quantity, associated to each vertex i, indicating if Ng(i) is an odd number or not: Od(i) = 1 2[Ng(i) mod(2)] and Od(g),X i∈I Od(i). Multiple Overlap Function 169 Associated to these notions, some subgraphs of graphs in Gkwill play a special role in the sequel: for each g∈ Gkwith N(g) = mand any u≤m, we define Su(g),{h|his a subgraph of g, and N(h) = u}. Notice that the definitions of the current subsection won’t be used until Proposition 4.4. However, we have already introduced them at this point, since they are at the core of our method. 2.3. Strategy of the proof for Theorem 1.2. The proof of our main result Theorem 1.2 is built upon a series of lemmas and propositions which will be stated and proved throughout Sections 3 and 4. Since the reader may get lost during these preliminary steps, here is a brief sketch of the methodology we will follow in order to estimate ν(R2 1,...,s). (1) Using the symmetry property among sites, we will check that νR2 1,2,...,s=1 N+νǫ1ǫ2...ǫsR− 1,2,...,s. With this relation in mind, our main task will be obviously to estimate the term ν(ǫ1ǫ2. . .ǫsR− 1,2,...,s). We will see that, whenever sis an odd number, the estimation is quite easy, and thus, we will concentrate mainly on the case s= 2k. (2) In order to get an equivalent of ν(ǫ1ǫ2...ǫ2kR− 1,2,...,2k), we will perform a Taylor expansion of this quantity along the smart path defined by νt. Then, due to the presence of the products of ǫ’s, we will be able to show that many terms of the expansion vanish, or can be neglected. These preliminary considerations will be developed at Section 3.1, and will lead us to focus essentially on some terms of the form ν0U− kS− αwith U− k=ǫ1ǫ2...ǫ2kR− 1,2,...,2k, where the multi-index α= (ℓ1, j1,...,ℓm, jm) lies in a certain class which will be determined throughout Section 3. (3) Recall that Ckhas been defined in (8). Then we will prove that, whenever the multi-index α= (ℓ1, j1,...,ℓm, jm) belongs to Ck, we have (9) ν0(U− kS− α) = ν(Sα) + O(2k+ 1). 170 S. de Carvalho Bezerra, S. Tindel This will be achieved at Section 4.2, through the introduction of a family of functions, called R-systems, allowing an operational backward induction on the order of multi-indexes defined in (8). (4) By looking at relation (9), we see that we are left with with the evaluation of the quantities S− α. Equivalently, since the random variables S− α are stable by multiplication, we have to deal with their covariance structure. This depend mainly on the form of the multi-index α, and after some rather standard computations, we will base our estimates on: 1) An equivalence relation between multi-indexes (see Proposition 3.8). 2) A graph structure on these multi-indexes, which will be used mainly at Section 4.1. Thanks to the two tools mentioned above, we will be able to analyze precisely the covariance structure of the random variables S− α, leading then to the conclusion of our proof by a series of elementary considerations. 3. Some general Taylor expansions In this section, we will first establish a general expression for the Taylor expansion of the function t7→ νt(f) around 0, for a given f: Σn N→R. Then we will identify some negligible terms and give a more explicit expression for the typical term of this expansion. Eventually, we will examine the special case where fis the function S− ℓ1,j1,...,ℓm,jm, and using an induction argument, we will evaluate ν(Sℓ1,j1,...,ℓm,jm). 