aa r X i v : . [ m a t h . P R ] N ov A unified stability property in spin glasses.
Dmitry Panchenko ∗ November 19, 2018
Abstract
Gibbs’ measures in the Sherrington-Kirkpatrick type models satisfy two asymptoticstability properties, the Aizenman-Contucci stochastic stability and the Ghirlanda-Guerra identities, which play a fundamental role in our current understanding of thesemodels. In this paper we show that one can combine these two properties very naturallyinto one unified stability property.
Key words: Gibbs’ measures, spin glass models, stability.
In the Sherrington-Kirkpatrick (SK) model [29] or, more generally, in the mixed p -spinmodel one considers a random process (Hamiltonian) H N ( σ ) indexed by spin configurations σ ∈ Σ N = {− , +1 } N given by a linear combination P p ≥ β p H N,p ( σ ) with some coefficients β p ≥ p -spin Hamiltonians, H N,p ( σ ) = 1 N ( p − / X ≤ i ,...,i p ≤ N g i ,...,i p σ i . . . σ i p , (1.1)where ( g i ,...,i p ) are standard Gaussian independent for all p ≥ i , . . . , i p ) . TheGibbs measure G N corresponding to the Hamiltonian H N is defined as a random probabilitymeasure on Σ N given by G N ( σ ) = 1 Z N exp H N ( σ ) , (1.2)where the normalizing factor Z N is called the partition function. The Gibbs measure G N in(1.2) is the central object of interest in spin glass models and the answers to many importantquestions follow from the conjectured properties of G N . These properties can be expressed in ∗ Department of Mathematics, Texas A&M University, email: [email protected]. Partially sup-ported by NSF grant. σ l ) l ≥ from G N , for example, the normalized Grammatrix R = ( R l,l ′ ) l,l ′ ≥ = N − ( σ l · σ l ′ ) l,l ′ ≥ . It is easy to see that knowing R is equivalentto knowing G N up to orthogonal transformations since one can reconstruct G N from R upto orthogonal transformations (as we shall see in the proof of Theorem 1 below) and theinformation encoded in the distribution of R turns out to be sufficient for most purposesin the setting of the SK model due to the fact that the Hamiltonian H N ( σ ) is a Gaussianprocess and its covariance E H N ( σ ) H N ( σ ) is the function of exactly the normalized scalarproduct R , = N − σ · σ , called the overlap of σ and σ . Given any limiting distributionof the Gram matrix R in the thermodynamic limit N → ∞ , one can use the Dovbysh-Sudakov representation ([10],[19]) to define the asymptotic analogue of the Gibbs measureas a random probability measure G on the unit ball of the Hilbert space ℓ (see [5], [3],[15]). This means that in the limit the matrix R is still generated as the Gram matrix ofthe sample from some random measure G. For simplicity, let us assume that this asymptoticGibbs measure G is atomic, G = X l ≥ w l δ ξ l , (1.3)with the weights arranged in non-increasing order, w ≥ w ≥ . . . , and let us denote by Q = ( ξ l · ξ l ′ ) l,l ′ ≥ the matrix of scalar products of the points in the support of G . Let ( σ l ) l ≥ again be an i.i.d. sample from this measure and let R l,l ′ = σ l · σ l ′ be the scalar product in ℓ of σ l and σ l ′ . For any n ≥ f = f ( σ , . . . , σ n ) of n configurations we willdenote its average with respect to G ⊗∞ by h f i = X l ,...,l n ≥ w l · · · w l n f ( ξ l , . . . , ξ l n ) . (1.4)We will denote by E the expectation with respect to the randomness of G .In the mixed p -spin models, one expects the asymptotic Gibbs measures (1.3) to bedescribed precisely by the Parisi ultrametric ansatz (see [24], [14]). So far, most of the progressin the direction of proving structural results about G was based on the idea that certaininformation about its geometry can be recovered from the asymptotic stability properties ofthe Gibbs measure G N under small perturbations of the Hamiltonian H N ( σ ). In particular,two such stability properties in the setting of mixed p -spin model are well known - theAizenman-Contucci stochastic stability [1] and the Ghirlanda-Guerra identities [13]. Theycan