We show that if a discrete random measure on the unit ball of a separable
Hilbert space satisfies the Ghirlanda-Guerra identities then by randomly
deleting half of the points and renormalizing the weights of the remaining
points we obtain the same random measure in distribution up to rotations

Panchenko, Dmitry

English

arXiv

Dmitry Panchenko

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 579–581Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comProbability Theory/Mathematical PhysicsA deletion-invariance property for random measures satisfyingthe Ghirlanda–Guerra identitiesUne propriété d’invariance-suppression des mesures aléatoires vérifiant les identitésde Ghirlanda–GuerraDmitry Panchenko 1Department of Mathematics, Texas A&M University, College Station, TX 77843, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 29 March 2011Accepted 1 April 2011Available online 20 April 2011Presented by Michel TalagrandWe show that if a discrete random measure on the unit ball of a separable Hilbert spacesatisfies the Ghirlanda–Guerra identities then by randomly deleting half of the points andrenormalizing the weights of the remaining points we obtain the same random measure indistribution up to rotations.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons que si une mesure aléatoire discrète sur la boule unité d’un espacede Hilbert séparable satisfait aux identités de Ghirlanda–Guerra, alors en suprimantaléatoirement la moitié des points et en renormalisant les poids des points restants, onobtient une mesure de même distribution à une rotation près.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. Introduction and main resultLet us consider a countable index set A and random probability measure μ on a unit ball B of a separable Hilbert spaceH such that μ = ∑α∈A wαδξα for some random points ξα ∈ B and weights (wα). We will call indices α from the set A“configuration” and for a function f = f (α1, . . . ,αn) of n configurations we will denote its average with respect to themeasure μ by〈 f 〉 =∑α1,...,αnwα1 · · · wαn f(α1, . . . ,αn). (1)We say that the random measure μ satisfies the Ghirlanda–Guerra identities [3] if for any n  2 and any function f thatdepends on the configurations α1, . . . ,αn only through the scalar products, or overlaps, R,′ = ξα · ξα′ for , ′  n wehaveE〈f R p1,n+1〉 = 1nE〈 f 〉E〈R p1,2〉 + 1nn∑=2E〈f R p1,〉(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.2011.04.001580 D. Panchenko / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 579–581for any integer p  1. Random measures satisfying the Ghirlanda–Guerra identities arise as the directing measures (ordeterminators in the terminology of [8]) of overlap matrices in the Sherrington–Kirkpatrick model where they can be seen asthe asymptotic analogues of the Gibbs measure. The importance of the asymptotic point of view provided by these measureswas brought to light in [2], even though it was the Aizenman–Contucci stochastic stability [1] and not the Ghirlanda–Guerraidentities that played the main role there. However, subsequently, such random measures satisfying the Ghirlanda–Guerraidentities played an equally important role in the results of [4] and to a less extent of [6].Our main result is based on a simple observation which extends the invariance theorem from [4]. Consider independentsymmetric Bernoulli random variables (εα)α∈A (taking values ±1 with probability 1/2) and for t ∈ R let us define a randommeasure μt = ∑α∈A wα,tδξα with weights defined by the random change of densitywα,t = wα exp tεα∑γ ∈A wγ exp tεγ, (3)and as in (1) let us denote the average with respect to this measure by〈 f 〉t =∑α1,...,αnwα1,t · · · wαn,t f(α1, . . . ,αn). (4)The following holds:Theorem 1.1. If a random measure μ satisfies the Ghirlanda–Guerra identities (2) then for any t ∈ R, any n  2 and any boundedfunction f of the overlaps on n configurations we have E〈 f 〉t = E〈 f 〉.The main difference here is that the result holds for all t ∈ R compared to |t| < 1/2 as stated in Theorem 4 in [4] whichwas sufficient for the main argument there. However, letting t go to infinity we now obtain the following new invarianceproperty. Let ηα = (εα + 1)/2 be independent random variables, now taking values 1 and 0 with probability 1/2 and letμ′ = ∑α∈A w ′αδξα be the random measure defined by the change of densityw ′α =wαηα∑γ ∈A wγ ηγ. (5)In other words, we randomly delete half of the point in the support of measure μ and renormalize the weights to definea probability measure μ′ on the remaining points. The denominator