{'fraction_non_alphanumeric': 0.053403431101272825, 'fraction_numerical': 0.013558384061981184, 'mean_word_length': 3.8451742627345844, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Ghirlanda-Guerra identities without averaging

The Ghirlanda-Guerra identities are one of the most mysterious features of spin glasses. We prove the GG identities in a large class of models that includes the Edwards-Anderson model, the random field Ising model, and the Sherrington-Kirkpatrick model in the presence of a random external field. Previously, the GG identities were rigorously proved only `on average' over a range of temperatures or under small perturbations.


Introduction
Consider the Sherrington-Kirpatrick model of spin glasses [11] in the presence of an external field, defined as follows. Let N be a positive integer. Let (g ij ) 1≤i<j≤N and (g i ) 1≤i≤N be collections of i.i.d. standard Gaussian random variables. Fix three real numbers β, γ, h. Define a Hamiltonian H N on {−1, 1} N as Let G N be the (random) probability measure on {−1, 1} N that puts mass at each configuration σ. This is the Gibbs measure of the SK model. As usual, for a function f : ({−1, 1} N ) n → R we write f := σ 1 ,...,σ n Let σ 1 , σ 2 , . . . be i.i.d. configurations from the Gibbs measure. In the spin glass parlance, these are known as replicas. The overlap between two replicas σ ℓ and σ ℓ ′ is defined as Now fix a number n, and a bounded measurable function f : The Ghirlanda-Guerra identities [7] claim that for any such n and f , for almost all β, γ, h, There is an extended form of the GG identities that claims more: in the infinite volume limit, the conditional distribution of R 1,n+1 given R n is where L R 1,2 is the (unconditional) law of R 1,2 , and δ R 1,ℓ is the point mass at R 1,ℓ . Clearly, (2) is a special case of this extended version. There is an early version of the GG identities that are sometimes called the Aizenman-Contucci identities; these appeared in [1]. The paper [1] also introduced the important notion of stochastic stability in spin glass models. An interesting connection between stochastic stability and the GG identities was discovered by Arguin [2].
The importance of the GG identities stems from their universal nature. The identities are supposed to hold in a wide array of spin glass models, both mean field and short range. A striking recent discovery of Panchenko [9], following on the insightful work of Arguin and Aizenman [3], indicates that the GG identities may actually solve the long-standing mystery of ultrametricity in spin glasses.
It is not difficult to show (see e.g. Talagrand [12], Section 2.12) that for any a > 0, and any β, h, Similar results were proved for a general class of spin glass models, including the short range Edwards-Anderson model, by Contucci and Giardinà [5]. However, this does not prove (2). The best advance till date is due to Talagrand [13] who showed that for any β, γ, h, there exists a sequence Let ψ N be the free energy per particle, defined as . As easy extension of the Guerra-Toninelli argument [8] shows that lim N →∞ p N (β, γ, h) exists for all β, γ, h. Let us call this limit p(β, γ, h). Then p is a convex function of each argument.
Clearly, the major drawback of the above result is that it only works for γ = 0. This is problematic since γ = 0 is the most interesting case. A minor drawback is that the theorem holds for almost every value of γ instead of all values. But this is not a serious problem since it is not clear whether the GG identities hold at points of phase transition, and the original paper of Ghirlanda and Guerra [7] does not make such a claim either. And lastly, it will be an important breakthrough if one can prove a similar statement for the extended GG identities, but that seems to be out of reach at present.
The proof of Theorem 1.1 admits a vast generalization, which we state below.
For each N , suppose µ N is a probability measure on {−1, 1} N . Suppose Let ψ N be the free energy per particle in this model, that is, Let p N = E(ψ N ). Assume that p(γ) := lim N →∞ p N (γ) exists. Next, for two replicas σ 1 and σ 2 , define the generalized overlap as Assume that R 1,1 is a deterministic constant for all σ 1 . Given n and a bounded measurable function f : R n(n−1)/2 → R, let δ N (γ) be defined as in (1), in terms of the generalized overlaps defined above.
Note that this result includes the Sherrington-Kirkpatrick model and Derrida's p-spin models under a random external field. It also covers lattice models like the random field Ising model and the Edwards-Anderson model [6]. For example, in the Edwards-Anderson model, the spins are located on a subset of the d-dimensional integer lattice; A N is the set of edges in this subset, and for an edge α = (i, j) ∈ A N , f α (σ) = σ i σ j . The measure µ N is just the uniform distribution on {−1, 1} N in this case. In this model, our generalized overlap is simply what's known as the bond overlap, and γ is the inverse temperature parameter. Here we do not need to apply a random external field. The existence of the limit of the free energy per particle in the lattice models follow from simple arguments, e.g. those in [10], Chapter 2.

Proof
We will only prove Theorem 1.2, since Theorem 1.1 is a special case of Theorem 1.

Let
It is easy to prove via integration by parts (see e.g. [12], Section 2.12) that the Ghirlanda-Guerra identities hold if We prove (5) in two steps. First, we show that This part requires no new ideas. The proof goes as follows. Note that Fix some γ ′ > γ. By the convexity of ψ N and (7), we have Again, by standard concentration of measure (see e.g. [12], Theorem 2.2.4) we know that for any γ, where C is a universal constant. Thus, Since |A N | = O(N ) and p N → p pointwise, we get Combining with (8), we see that Here x + denotes the positive part of a real number x. Since this bound holds for any γ ′ > γ and p is differentiable at γ, we get Similarly, considering γ ′ < γ and repeating the steps, we can show that the limit of the negative part is zero as well. Thus, Combining the last two identities, we get (6). This completes the first step.
In the next part of the proof, we show that Combined with (6), this will complete the proof of (5) and hence the theorem. To show (10), it suffice to prove that The key new idea at this stage is the use of the following result. Essentially, this is the Parseval identity for the L 2 norm of a Gaussian functional expressed as a sum of squares of its Fourier coefficients in the orthogonal basis of multidimensional Hermite polynomials.

Theorem 2.1 ([4]
). Let g = (g 1 , . . . , g n ) be a vector of i.i.d. standard Gaussian random variables, and let f be a C ∞ function of g with bounded derivatives of all orders. Then The convergence of the infinite series is part of the conclusion.
The formula in its above form is stated and proved in [4]. Different versions of the identity were observed by a number of authors; let us refer to Section 3.8 of [4] for a discussion of these observations and alternate proofs.
We will now prove (11) using the above theorem. In the following, we will simply write f α instead of f α (σ). Integration by parts gives where ψ N is the free energy per particle defined in (4). Thus, by the Cauchy-Schwarz inequality and Theorem 2.1, we get The proof is now completed by applying the variance inequality (9).