be written down in terms of the asymptotic Gibbs measure G as follows.1. (Ghirlanda-Guerra identities) Random measure G is said to satisfy the Ghirlanda-Guerraidentities [13] if for any n ≥ , any bounded measurable function f of the overlaps ( R l,l ′ ) l,l ′ ≤ n and any integer p ≥ E h f R p ,n +1 i = 1 n E h f i E h R p , i + 1 n n X l =2 E h f R p ,l i . (1.5)These constraints on the distribution of R look very mysterious but they arise from a verynatural and very general principle of the concentration of the Hamiltonian H N ( σ ) (see [13]).2. (Aizenman-Contucci stochastic stability) Given integer p ≥ , let ( g p ( ξ l )) l ≥ be a Gaussiansequence conditionally on G indexed by the points ( ξ l ) l ≥ with covarianceCov (cid:0) g p ( ξ l ) , g p ( ξ l ′ ) (cid:1) = ( ξ l · ξ l ′ ) p . (1.6)Given t ∈ R , consider a new sequence of weights w tl = w l e tg p ( ξ l ) P j ≥ w j e tg p ( ξ j ) (1.7)defined by a random change of density proportional to e tg p ( ξ l ) . Let ( w πl ) be the weights ( w tl )arranged in the non-increasing order and let π : N → N be the permutation keeping track ofwhere each index came from, w πl = w tπ ( l ) . Let us define by G π = X l ≥ w πl δ ξ π ( l ) and Q π = (cid:0) ξ π ( l ) · ξ π ( l ′ ) (cid:1) l,l ′ ≥ (1.8)the probability measure G after the change of density proportional to e tg p ( ξ l ) and the matrix Q rearranged according to the reordering of weights. Measure G is said to satisfy the Aizenman-Contucci stochastic stability [1] if for any p ≥ t ∈ R , (cid:0) ( w πl ) l ≥ , Q π (cid:1) d = (cid:0) ( w l ) l ≥ , Q (cid:1) (1.9)where equality in distribution is in the sense of finite dimensional distributions of these arrays.This property represents the invariance of the distribution of measure G up to orthogonaltransformations under the random changes of density (1.7) and it arises from the continuityof the Gibbs measure G N under small changes in the inverse temperature parameters β p inthe Hamiltonian H N ( σ ) (see [1]).Originally, the proofs of the Ghirlanda-Guerra identities (1.5) in [13] and the Aizenman-Contucci stochastic stability (1.9) in [1], [8] obtained these results for each p ≥ β p over any non-trivial interval. In a closely related formulation, one canalways perturb the parameters ( β p ) slightly (for example, one can find ( β N,p ) such that | β N,p − β p | ≤ − p N − / ) so that the sequence of Gibbs measures G N corresponding to theseslightly perturbed parameters satisfies the above properties in the limit (see [33], [34]). Bothof these formulations hold for mixed p -spin models with arbitrary subset of p -spin terms(1.1) present in the model, i.e. for which β p = 0. However, if the model contains terms for p = 1 and all even p ≥ p ≥
1. The proofwas based on the validity of the Parisi formula for the free energy proved by M. Talagrand in[31] following the discovery of the replica symmetry breaking bound by F. Guerra in [12], onthe differentiability properties of the Parisi formula (see [32], [17]) and on the positivity ofthe overlap (see [34]). A similar strong version of the Aizenman-Contucci stochastic stabilitywas proved in [6]. 3s we mentioned above, the importance of the stability properties (1.5) and (1.9) inthe SK model comes from their many applications (see e.g. [4], [5], [6], [15], [16], [18], [21],[22], [23], [25], [28], [33], [30], [34]). Of course, the ultimate goal would be to prove the Parisiultrametricity conjecture which states that the support of the asymptotic Gibbs measure G must be ultrametric in ℓ with probability one, which would allow us to identify G with theRuelle Probability Cascades in [27]. At the moment, the results that come closest to provingthis conjecture are based either on the Aizenman-Contucci stochastic stability or on theGhirlanda-Guerra identities. First such result was proved by L.