in (5) is non-zero with probability one since it is wellknown that unless the measure μ is concentrated at 0 ∈ B (a case we do not consider) it must have infinitely many differentpoints in the support in order to satisfy (2). Let us define by 〈 f 〉′ the average with respect to μ′ .Theorem 1.2 (Deletion invariance). If a random measure μ satisfies the Ghirlanda–Guerra identities (2) then E〈 f 〉′ = E〈 f 〉.Remark 1. In particular, this implies that the measure μ′ also satisfies the Ghirlanda–Guerra identities (2) and, hence, wecan repeat the random deletion procedure as many times as we want. This means that the deletion invariance also holdswith random variables (ηα) taking values 1 and 0 with probabilities 1/2s and 1 − 1/2s correspondingly, for any integers  1.Remark 2. It is well known that invariance for the averages as in Theorem 1.2 implies that the random measures μ and μ′have the same distribution, up to rotations. Let (w)1 be the weights (wα) arranged in the non-increasing order and let(ξ) be the points (ξα) rearranged accordingly, so that μ = ∑1 wδξ . Similarly, let μ′ =∑1 w′δξ ′ . Then arguing as atthe end of the proof of Theorem 4 in [4] (or Lemma 4 in [5]) one can show that((w)1, (ξ · ξ′),′1) d= ((w ′)1,(ξ ′ · ξ ′′),′1)(6)which means that up to rotations the configurations of the random measures μ and μ′ have the same distributions.Proof of Theorem 1.1. Suppose that | f |  1. Let ϕ(t) = E〈 f 〉t and by symmetry we only need to consider t  0. Givenconfigurations α1,α2, . . . let us denoteDn = εα1 + · · · + εαn − nεαn+1and a straightforward computation shows that ϕ′(t) = E〈 f Dn〉t and similarly for all k  1,ϕ(k)(t) = E〈 f Dn · · · Dn+k−1〉t .It was proved in Theorem 4 in [4] (a more streamlined proof was given in Theorem 6.3 in [6]) that if the measure μ satisfiesthe Ghirlanda–Guerra identities (2) thenϕ(k)(0) = 0 for all k  1. (7)D. Panchenko / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 579–581 581It is also easy to see that |Dn|  2n so that for all t,∣∣ϕ(k)(t)∣∣  2kn(n + 1) · · · (n + k − 1). (8)This is all one needs to show that ifϕ(t) = ϕ(0) and ϕ(k)(t) = 0 for all k  1 (9)holds for all t  t0 for some t0  0 then it also holds for all t < t0 + 1/2. This will finish the proof of the theorem sinceby (7) this holds for t0 = 0. Take any k  0. Using (8) and (9) for t = t0 and using Taylor’s expansion for a function ϕ(k)(t)around the point t = t0 we get for any m  1∣∣ϕ(k)(t) − ϕ(k)(t0)∣∣  supt0st|ϕ(k+m)(s)|m! |t − t0|m  2k+mn(n + 1) · · · (n + k + m − 1)m! |t − t0|m.If |t −t0| < 1/2 then letting m → ∞ proves that ϕ(k)(t) = ϕ(k)(t0) for all k  0 and therefore (9) holds for all t < t0 +1/2. Proof of Theorem 1.2. Let I = {α ∈ A: εα = 1} and letZt =∑α∈Iwα + e−2t∑α∈Icwαso thatwα,t = wαZt(I(α ∈ I) + e−2t I(α ∈ Ic)). (10)Then the sum on the right-hand side of (4) can be broken into 2n groups depending on which of the indexes α1, . . . ,αnbelong to I or its complement Ic, for example, the terms corresponding to all indices belonging to I will give1Znt∑α1,...,αn∈Iwα1 · · · wαn f(α1, . . . ,αn). (11)This sum is bounded by one and when t → +∞ it obviously converges to 〈 f 〉′ while the sums corresponding to othergroups, when at least one of the indices belongs to Ic , will converge to zero because of the factor e−2t in (10). By dominatedconvergence theorem we get convergence of expectations. 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] 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.[4] D. Panchenko, A connection between Ghirlanda–Guerra identities and ultrametricity, Ann. Probab. 38 (1) (2010) 327–347.[5] D. Panchenko, On the Dovbysh–Sudakov representation result, Electron. Commun. Probab. 15 (2010) 330–338.[6] M. Talagrand, Construction of pure states in mean-field models for spin glasses, Probab. Theory Related Fields 148 (3–4) (2010) 601–643.[7] M. Talagrand, Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models, Springer-Verlag, Berlin, 2003.[8] M. Talagrand, Mean-Field Models for Spin Glasses, second edition of [7], in press. First volume was published: http://www.springer.com/mathematics/probability/book/978-3-642-15201-6. Vol. 2 will appear in Ergebnisse Series, vol. 55.

Comptes Rendus Mathematique

A deletion-invariance property for random measures satisfying the Ghirlanda–Guerra identities

Numérisation de Documents Anciens Mathématiques

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A deletion-invariance property for random measures satisfying the Ghirlanda-Guerra identities

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