3.1. General and error term. Let us start this section by giving an extension of Proposition 2.1: for k, n ≥1, define the set Dn,k as (10) Dn,k ={α=(ℓ1, j1,...,ℓk, jk); ℓi, ji≤n+2k, ℓi< jifor all i≤k}. Proposition 3.1. Let fbe a function on Σn Nand consider t≥0. Then the kth derivative of νt(f)can be written as (11) ν(k) t(f) = X α=(ℓ1,j1,...,ℓk,jk)∈Dn,k c(n, k, α)β2kνtfS− ℓ1,j1,...,ℓk,jk, where the family {c(n, k, α) ; α∈ Dn,k}is just a family of Z-valued coefficients. Proof: The approach used to show the result is an induction argument on k: the case k= 1 can be easily shown thanks to Proposition 2.1, and in order to advance the induction, we assume that the result holds Multiple Overlap Function 171 for k=u−1. Let us differentiate now a typical term of ν(u−1) t(f), of the form cβ2(u−1)νt(g),with g=fS− ℓ1,j1,...,ℓu−1,ju−1, where ℓ1, j1,...,ℓu−1, ju−1≤n′, with n′≡n+ 2(u−1). Thus we get, by means of Proposition 2.1, that ν′ t(g) = β2X 1≤l<l′≤n′ νt(gS− l,l′) −β2n′X l≤n′ νt(gS− l,n′+1) + β2n′(n′+ 1) 2νt(gS− n′+1,n′+2), which is easily seen to be of the form given by (11). Let us recall now an estimate for ν(i) t(f) which can be found in [8]: Proposition 3.2. If fis a function defined on Σn Nand β < 1, then for all t∈[0,1) we have ν(i) t(f)≤K(β, i, n) Ni 2 ν(f2)1 2. The following estimations for the variables Sℓ1,j1,...,ℓs,jswill be also be used several times along the article: Proposition 3.3. Given s≥1and a family of integers ℓ1, j1,...,ℓs, js, we have, for all β < 1: (a) ν(S− ℓ1,j1,...,ℓs,js) = O(s). (b) ν(Sℓ1,j1,...,ℓs,js) = O(s). (c) ν0(S− ℓ1,j1,...,ℓs,js) = O(s). (d) ν(u) t(S− ℓ1,j1,...,ℓs,js) = O(u+s)for all t∈[0,1]. Proof: Relations (a)–(c) are proved in [1]. The last relation follows easily from the previous ones, together with Proposition 3.2. 3.2. Negligible terms. We will try now to find a class of terms in (11) for which the coefficient c(n, k, α) vanishes. And a basic tool for this kind of identification can be found again in [8]: 178 S. de Carvalho Bezerra, S. Tindel and thanks to Proposition 3.7, item ii) it follows that ν′ 0S− ℓ2,j2,...,ℓ2k,j2k=β2ν(Sℓ1,j1,...,ℓ2k,j2k) + O(2k+ 1), which gives our claim (30). Step 3: We will prove that K3=O(2k+ 1). Notice that for each a1,...,auwe have 1 N2k−u−1νS− ℓa1,ja1,...,ℓau,jau=O(2(2k−u−1) + u), where we have just applied Proposition 3.3. Since 2(2k−u−1) + uis greater than 2k+ 1 whenever u≤2k−3, it follows that (32) K3=O(2k+ 1). Step 4: Study of K2. We claim that (33) K2=(1 NνS˜ l1,˜ k1,...,˜ l2k−2,˜ k2k−2+O(2k+ 1),if r∼w; O(2k+ 1),otherwise, where (˜ l1,˜ k1,...,˜ l2k−2,˜ k2k−2) = (˜r, ˜w) for a certain couple with ˜r, ˜w∈ Ω2(k−1) satisfying ˜r∼˜w. Indeed, using Proposition 3.7, for any family (a1,...,a2k−2) such that i2≤a1<···< a2k−2≤i2k, we have (34) 1 NνS− ℓa1,ja1,...,ℓa2k−2,ja2k−2=1 Nν0S− ℓa1,ja1,...,ℓa2k−2,ja2k−2 +O(2k+ 1). In the case r≁w, since r∈Ω2k, there exists an index i∈ {1,...,2k} such that for all u∈ {1,...,2k}\{i}, we have (ℓi, ji)6= (ℓu, ju) (remember that, by definition, the l’s and j’s are the elements of rand w). We denote this index iby i1. In this case the index i2such that the product Q2k−2 v=1 ǫℓavǫjav=ǫℓi1ǫji1ǫℓi2ǫji2satisfies {ǫℓi1, ǫji1} ∩ {ǫℓi2, ǫji2} 6= {ǫℓi1, ǫji1}. Thus, applying Proposition 3.5, the first term on the right side in (34) vanishes and we obtain 1 