In this paper we give another proof of the fact that a random overlap array,
which satisfies the Ghirlanda-Guerra identities and whose elements take values
in a finite set, is ultrametric with probability one. The new proof bypasses
random change of density invariance principles for directing measures of such
arrays and, in addition to the Dobvysh-Sudakov representation, is based only on
elementary algebraic consequences of the Ghirlanda-Guerra identities

Panchenko, Dmitry

English

arXiv

Dmitry Panchenko

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 813–816Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comProbability Theory/Mathematical PhysicsGhirlanda–Guerra identities and ultrametricity: An elementary proof inthe discrete caseUltramétricité et identités de Ghirlanda–Guerra : une preuve élémentaire du cas discretDmitry Panchenko 1Department of Mathematics, Texas A&M University, 77843 College Station, TX, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 20 June 2011Accepted 24 June 2011Available online 20 July 2011Presented by Michel TalagrandIn this Note we give another proof of the fact that a random overlap array, whichsatisfies the Ghirlanda–Guerra identities and whose elements take values in a finiteset, is ultrametric with probability one. The new proof bypasses random change ofdensity invariance principles for directing measures of such arrays and, in addition to theDovbysh–Sudakov representation, is based only on elementary algebraic consequences ofthe Ghirlanda–Guerra identities.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette Note nous donnons une nouvelle preuve du fait qu’une matrice aléatoire infinie,qui satisfait l’identité Ghirlanda–Guerra et dont les coefficiants prennent leurs valeurs dansun ensemble fini, est ultramétrique avec probabilité un. La preuve utilise uniquement desconséquences algébriques élémentaires des identités Ghirlanda–Guerra et la représentationde Dovbysh–Sudakov.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. Introduction and main resultIn this Note we will give a simplified proof of the main result in [5]. Let us consider an infinite random array R =(Rl,l′ )l,l′1 which is symmetric, non-negative definite and weakly exchangeable, i.e. for any n  1 and for any permutation ρof {1, . . . ,n} the matrix (Rρ(l),ρ(l′))l,l′n has the same distribution as (Rl,l′ )l,l′n. We assume that the diagonal elements areidentically 1, Rl,l = 1 and that the non-diagonal elements take finitely many values,P(R1,2 = ql) = pl (1)for some −1  q1 < q2 < · · · < qk  1 and pl > 0, p1 + · · · + pk = 1. The array R is said to satisfy the Ghirlanda–Guerraidentities [4] if for any n  2 and any bounded measurable functions f = f ((Rl,l′ )l,l′n) and ψ : R → R,E f ψ(R1,n+1) = 1nE f Eψ(R1,2) + 1nn∑l=2E f ψ(R1,l). (2)E-mail address: panchenko.math@gmail.com.1 Partially supported by NSF grant.1631-073X/$ – see front matter © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2011.06.021814 D. Panchenko / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 813–816By the positivity principle of Talagrand [8,10], the Ghirlanda–Guerra identities imply that R1,2  0 with probability one and,therefore, we can assume that q1  0. The following result was proved in [5]:Theorem 1.1. Under assumptions (1) and (2), the array R is ultrametric,P(R2,3  min(R1,2, R1,3)) = 1. (3)Another way to express the event in (3) is to say thatR1,2  ql, R1,3  ql ⇒ R2,3  ql for all 1  l  k. (4)Infinite arrays that satisfy the Ghirlanda–Guerra identities arise as the limits of the overlap arrays in the Sherrington–Kirkpatrick spin glass models (see e.g. [10,7]). The assumption (1) is purely technical (and unfortunately is not satisfiedin the most important situations). The first ultrametricity result was proved in [2] under different conditions which alsoincluded (1), but instead of (2) the authors worked with the Aizenman–Contucci stochastic stability [1]. The original proofof Theorem 1.1 in [5] utilized a key idea from [2], namely, the existence of directing measures guaranteed by the Dovbysh–Sudakov representation result in [3], and we will still rely on this representation here. However, we will completely avoidproving any invariance principles under random changes of density for the directing