-P. Arguin and M. Aizenmanin [5] using the Aizenman-Contucci stochastic stability under a technical assumption that thescalar products ξ l · ξ l ′ of points in the support of measure G take finitely many non-randomvalues. Following their work, a similar result was proved by the author in [15] (see also [22]for a recent elementary proof) and by M. Talagrand in [33] using the Ghirlanda-Guerraidentities instead. Some modest progress toward the general case was made in [23] but theconjecture still remains open. Once this conjecture is proved, the Parisi formula for the freeenergy will easily follow using the Aizenman-Sims-Starr scheme developed in [2] which wouldnaturally complete the mathematical justification of the Parisi ansatz in the SK model. It isworth mentioning that stability properties of the Gibbs measure under small perturbationsof the Hamiltonian play very important role in other spin glass models as well (see e.g. [20]).In the main result of this paper we will show that one can combine the Ghirlanda-Guerra identities (1.5) and the Aizenman-Contucci stochastic stability (1.9) into a jointstability property as follows. It is known (Theorem 2 in [15]) that if the measure G satisfiesthe Ghirlanda-Guerra identities and if q ∗ is the supremum of the support of the distributionof the overlap R , under E G ⊗ then with probability one G is concentrated on the sphereof radius √ q ∗ . Let b p = ( q ∗ ) p − E h R p , i . (1.10)Then the following holds. Theorem 1
A random measure G satisfies the Ghirlanda-Guerra identities (1.5) and theAizenman-Contucci stochastic stability (1.9) if and only if it is concentrated on the sphereof constant radius √ q ∗ with probability one and for any p ≥ and t ∈ R , (cid:16)(cid:0) w πl (cid:1) l ≥ , (cid:0) g p ( ξ π ( l ) ) − b p t (cid:1) l ≥ , Q π (cid:17) d = (cid:16)(cid:0) w l (cid:1) l ≥ , (cid:0) g p ( ξ l ) (cid:1) l ≥ , Q (cid:17) . (1.11) where equality in distribution is in the sense of finite dimensional distributions. One can see that the Ghirlanda-Guerra identities are now replaced by the statement thatthe Gaussian field ( g p ( ξ l )) after permutation π corresponding to the reordering of weightsin (1.7) will only differ by a constant shift b p t in distribution. In the language of competingparticle systems ([28] and [5]), (1.11) means that the past increments of the dynamics afterre-centering have the same law as the future or forward increments. The stability property(1.11) is well-known for the ultrametric Ruelle Probability Cascades (see Theorem 4.2 in [7]or Theorem 15.2.1 in [34]) and, of course, the big question is whether it holds only for thesemeasures and whether (1.11) implies ultrametricity.4he Ghirlanda-Guerra identities do not require the random measure G to be discreteand, in fact, the Aizenman-Contucci stochastic stability can be formulated not only fordiscrete measures as well. We will mention this more general formulation in the next section.However, we prefer to state our main result in the setting of discrete measures since it allowsfor a particularly attractive formulation (1.11) in the spirit of competing particle systems,as in [28] and [5]. Moreover, from the point of view of studying structural properties of suchmeasures one can without loss of generality start with discrete measures since it is easy toshow that sampling an i.i.d. sequence of points from the original measure and assigning themnew independent weights from the Poisson-Dirichlet distribution creates a discrete measurewhich still satisfies both properties. On the other hand, almost any geometric property ofthe original measure will be encoded into a countable i.i.d. sample and, therefore, this newdiscrete measure.The unified stability property (1.11) inspired a new representation of the Ghirlanda-Guerra identities in [23] which yielded some interesting applications. For another recentstability property that reproduces the Ghirlanda-Guerra identities on average see [9]. Acknowledgment.
The author would like to thank the referees for making many importantsuggestions that helped improve the paper.