NνS− ℓa1,ja1,...,ℓa2k−2,ja2k−2=O(2k+ 1). Multiple Overlap Function 179 On the other hand, in the case r∼w, there exists a unique sequence (ˆa1,...,ˆa2k−2) which satisfies ˆa1<ˆa2<···<ˆa2k−2and 2k−2 Y i=1 ǫℓˆaiǫjˆai= 1. Notice that in our example (21), we have (ˆa1,ˆa2,ˆa3,ˆa4) = (2,3,4,6). Then it is easily seen, with the same kind of arguments as in the previous steps, that K2=1 NνS− ℓˆa1,jˆa1,...,ℓˆa2k−2,jˆa2k−2+O(2k+ 1). Thanks to Proposition 3.7, we thus get K2=1 NνSℓˆa1,jˆa1,...,ℓˆa2k−2,jˆa2k−2+O(2k+ 1), and we remark that (ℓˆa1, jˆa1,...,ℓˆa2k−2, jˆa2k−2) = (˜r, ˜w) with ˜r, ˜w∈ Ω2k−2and ˜r∼˜w, since r∼w(in our example (21), r=w= (4,7,3,5)). Our claim is now proved. Step 5: Conclusion. Plugging (30), (32) and (33) into (26), and invoking Proposition 3.7, item ii), we obtain, for any β < 1, νS− ℓ1,j1,...,ℓ2k,j2k=(1 (1−β2)NνS˜ l1,˜ k1,...,˜ l2k−2,˜ k2k−2+O(2k+1),if r∼w; O(2k+ 1),otherwise, and equation (20) follows now easily by induction on k. Indeed, the case k= 1 has been shown by Talagrand in [8], under the following form: for β < 1, we have ν(R2 1,2) = 1 N(1 −β2)+O(3). The induction is now a trivial fact. As a consequence of the previous properties, we can evaluate the following general term: Proposition 3.9. Let (ℓ1, j1,...,ℓk, jk)∈Ω2k. Then, for all β < 1, we have (35) 1 k!ν(k) 0S− ℓ1,j1,...,ℓk,jk=β2 N(1 −β)2k +O(2k+ 1). 180 S. de Carvalho Bezerra, S. Tindel Proof: Applying Proposition 3.6 with f=S− ℓ1,j1,...,ℓk,jkand w=(ℓ1, j1,..., ℓk, jk), we obtain that ν(k) 0S− ℓ1,j1,...,ℓk,jk=k!β2kν0S− ℓ1,j1,...,ℓk,jkS− ℓ1,j1,...,ℓk,jk +β2kX r≁w c(r)ν0S− ℓ1,j1,...,ℓk,jkS− r1,r2,...,r2k−1,r2k. Hence, according to Proposition 3.8, we can conclude that 1 k!ν(k) 0S− ℓ1,j1,...,ℓk,jk=β2k1 N(1 −β)2k +O(2k+ 1). Eventually, we will end the section by the evaluation of the first term in the expansion of ν(Sℓ1,j1,...,ℓk,jk): Lemma 3.10. Let r= (ℓ1, j1,...,ℓk, jk)∈Ω2k. Then, for all β < 1, the following relation holds true: ν(Sℓ1,j1,...,ℓk,jk) = 1 Nk1 1−β2k +O(2k+ 1). Proof: First remark that Sℓ1,j1,...,ℓk,jk=Qk i=1 Sℓi,ji, and thanks to the relation Sl,l′=S− l,l′+1 N, we obtain Sℓ1,j1,...,ℓk,jk= k X u=1 X 1≤i1<···<iu≤k 1 Nk−u u Y v=1 S− ℓiv,jiv+1 Nk = k X u=1 X 1≤i1<···<iu≤k 1 Nk−uS− ℓi1,ji1,...,ℓiu,jiu+1 Nk. (36) Hence (37) ν(Sℓ1,j1,...,ℓk,jk)= k X u=1 X 1≤i1<···<iu≤k 1 Nk−uνS− ℓi1,ji1,...,ℓiu,jiu+1 Nk. Notice that r∈Ω2kiff for any u≤kand any sequence (i1,...,iu) such that 1 ≤i1<···< iu≤k, we have (ℓi1, ji1,...,ℓiu, jiu)∈Ω2u. Whence, expanding the Taylor series, we get 1 Nk−uνS− ℓi1,ji1,...,ℓiu,jiu=1 Nk−u u X v=0 1 v!ν(v) 0S− ℓi1,ji1,...,ℓiu,jiu +1 Nk−u 1 (u+ 1)!ν(u+1) ξS− ℓi1,ji1,...,ℓiu,jiu, (38) Multiple Overlap Function 181 for a certain ξ∈[0,1]. Now, Invoking Proposition 3.5, all the derivative terms of order smaller than uvanish, and by Proposition 3.3, item (d), the error term can be estimated as follows: 1 Nk−u 1 (u+ 1)!ν(u+1) ξS− ℓi1,ji1,...,ℓiu,jiu=O(2(k−u)+2u+1)=O(2k+1). Hence, we get the following expression: (39) 1 Nk−uνS− ℓi1,ji1,...,ℓiu,jiu=1 u!Nk−uν(u) 0S− ℓi1,ji1,...,ℓiu,jiu +O(2k+ 1). On the other hand, the derivative term of order ucan be evaluated by means of Proposition 3.9: since (ℓi1, ji1,...,ℓiu, jiu)∈Ω2u, by plugging (35) into (39), we get 1 Nk−uνS− ℓi1,ji1,...,ℓiu,jiu=1 Nk−u 1 u!u!β2 N(1−β2)u +O(2u+1)  =1 Nkβ2 1−β2u +O(2k+ 1). (40) Moreover, Card {(i1,...,iu)|1≤i1<···< iu≤k}=k u, and thus we can recast equation (37) into ν(Sℓ1,j1,...,ℓk,jk) = k X u=1 1 Nkk u β2 (1 −β2)u +1 Nk+O(2k+ 1) =1 Nk1 + β2 1−β2k +O(2k+ 1). This completes the proof. 