measure, which played crucial rolesboth in [2] and [5] and our new induction will be quite elementary in nature. M. Talagrand gave a proof of Theorem 1.1in [9] that did not use the Dovbysh–Sudakov representation but still used the invariance principle from [5]. The Dovbysh–Sudakov representation [3] (for detailed proof see [6]) states that given a symmetric, non-negative definite and weaklyexchangeable array R, there exists a random measure μ on H × [0,∞), where H is a separable Hilbert space, such that Ris equal in distribution to the array(σ l · σ l′ + alδl,l′)l,l′1 (5)where (σ l,al) is an i.i.d. sequence from μ and σ · σ ′ denotes the scalar product on H . Let us denote by G the marginal ofμ on H . The following simple consequence of the Ghirlanda–Guerra identities (2) was proved in Theorem 2 in [5]:Proposition 1.1. Under (1) and (2), the random measure G is (countably) discrete and is concentrated on the sphere of radius√qkwith probability one.In particular, this implies that al = 1 −qk in (5) and without loss of generality we can redefine the array by Rl,l′ = σ l ·σ l′for an i.i.d. sequence (σ l) from G. Since Rl,l′ = qk if and only if σ l = σ l′ , we haveP(R1,2 = qk, R1,3 = qk, R2,3 < qk) = 0,which proves “ultrametricity at the level k” in the sense of (4). As in [2] and [5], we would like to find a way to make aninduction step and prove “ultrametricity at the level k −1”. The main new idea of the paper will be to consider the distribu-tion of the array (Rl,l′ ) conditionally on the event that all replicas (σ l) are different and prove that this new distribution iswell-defined and satisfies all the conditions of the Dovbysh–Sudakov representation. Since on the above event the elementsof the new array cannot take value qk , the induction step will follow.2. ProofBy Proposition 1.1, G = ∑l1 wlδξl for some random weights (wl) and random sequence (ξl) in H such that ξl · ξl = qk.Let us denote by 〈·〉 the average with respect to G⊗∞ and by E the expectation with respect to the randomness of G. Withthese notations, the Ghirlanda–Guerra identities (2) can be rewritten asE〈fnψ(R1,n+1)〉 = 1nE〈 fn〉E〈ψ(R1,2)〉 + 1nn∑l=2E〈fnψ(R1,l)〉. (6)For each n  2, let us consider the eventAn ={Rl,l′ = qk, ∀1  l < l′  n}(7)and let Pn be the distribution of the n × n matrix Rn = (σ l · σ l′ )l,l′n conditionally on An,Pn(B) = E〈I(Rn ∈ B)I An 〉E〈I An 〉. (8)It is obvious that Pn is concentrated on the symmetric non-negative definite matrices with off-diagonal elements now takingvalues {q1, . . . ,qk−1} and Pn is invariant under the permutation of replica indices since the set An is. We will now showthat Pn+1 restricted to the first n replica coordinates coincides with Pn and, thus, the sequence (Pn) defines a law of theinfinite overlap array.D. Panchenko / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 813–816 815Lemma 2.1. For any measurable function f of the overlaps on n replicas,E〈f(Rn)I An+1〉 = (1 − pk)E〈f(Rn)I An〉. (9)Proof. Notice that An = {σ 1, . . . , σ n are all different} by Proposition 1.1 and, therefore,I An+1 = I An −∑lnI An∩{Rl,n+1=qk}. (10)This implies thatE〈f(Rn)I An+1〉 = E〈 f (Rn)I An〉 −∑lnE〈f(Rn)I An I(Rl,n+1 = qk)〉.Using the Ghirlanda–Guerra identities (6), for each l  n,E〈f(Rn)I An I(Rl,n+1 = qk)〉 = pknE〈f(Rn)I An〉 + 1nn∑l′ =lE〈f(Rn)I An I(Rl,l′ = qk)〉= pknE〈f(Rn)I An〉since An ⊆ {Rl,l′ = qk} and, thus, I An I(Rl,l′ = qk) = 0. Adding up over l  n finishes the proof. First, using (9) inductively for f ≡ 1 we get E〈I An 〉 = (1 − pk)n−1 and then dividing (9) by (1 − pk)n ,E〈 f (Rn)I An+1〉E〈I An+1〉= E〈 f (Rn)I An 〉E〈I An 〉. (11)This means that the family (Pn) is consistent and by Kolmogorov’s theorem we can define the distribution of the infinitearray with the corresponding marginals given by Pn. Let us consider an array Q = (Q l,l′ )l,l′1 with this distribution.Proof of Theorem 1.1. By construction, Q is a symmetric, non-negative definite and weakly exchangeable array with diago-nal elements equal to qk and off-diagonal elements taking values {q1, . . . ,qk−1} with probabilities P(Q 1,2 = ql) = pl/(1− pk).Using the Dovbysh–Sudakov representation for the array Q implies that there exists a random measure G ′ on H such thatQ can be generated asQ l,l′ = σ l · σ l′ + δl,l′(qk − σ l · σ l)for an i.i.d. sequence (σ l) from G ′ . Since σ l · σ l′ ∈ {q1, . . . ,qk−1}, it is easy to see that the support of G ′ must be inside thesphere of radius√qk−1 for, otherwise, with positive probability we could sample two points σ 1, σ 2 arbitrarily close to apoint σ such that ‖σ‖ > √qk−1 which would contradict that σ 1 · σ 2  qk−1 (see [2] or [5] for details). Because of this, thetruncated array (Q l,l′ ∧ qk−1)l,l′1 can be computed asQ l,l′ ∧ qk−1 = σ l · σ l′ + δl,l′(qk−1 − σ l · σ l)(12)and it is non-negative definite as the sum of two non-negative definite arrays. If we recall the definition (8), the matrix(Q l,l′ )l,l′n is obtained by sampling n configurations from the measure G = ∑l1 wlδξl conditionally on the event thatthese configurations are different. Since with positive probability we can sample ξ1, . . . , ξn, we must have that the matrix(ξl · ξl′ ∧ qk−1)l,l′n is non-negative definite and, therefore, (ξl · ξl′ ∧ qk−1)l,l′1 is non-negative definite with probability one.This of course means that (Rl,l′ ∧ qk−1)l,l′1 is also non-negative definite. Since the function x ∧ qk−1 can be approximatedby polynomials, the truncated overlap array also satisfies the Ghirlanda–Guerra identities and its elements now take valuesin {q1, . . . ,qk−1}. One can proceed by induction on k. Even though we did not need it in the proof, one can show that the measure G ′ is actually concentrated on the sphereof radius√qk−1 by using Proposition 1.1 and the following observation:Lemma 2.2. The distribution of Q satisfies the Ghirlanda–Guerra identities,E f(Q n)ψ(Q 1,n+1) = 1nE f(Q n)Eψ(Q 1,2) + 1nn∑l=2E f(Q n)ψ(Q 1,l). (13)816 D. Panchenko / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 813–816Proof. The proof is a straightforward computation using (10) and the Ghirlanda–Guerra identities. The idea of the proof of Theorem 1.1 suggests the following criterion of ultrametricity in the general case without theassumption (1). Given q ∈ [0,1] such that P(R1,2 < q) > 0, consider the eventsAn,q ={Rl,l′ < q, ∀1  l < l′  n}(14)and let Pn,q be the distribution of Rn = (Rl,l′ )l,l′n conditionally on An,q.Theorem 2.3. Under (2), the array R is ultrametric if and only if for any q such that P(R1,2 < q) > 0 and any set B of 3 × 3 matricessuch that P3,q(R3 ∈ B) > 0 we have lim supn→∞ Pn,q(R3 ∈ B) > 0.One can check that, in one direction, ultrametricity yields a relationship of the type (10) which implies the consistency ofthe sequence (Pn,q) as in Lemma 2.1 and Pn,q(R3 ∈ B) = P3,q(R3 ∈ B). In the other direction, for any B with P3,q(R3 ∈ B) > 0we can choose the limit Pq over a subsequence of Pn,q such that Pq(R3 ∈ B) > 0. If ultrametricity fails, one can makea choice of a subset of non-ultrametric configurations B and q ∈ [0,1] that will lead to contradiction with the Dovbysh–Sudakov representation for Pq .References[1] M. Aizenman, P. Contucci, On the stability of the quenched state in mean-field spin-glass models, J. Stat. Phys. 92 (5–6) (1998) 765–783.[2] L.-P. Arguin, M. Aizenman, On the structure of quasi-stationary competing particles systems, Ann. Probab. 37 (3) (2009) 1080–1113.[3] L.N. Dovbysh, V.N. Sudakov, Gram–de Finetti matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 119 (1982) 77–86.[4] S. Ghirlanda, F. Guerra, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity, J. Phys.A 31 (46) (1998) 9149–9155.[5] D. Panchenko, A connection between Ghirlanda–Guerra identities and ultrametricity, Ann. Probab. 38 (1) (2010) 327–347.[6] D. Panchenko, On the Dovbysh–Sudakov representation result, Electron. Commun. Probab. 15 (2010) 330–338.[7] D. Panchenko, The Ghirlanda–Guerra identities for mixed p-spin model, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 189–192.[8] M. Talagrand, Spin Glasses: A Challenge for Mathematicians, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveysin Mathematics, vol. 43, Springer-Verlag, 2003.[9] M. Talagrand, Construction of pure states in mean-field models for spin glasses, Probab. Theory Related Fields 148 (3–4) (2010) 601–643.[10] M. Talagrand, Mean-Field Models for Spin Glasses, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathe-matics, vols. 54, 55, Springer-Verlag, 2011.

Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case

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