Let ( ρ l ) l ≥ be an i.i.d. sequence from measure G π defined in (1.8) and denote by S l,l ′ = ρ l · ρ l ′ the overlap of ρ l and ρ l ′ . Analogously to (1.4), for any n ≥ f = f ( ρ , . . . , ρ n )of n configurations we will denote its average with respect to ( G π ) ⊗∞ by h f i π = X l ,...,l n ≥ w πl · · · w πl n f ( ξ π ( l ) , . . . , ξ π ( l n ) ) . (2.1)We now will denote by E the expectation with respect to the randomness of G and the Gaus-sian sequence ( g p ) . Let us first make a simple observation that equality of finite dimensionaldistributions in (1.9) and (1.11) implies equality of averages with respect to the randommeasures in the following sense.
Lemma 1
If (1.11) holds then for any k ≥ , any bounded measurable function f of theoverlaps on k replicas and any integers n , . . . , n k ≥ , E DY l ≤ k (cid:0) g p ( ρ l ) − b p t (cid:1) n l f (cid:0) ( S l,l ′ ) l,l ′ ≤ k (cid:1)E π = E DY l ≤ k g p ( σ l ) n l f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E (2.2) Under (1.9), this holds with all n l = 0 . Remark.
One can consider (2.2) with all n l = 0 as the definition of the Aizenman-Contuccistochastic stability for non-atomic measures in which case ( g p ( ξ )) is the Gaussian field withcovariance (1.6). Moreover, in this case (2.2) should be considered as the analogue of (1.11).5 roof. This is obvious by separating the sum in (1.4) and (2.1) into finitely many termscorresponding to the largest weights and the remaining small weights. For example, E DY l ≤ k g p ( σ l ) n l f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E = E X j ,...,j k ≥ w j · · · w j n Y l ≤ k g p ( ξ j l ) n l f (cid:0) ( ξ j l · ξ j l ′ ) l,l ′ ≤ k (cid:1) = E X j ,...,j k ≤ N w j · · · w j n Y l ≤ k g p ( ξ j l ) n l f (cid:0) ( ξ j l · ξ j l ′ ) l,l ′ ≤ k (cid:1) + R N , where the remainder R N consists of the terms with at least one index j , . . . , j k > N . The lefthand side of (2.2) can be similarly broken into two sums. The finite sums are equal becausethey involve only finitely many elements of the arrays (1.11) which are equal in distributionby assumption. Thus, we only need to show that R N becomes small for large N . First takingexpectation in the Gaussian random variables ( g p ( ξ l )) conditionally on ( w l ) and Q and usingthat E (cid:16)Y l ≤ k | g p ( ξ j l ) | n l (cid:12)(cid:12)(cid:12) ( w l ) , Q (cid:17) ≤ L ( n , . . . , n k )we get that |R N | ≤ L ( n , . . . , n k ) k f k ∞ E X ( j ,...,j k ≤ N ) c w j · · · w j n ≤ Lk E X j>N w j which goes to zero as N → ∞ . The remainder R πN for the left hand side of (2.2) is controlledby exactly the same bound because, by (1.11), the corresponding terms in R πN and R N areequal in distribution.The “if” part of the Theorem 1 is easy since assuming (1.11) we only need to prove (1.5)and this follows from integration by parts of (2.2) with n = 1 , n = . . . = n k = 0 . In thiscase the right hand side is zero by averaging g p ( σ ) first and the left hand side is E D ( g p ( ρ ) − b p t (cid:1) f (cid:0) ( S l,l ′ ) l,l ′ ≤ k (cid:1)E π = t E D(cid:0) k X l =1 S p ,l − b p − kS p ,k +1 (cid:1) f (cid:0) ( S l,l ′ ) l,l ′ ≤ k (cid:1)E π = t E D(cid:0) k X l =1 R p ,l − b p − kR p ,k +1 (cid:1) f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E = t E D(cid:0) k X l =2 R p ,l + E h R p , i − kR p ,k +1 (cid:1) f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E where in the second line we used (1.9) part of (1.11) and Lemma 1, and in the third line weused (1.10) and the fact that ξ l · ξ l = q ∗ . The fact that the last sum is zero is exactly (1.5).To prove the ”only if” part we need the following key lemma. Lemma 2