4. R-systems and graphs In this section, we will make an essential step towards the evaluation of multiple overlaps of the form R1,...,s defined at (2). Indeed, we will prove an important preliminary result involving the functional U− kS− ℓ1,j1,...,ℓm,jm, where U− k=ǫ1ǫ2. . . ǫ2kR− 1,2,...,2k. We will also evaluate ν(Sℓ1,j1,...,ℓm,jm) for some special cases of indexes (ℓ1, j1,...,ℓm, jm). More specifically, this section is devoted to the proof of the following result: 182 S. de Carvalho Bezerra, S. Tindel Proposition 4.1. Let kbe a positive integer, β < 1and recall that Ckhas been defined at (8). Then, for any m≥kand (ℓ1, j1,...,ℓm, jm)∈ Ck, we have: i) ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(Sℓ1,j1,...,ℓm,jm) + O(2k+ 1). ii) ν(Sℓ1,j1,...,ℓm,jm) = O(2k+ 1) if m≥k+ 1. Notice that the proof of this result will require two kind of tools: first a graph representation that will help us to identify the main contribution in our expansions, and then the introduction of some families of functions whose role is to avoid a cumbersome recursive procedure. 4.1. Graph tools: Proof of Proposition 4.1, item (ii). We will include in fact Proposition 4.1, item (ii) into a more general statement: Proposition 4.2. Consider a positive integer kand β < 1. Assume that the sequence (ℓ1, j1,...,ℓm, jm)belongs to Ckwith m≥k+1. Then the following estimations hold true: i) ν(Sℓ1,j1,...,ℓm,jm) = O(2k+ 1). ii) For all u≥1and 1≤i1<···< iu≤m, we have 1 Nm−uν(S− ℓi1,ji1,...,ℓiu,jiu) = O(2k+ 1). iii) For all u≥1and 1≤i1<···< iu≤m, we have 1 Nm−uν(Sℓi1,ji1,...,ℓiu,jiu) = O(2k+ 1). Proof: Let (ℓ1, j1,...,ℓm, jm)∈ Ck. Using the same kind of calculation as in relation (36), we obtain (41) ν(Sℓ1,j1,...,ℓm,jm) = m X u=1 X 1≤i1<···<iu≤m 1 Nm−uν(S− ℓi1,ji1,...,ℓiu,jiu) +1 Nm. Multiple Overlap Function 183 For each u≤mand 1 ≤i1<··· < iu≤m, let us expand the term ν(S− ℓi1,ji1,...,ℓiu,jiu) up to an order v∈N. We get (42) ν(S− ℓi1,ji1,...,ℓiu,jiu) = v X r=1 1 r!ν(r) 0(S− ℓi1,ji1,...,ℓiu,jiu) +1 (v+ 1)!ν(v+1) ζ(S− ℓi1,ji1,...,ℓiu,jiu) for a certain ζ∈R. Let us admit for the moment the following proposition, whose proof will require the introduction of the graph tools mentioned above: Proposition 4.3. Given a positive integer kand (ℓ1, j1,...,ℓm, jm)∈ Ck, the following holds true for any u≥1and 1≤i1<···< iu≤m: i) There exists a positive integer ˆa= ˆa(ℓi1, ji1,...,ℓiu, jiu) such that Qu p=1 ǫℓipǫjip=ǫc1...ǫc2ˆa, where all the indexes c′s are different. ii) u−ˆais bounded by m−k. Let us apply now this last proposition: set v= ˆain equation (42). Then, invoking Proposition 3.5, item (i), it is easily seen that ν(S− ℓi1,ji1,...,ℓiu,jiu) = 1 ˆa!ν(ˆa) 0(S− ℓi1,ji1,...,ℓiu,jiu) +1 (ˆa+ 1)!ν(ˆa+1) ζ(S− ℓi1,ji1,...,ℓiu,jiu). Furthermore, according to Proposition 3.3, item (d), ν(S− ℓi1,ji1,...,ℓiu,jiu) is of order O(ˆa+u). We thus get the following estimation: 1 Nm−uν(S− ℓi1,ji1,...,ℓiu,jiu) = O(2m−(u−ˆa)) . Eventually, thanks to item ii) in Proposition 4.3, and since we have assumed m≥k+ 1, we get 2m−(u−ˆa)≥2m−(m−k) = m+k≥2k+ 1, and hence (43) 1 Nm−uν(S− ℓi1,ji1,...,ℓiu,jiu) = O(2k+ 1), which proves item ii) of our Proposition 4.2. Moreover, putting together (43) and (41), item i) of Proposition 4.2 is also easily shown. 184 S. de Carvalho Bezerra, S. Tindel In order to obtain iii) in Proposition 4.2 we perform again the same expansion as in (36), and we get 1 Nm−uν(Sℓi1,ji1,...,ℓiu,jiu) =1 Nm−u u X q=0 X i1≤a1<···<aq≤iu 1 Nu−qν(S− ℓa1,ja1,...,ℓaq,jaq) =O(2k+ 1), where we used ii) for each qand (a1,...,aq) and m≥k+ 1. The remainder of this section will now be devoted to prove Proposition 4.3, starting with item (i), for which we will use the graph definitions of Section 2.2: Proposition 4.4. Let k,ube two positive integers such that u≤k. Let also (ℓ1, j1,...,ℓm, jm)∈ Ckwith m≥k+ 1 and 1≤i1<···< iu≤m. Consider g=G((ℓ1, j1,...,ℓm, jm)) and h,G((ℓi1, ji1,...,ℓiu, jiu). Then i) hbelongs to Su(g). ii) There exists an integer tsuch that Qu i=1 ǫℓiiǫjii=ǫc1...ǫctfor any value of ǫ, where all the indexes (c1,...,ct)are different. Furthermore, Od(h) = t 2. Remark 4.5.Item i) justifies our interest for the class Su(g), while item ii) implies item i) of Proposition 4.3, with ˆa(ℓi1,...,jiu) = Od(h). Proof of Proposition 4.4: i) This is a straightforward consequence of the definitions given at Section 2.2. ii) The quantities ǫciare just the elements which appear an odd number of times in Qu p=1 ǫℓipǫjip, and as a consequence, Od(h) = X i∈I;Nh(i) is odd 1 2=t 2. Given a graph gsuch that N(g) = m, another quantity of interest for us will be an upper bound on maxh∈Su(g)u−Od(h). Define then, for each u∈ {1,...,m}, the function Mg u:Su(g)−→ N h7−→ u−Od(h). Multiple Overlap Function 185 In order to simplify the notations we will use during the proof of the next proposition, we define an operation with graphs that we call the juxtaposition: given two graphs g1= (I1, E1,Υ1) and g2= (I2, E2,Υ2), we denote by g=g1+g2the graph defined by g= (I, E, Υ), such that I=I1∪I2,E=E1∪E2and Υ = Υ1+ Υ2(we consider that Υ1(e) and Υ2(e) are equal to zero when they are not defined in Υ1and Υ2 separately). Recall that the class of graphs Gkis defined by relation (8). Then the next lemma asserts an inner characteristic of monotonicity for Gk. Lemma 4.6. Given k∈Nand a graph g= (I, E, Υ) ∈ Gk, the following holds true: i) maxh∈Su(g)Mg uis increasing with u. ii) maxu≤N(g)maxh∈Su(g)Mg u≤N(g)−k. Remark 4.7.Item (ii) of Proposition 4.3 is an easy consequence of item (ii) in Lemma 4.6. Proof of Lemma 4.6: i) Let h= (I1, E1,Υ1)∈Su(g) such that u < N(g) and max h∈Su(g)Mg u=u−Od(h). Let e= (p, q)∈E\E1(eexists because u < m). One defines h1= ({p, q},{e},Υ2) such that Υ2(e) = 1, and let ˜ hbe the graph h+h1. Then ˜ h∈Su+1 because N(˜ h) = N(h) + N(h1) = u+ 1, I1∪{p, q} ⊆ I, E1∪{e} ⊆ Eand Υ1(e) + Υ2(e)≤Υ(e).We will show that Mg u+1(˜ h)≥ Mg u(h) which, in turn, implies statement i). There are three possible cases for pand q: •p, q /∈I1. In this case N˜ h(p) = N˜ h(q) = 1, and then Od(˜ h) = Od(h) + 1 2+1 2, which gives Mg u+1(˜ h) = u+ 1 −(Od(h) + 1) = Mg u(h). •p∈I1,q /∈I1(or q∈I1,p /∈I1). One has N˜ h(q) = 1, and if N˜ h(p) is odd, then Od(˜ h) = Od(h) + 1 2+1 2and thus one obtains the same result than in the previous item. If N˜ h(p) is even, which gives Od(˜ h) = Od(h) + 1 2−1 2then Mg u+1(˜ h) = u+ 1 −Od(h)> Mg u(h). 