If (1.5) and (1.9) hold then (2.2) holds. roof. The proof is by induction on N = n + . . . + n k . When N = 0 , (2.2) is the consequenceof (1.9) by Lemma 1. Suppose (2.2) holds for all k ≥
1, all f and for all N ≤ N . Clearly,we only need to prove the case of powers n + 1 , n , . . . , n k . Writing g p ( σ ) n +1 = g p ( σ ) g p ( σ ) n and using Gaussian integration by parts for g p ( σ ) we can rewrite the right hand side of(2.2) with n + 1 instead of n as X l ≤ k n l E D g p ( σ ) n . . . g p ( σ l ) n l − . . . g p ( σ k ) n k R p ,l f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E . (2.3)Again, writing ( g p ( ρ ) − b p t ) n +1 = ( g p ( ρ ) − b p t )( g p ( ρ ) − b p t ) n and using Gaussian integration by parts for g p ( ρ ) we can rewrite the left hand side of (2.2)with n + 1 instead of n as I + II where I is given by X l ≤ k n l E D(cid:0) g p ( ρ ) − b p t (cid:1) n . . . (cid:0) g p ( ρ l ) − b p t (cid:1) n l − . . . (cid:0) g p ( ρ k ) − b p t (cid:1) n k S p ,l f (cid:0) ( S l,l ′ ) l,l ′ ≤ k (cid:1)E π (2.4)and II is given by t E DY l ≤ k (cid:0) g p ( ρ l ) − b p t (cid:1) n l (cid:16)X l ≤ k S p ,l − b p − kS p ,k +1 (cid:17) f (cid:0) ( S l,l ′ ) l,l ′ ≤ k (cid:1)E π . (2.5)By induction hypothesis, (2.4) is equal to (2.3) and (2.5) is equal to t E DY l ≤ k g p ( σ l ) n l (cid:16)X l ≤ k R p ,l − b p − kR p ,k +1 (cid:17) f (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E . (2.6)Since h·i does not depend on the Gaussian sequence ( g p ( ξ l )) we can take expectation E g withrespect to the randomness of this sequence conditionally on G first and notice that E g Y l ≤ k g p ( σ l ) n l = f ′ (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1) for some function f ′ of the overlaps of k configurations σ , . . . , σ k . Therefore, (2.6) equals t E D(cid:16) k X l =1 R p ,l − b p − kR p ,k +1 (cid:17) ( f f ′ ) (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E (2.7)= t E D(cid:16) k X l =2 R p ,l + E h R p , i − kR p ,k +1 (cid:17) ( f f ′ ) (cid:0) ( R l,l ′ ) l,l ′ ≤ k (cid:1)E = 0 , where in the first equality we again used (1.10) and the fact that ξ l · ξ l = q ∗ and the secondequality is by the Ghirlanda-Guerra identities (1.5). This finishes the proof.The equality of joint moments (2.2) proved in Lemma 2 implies the following.7 emma 3 If (1.5) and (1.9) hold then (cid:16)(cid:0) g p ( ρ l ) − b p t (cid:1) l ≥ , ( S l,l ′ ) l,l ′ ≥ (cid:17) d = (cid:16)(cid:0) g p ( σ l ) (cid:1) l ≥ , ( R l,l ′ ) l,l ′ ≥ (cid:17) (2.8) where equality in distribution is in the sense of finite dimensional distributions. Remark.
Let us recall that the i.i.d. sequences ( σ l ) and ( ρ l ) are sampled from G and G π correspondingly and, therefore, the distributions of the right-hand side and left-hand side in(2.8) are under E G ⊗∞ and E ( G π ) ⊗∞ . Proof.
By choosing f to be monomials, (2.2) gives the equality of joint moments of thecorresponding elements of the two arrays in (2.8). In our case the joint moments uniquelydetermine joint distributions, for example, by the main result in [26] which states that weonly need to ensure the uniqueness of one dimensional marginals and the fact that the onedimensional marginals are either bounded or Gaussian. Proof of Theorem 1.