186 S. de Carvalho Bezerra, S. Tindel •p, q ∈I1. If both N˜ h(p) and N˜ h(q) are odd, then Od(˜ h) = Od(h) + 1; if N˜ h(p) is even and N˜ h(q) is odd, then Od(˜ h) = Od(h), and these two cases have already been studied. In the case where both N˜ h(p) and N˜ h(q) are even, then Od(˜ h) = Od(h)−1, and Mg u+1(˜ h) = u+ 1 −(Od(h)−1) > Mg u(h). The proof of point i) is now clear. The statement ii) follows from item i), because maxumaxh∈Su(g)Mg u= Mg m=m−k, where in the last step, we have used the fact that gis the only subgraph of gwith medges such that g∈ Gk. Let us recall that, at that point, we have proved Proposition 4.3, and thus Proposition 4.1, item (ii). 4.2. R-systems: Proof of Proposition 4.1, item (i). The aim of this subsection is to finish the proof of Proposition 4.1, item (i), which amounts to prove (44) ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(Sℓ1,j1,...,ℓm,jm) + O(2k+ 1). The general strategy we will use here is a backward induction principle on m. However, in order to simplify the cumbersome procedure one is faced with at first sight, we will introduce a family of function that we call R-systems. Let us delve now into the details of the proof: Step 1: First step of the induction. In the case m≥2k+ 2, thanks to Schwarz inequality and Proposition 3.2, we easily get ν0(U− kS− ℓ1,j1,...,ℓm,jm)≤K(β, m)νS− ℓ1,j1,...,ℓm,jm21 2 =O(m) for a positive constant K(β, m), where in the last step, we have used Proposition 3.3. The statement follows because m≥2k+ 2 >2k+ 1 and Proposition 4.2 yields ν(Sℓ1,j1,...,ℓm,jm) = O(2k+ 1). Whence the difference ν0(U− kS− ℓ1,j1,...,ℓm,jm)−ν(Sℓ1,j1,...,ℓm,jm) is also O(2k+1), which finishes the proof. Step 2: We will start our induction procedure. Let us pick a m < 2k+2, and we assume the result holds true for all r > m. First, we will show that ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(U− kS− ℓ1,j1,...,ℓm,jm) + O(2k+ 1). Multiple Overlap Function 187 Indeed, performing an inverse Taylor expansion, we get, for a certain ζ∈ [0,1], (45) ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(U− kS− ℓ1,j1,...,ℓm,jm) − k X r=1 1 r!ν(r) 0(U− kS− ℓ1,j1,...,ℓm,jm)−1 (k+ 1)!ν(k+1) ζ(U− kS− ℓ1,j1,...,ℓm,jm). Let us bound now the last term of this inequality: applying Schwarz’ inequality and Propositions 3.2 and 3.3, we have ν(k+1) ζ(U− kS− ℓ1,j1,...,ℓm,jm)≤1 Nk+1 2 νU− kS− ℓ1,j1,...,ℓm,jm21 2 ≤1 Nk+1 2 νS− ℓ1,j1,...,ℓm,jm21 2 =O(2k+ 1), according to the fact that m≥k. Consequently, equation (45) becomes ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(U− kS− ℓ1,j1,...,ℓm,jm) − k X r=1 1 r!ν(r) 0(U− kS− ℓ1,j1,...,ℓm,jm) + O(2k+ 1). However, Proposition 3.1 asserts that each term ν(r) 0(U− kS− ℓ1,j1,...,ℓm,jm) can be evaluated as a finite sum of terms of the form c(β, r)ν0(U− kS− ℓ1,j1,...,ℓm,jmS− ℓ1,j1,...,ℓr,jr), which can be rewritten as c(β, r)ν0(U− kS− ℓ1,j1,...,ℓm,jmS− ℓ1,j1,...,ℓr,jr) =c(β, r)ν0(U− kS− ℓ1,j1,...,ℓm+r,jm+r). (46) By Proposition 3.4, if (ℓ1, j1, . . . , ℓm+r, jm+r)/∈ Ck, the expression (46) vanishes. Otherwise, by backward induction hypothesis, we get c(β, r)ν0(U− kS− ℓ1,j1,...,ℓm+r,jm+r) = c(β, r)ν(Sℓ1,j1,...,ℓm+r,jm+r)+O(2k+1). Thus, since r≥1, it is readily checked from Proposition 4.2 that (47) ν0(U− kS− ℓ1,j1,...,ℓm,jm) = ν(U− kS− ℓ1,j1,...,ℓm,jm) + O(2k+ 1), which was the claim to be proved. 194 S. de Carvalho Bezerra, S. Tindel and Uk=ǫ1ǫ2...ǫ2kR1,2,...,2k. When we choose r= 2k, equation (58) becomes νU− k=ν0U− k+ 2k X u=1 1 u!ν(u) 0U− k+1 (2k+ 1)!ν(2k+1) ζU− k. Furthermore, applying Propositions 3.5 and 3.2, we obtain νU− k= 2k X u=k 1 u!ν(u) 0U− k+1 (2k+ 1)!ν(2k+1) ζU− k = 2k X u=k 1 u!ν(u) 0U− k+O(2k+ 1). However, Proposition 3.1 asserts that ν(u) 0(U− k) can be written as (59) ν(u) 0(U− k)= X α=(l1,j1,...,lu,ju)∈D2k,u c(2k, u, α)β2uν0U− kS− ℓ1,j1,...,ℓu,ju, and we are now in a position to identify the negligible terms in the above sum. Indeed, setting αfor a tuple (l1, j1,...,lu, ju), we have: (1) According to Proposition 3.4, if Qu i=1 ǫℓiǫji6=Qk i=1 ǫ2i−1ǫ2i, the term ν0(U− kS− ℓ1,j1,...,ℓu,ju) vanishes. This means in particular that, in relation (59), c(2k, u, α) = 0 unless α∈ Ck, and ν(u) 0(U− k) = X α=(l1,j1,...,lu,ju)∈D2k,u∩Ck c(2k, u, α)β2uν0U− kS− ℓ1,j1,...,ℓu,ju. (2) If u≥m+ 1, Proposition 4.1 yields ν0U− kS− ℓ1,j1,...,ℓu,ju=ν(Sℓ1,j1,...,ℓu,ju) + O(2k+ 1) = O(2k+ 1). Hence, the terms ν(u) 0(U− k) can be neglected for u > k, and we obtain ν(U− k) = 1 k!X α∈D2k,k∩Ck c(2k, k, α)β2kν0U− kS− ℓ1,j1,...,ℓk,jk+O(2k+ 1) =1 k!X α∈D2k,k∩Ck c(2k, k, α)β2kν(Sℓ1,j1,...,ℓk,jk) + O(2k+ 1), where we have applied again Proposition 4.1, item (i) for the last equality. Multiple Overlap Function 195 (3) Let us go back now to the Definitions (8) and (10) of Ckand D2k,k, to see that Ck∩D2k,k ={α= (l1, j1, . . . , lk, jk); αis a permutation of (1,...,2k), li< jifor all i≤k}. In particular, it is easily seen that, if α∈ Ck∩D2k,k,αis also an element of Ω2k. Thus, owing to Lemma 3.10, we obtain (60) ν(U− k) = 1 k!Nkβ2 1−β2k X α∈D2k,k∩Ck c(2k, k, α) + O(2k+ 1). (4) Eventually, we will finish the proof by calculating the sum X α∈D2k,k∩Ck c(2k, k, α). A first step in that direction is to notice that Card (D2k,k ∩Ck) = 2k 22k−2 2...2 2=(2k)! 2k. Furthermore, it is easily seen that D2k,k ∩Ckcontains exactly (2k)! 2kk!classes for the relation ∼defined just before Proposition 3.6. Thus, Proposition 3.6 yields X α∈D2k,k∩Ck c(2k, k, α) = k!(2k)! 2kk!=(2k)! 2k, and plugging this relation into (60), we get ν(U− k) = (2k)! 2kk!β2 (1 −β2)Nk +O(2k+ 1). Putting this relation together with (57), we obtain νR2 1,2,...,s=1 N+(2k)! 2kk!β2 (1 −β2)Nk +O(2k+ 1), which is the announced result (5). 196 S. de Carvalho Bezerra, S. Tindel 6. CLT generalization for the overlap function We will now prove Theorem 1.1. This will be done along the same lines as in [8], except for the use of our asymptotic expansions (4) and (5). We include the proof here for sake of readability: first, we need to establish a result for the moments of R1,...,s which is a natural consequence of Theorem 1.2. Proposition 6.1. If k≥0,β < 1and s≥3then ν(Rk 1,...,s) = a(k) Nk 2 +O(k+ 1), where a(k)is the kth-moment of a standard Gaussian random variable. Proof: We use symmetry between sites to get ν(Rk 1,...,s) = ν(ǫ1...ǫsRk−1 1,...,s) =ν(ǫ1...ǫs(R− 1,...,s)k−1) + k−1 Nν((R− 1,...,s)k−2) +X l≥2 1 Nlν((ǫ1...ǫs)(l+1)(R− 1,...,s)k−l−1), (61) by writing R1,...,s =R− 1,...,s +N−1ǫ1...ǫsand expanding the power. However, using Theorem 1.2 and Schwarz inequality, we obtain 1 Nlν((ǫ1...ǫs)(l+1)(R1,...,s)k−l−1) = O(k+ 1), for l≥2. Writing now R− 1,...,s =R1,...,s −ǫ1. . . ǫs/N and expanding the quantity (a+b)k−2, we see in a similar manner that ν((R− 1,...,s)k−2) = ν(Rk−2 1,...,s) + O(k−1) and thus 1 Nν((R− 1,...,s)k−2) = 1 Nν(Rk−2 1,...,s) + O(k+ 1), so that (61) gives (62) ν(Rk 1,...,s)=ν(ǫ1...ǫs(R1,...,s)k−1)+k−1 Nν((R1,...,s)k−2)+O(k+1). Now, we perform a Taylor expansion for the first term on the right hand side of (62), which yields (63) ν(ǫ1. . . ǫs(R1,...,s)k−1) = ν0(ǫ1...ǫs(R1,...,s)k−1) +ν′ 0(ǫ1...