Finally, we will show that (2.8) implies (1.11). The procedure isvery similar to the one at the end of Theorem 4 in [15] or a more general argument inLemma 4 in [19]. First of all, by the well-known result of Talagrand (Section 1.2 in [30]),the Ghirlanda-Guerra identities imply that the weights ( w l ) must have a Poisson-Dirichletdistribution P D ( m ) where m is determined by E h I ( R , = q ∗ ) i = E X l ≥ w l = 1 − m. This means that if ( u l ) is a Poisson point process on (0 , ∞ ) with intensity measure x − m − dx then w l = u l / P j ≥ u j . In particular, all the weights are different with probability one. Thispoint is not crucial but it makes for an easier argument. The reason why (2.8) implies (1.11)is because one can easily reconstruct the arrays in (1.11) from the arrays (2.8) using that( σ l ) is an i.i.d. sample from ( ξ l ) according to weights ( w l ) and ( ρ l ) is an i.i.d. sample from( ξ π ( l ) ) according to weights ( w πl ) . The key observation here is that given arrays (2.8) weknow exactly when σ l = σ l ′ and ρ l = ρ l ′ since this is equivalent to R l,l ′ = q ∗ and S l,l ′ = q ∗ .Therefore, given N ≥ (cid:0)(cid:0) g p ( σ l ) (cid:1) l ≤ N , ( R l,l ′ ) l,l ′ ≤ N (cid:1) we can partition the set { , . . . , N } according to the equivalence relation l ∼ l ′ definedby R l,l ′ = q ∗ , let the sequence of weights ( w Nl ) l ≥ be the proportions of the sets in thispartition arranged in non-increasing order and extended by zeros and, given any integer j in the element of the partition corresponding to the weight w Nl , define ξ Nl = σ j . We let Q N = ( ξ Nl · ξ Nl ′ ) l,l ′ ≥ . The elements of ( ξ Nl ) and Q N with indices corresponding to zero weights w Nl can be set to some fixed values, and we break ties between w Nl by any pre-determinedrule. Similarly, given (cid:0)(cid:0) g p ( ρ l ) − b p t (cid:1) l ≤ N , ( S l,l ′ ) l,l ′ ≤ N (cid:1) we can construct sequences ( ˜ w Nl ) , ( ˜ ξ Nl ) and ˜ Q N = ( ˜ ξ Nl · ˜ ξ Nl ′ ). Equation (2.8) implies that forany fixed k ≥ , (cid:16)(cid:0) ˜ w Nl (cid:1) l ≤ k , (cid:0) g p ( ˜ ξ Nl ) − b p t (cid:1) l ≤ k , (˜ q Nl,l ′ ) l,l ′ ≤ k (cid:17) d = (cid:16)(cid:0) w Nl (cid:1) l ≤ k , (cid:0) g p ( ξ Nl ) (cid:1) l ≤ k , ( q Nl,l ′ ) l,l ′ ≤ k (cid:17) .
8t remains to observe that the right hand side converges (cid:16)(cid:0) w Nl (cid:1) l ≤ k , (cid:0) g p ( ξ Nl ) (cid:1) l ≤ k , ( q Nl,l ′ ) l,l ′ ≤ k (cid:17) → (cid:16)(cid:0) w l (cid:1) l ≤ k , (cid:0) g p ( ξ l ) (cid:1) l ≤ k , ( q l,l ′ ) l,l ′ ≤ k (cid:17) (2.9)almost surely and, similarly, the left hand side converges a.s. to the corresponding array fromthe left hand side of (1.11). To prove (2.9), we notice that by construction G N := X l ≥ w Nl δ ξ Nl = 1 N X i ≤ N δ σ i is the empirical measure based on the sample σ , . . . , σ N from the measure G = P l ≥ w l δ ξ l . By the strong law of large number for empirical measures (e.g. Theorem 11.4.1 in [11]), thelaws G N → G almost surely and since the Poisson-Dirichlet weights ( w l ) are all different a.s.,the largest k weights must converge ( w Nl ) l ≤ k → ( w l ) l ≤ k almost surely and for large enough N we must have ( ξ Nl ) l ≤ k = ( ξ l ) l ≤ k and, thus, (2.9) holds. References [1] Aizenman, M., Contucci, P. (1998) On the stability of the quenched state in mean-field spin-glassmodels.
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