ǫs(R1,...,s)k−1) + 1 2ν2 ζ(ǫ1...ǫs(R1,...,s)k−1). Multiple Overlap Function 197 Since s≥3, the first two terms on the right hand of (63) vanish, and from Theorem 1.2 the error term is of order O(k+1). So, we can conclude that ν(Rk 1,...,s) = k−1 Nν((R1,...,s)k−2) + O(k+ 1), and our claim follows by induction on k. Now, let us prove Theorem 1.1: Proof of Theorem 1.1: Without loss of generality, we can assume k(1,...,s)≥1. For each integer 1 ≤v≤kwe consider integers ℓ1(v),...,ℓs(v) such that Y ℓ1<···<ℓs Rk(ℓ1,...,ℓs) ℓ1,...,ℓs=Y v≤k Rℓ1(v),...,ℓs(v), and we set R(v) = Rℓ1(v),...,ℓs(v), R−(v) = R− ℓ1(v),...,ℓs(v), ǫ(v) = ǫℓ1(v)...ǫℓs(v), so that R(v) = R(v)−+ǫ(v)/N. Now we use symmetry between sites to write (64) ν Y ℓ1<···<ℓs Rk(ℓ1,...,ℓs) ℓ1,...,ℓs!=ν Y v≤k R(v) =ν ǫ(1) Y 2≤v≤k R(v) , and we expand the product Y 2≤v≤k R(v) = Y 2≤v≤kR−(v) + ǫ(v) N. In each of the k−1 factors, we can choose either the term R−(v) (henceforth called the big term) or the term ǫ(v)/N (henceforth called the small term). These k−1 choices result into 2k−1terms. When we choose the small term in at least two factors the resulting contribution is O(k+ 1), which is easily seen by repeating the argument of Proposition 6.1 and invoking H¨older’s inequality. If we choose the small term in exactly lfactors, the resulting contribution is O(2l)O(k−1−l) = O(k+ 1) for l≥2. 198 S. de Carvalho Bezerra, S. Tindel Thus, we only need to consider the contributions where we have chosen the small term in at most one factor, and this gives ν(ǫ(1) Y 2≤v≤k R(v)) = ν(ǫ(1) Y 2≤v≤k R−(v)) +1 NX 2≤v≤k ν(ǫ(1)ǫ(v)Y u R−(u)) + O(k+ 1), where the product is for 2 ≤u≤k,u6=v. Since there are k−2 terms in the product QuR−(u) and performing another expansion, it follows that (65) ν(ǫ(1)ǫ(v)Y u R−(u)) = ν0(ǫ(1)ǫ(v)Y u R−(u)) + O(k−1). However, ν0(ǫ(1)ǫ(v)QuR−(u)) is zero unless ǫ(1)ǫ(v)=1 i.e. {1,...,s}= {ℓ1(v),...,ℓs(v)}. So, the expression (65) is of order O(k−1) unless v≤k(1,...,s), and thus ν ǫ(1) Y 2≤v≤k R(v) =ν ǫ(1) Y 2≤v≤k R−(v)  +k(1,...,s)−1 Nν Y u R−(u)!+O(k+ 1). We then proceed as in Proposition 6.1 to get, using similar expansions as in (63), ν(ǫ(1) Y 2≤v≤k R−(v)) = O(k+ 1), because s > 2. Thus, putting together (64) and (65), we get ν Y ℓ1<···<ℓs Rk(ℓ1,...,ℓs) ℓ1,...,ℓs!=k(1,...,s)−1 Nν Y u R−(u)!+O(k+ 1). We can now establish our claim (3) by induction on k. References [1] X. Bardina, D. M´ arquez-Carreras, C. Rovira and S. Tindel, Higher order expansions for the overlap of the SK model, in: “Seminar on Stochastic Analysis, Random Fields and Applications IV”, Progr. Probab. 58, Birkh¨auser, Basel, 2004, pp. 21–43. 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Sherrington, Complexity due to disorder and frustration, in: “1989 lectures in complex systems” (Santa Fe, NM, 1989), Santa Fe Inst. Stud. Sci. Complexity Lectures II, Addison-Wesley, Redwood City, CA, 1990, pp. 415–453. [8] M. Talagrand,“Spin glasses: a challenge for mathematicians”, Cavity and mean field models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Folge 46, Springer-Verlag, Berlin, 2003. [9] M. Talagrand, The Parisi formula, Ann. of Math. (2) 163(1) (2006), 221–263. Institut ´ Elie Cartan Universit´e de Nancy 1 BP 239 54506-Vandoeuvre-l`es-Nancy France E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 4 de juliol de 2006, darrera versi´o rebuda el 13 de desembre de 2006.