A Simple Approach to Chaos For p -Spin Models
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A Simple Approach to Chaos For p-Spin Models Ronen Eldan1 Received: 11 May 2020 / Accepted: 13 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove that, in mixed p-spin models of spin glasses, the location of the ground state is chaotic under small Gaussian perturbations. For the case of even p-spin models, this was shown by Chen et al. (Probab. Theory Related Fields 171(1–2):53–95, 2018). We rely on a different approach which only uses the Parisi formula as a black box. Keywords Spin glasses · Disorder chaos · Sherrington–Kirkpatrick model
1 Introduction This paper concerns with the mixed p-spin model of spin-glasses, defined as follows. 2 Fix a dimension N ∈ N and fix non-negative constants (c p )∞ normalized so that p c p = 1. p=2 Define ξ(s) :=
∞
c2p s p .
p=2
Consider the discrete hypercube C N = {−1, 1} N . For x ∈ R N , define J (x) =
∞ p=2
cp x ⊗ p ∈ H, N ( p−1)/2
N ⊗ p . where H is the Hilbert space ∞ p=2 R The mixed p-spin model is a Gaussian process indexed by C N , defined as œ → H N (œ) with covariance structure
1 1 2 Cov(H N (σ 1 ), H N (σ 2 )) = J (σ 1 ), J (σ 2 ) = N ξ σ , σ , ∀σ 1 , σ 2 ∈ C N . N
This research was supported by a European Research Council Starting Grant (ERC StG) and by an Israel Science Foundation (ISF) Grant No. 715/16.
B 1
Ronen Eldan [email protected] Weizmann Institute of Science, Rehovot, Israel
123
R. Eldan
N ⊗ p Leting g be a vector of independent, standard Gaussian entries in ∞ , we will p=2 R define this Gaussian process more explicitly by setting H N (σ ) = H N (σ, g) where H N (σ ; x) := J (σ ), xH (using the scalar product notation is a slight abuse of notation since g is not in the Hilbert space H. However, since J (σ ) H = 1, the above is well-defined). The special case of this model, corresponding to c p = 1{ p=2} , is the so-called Sherrington Kirkpatrick spin glass which can be equivalently defined as H N (σ ) = i, j gi, j σi σ j where (gi, j )1≤i, j≤N are standard Gaussians. Consider the ground state σ ∗ (x) := arg max H N (σ ; x). σ ∈C N
This paper is concerned with the question: How stable is σ ∗ (g) with respect to small perturbations of g? To make√the question precise, let g be an independent copy of g and for t ≥ 0 set gt = e−t g + 1 − e−2t g , so that
Id e−t Id . (g, gt ) ∼ N 0, −t e Id Id For small t, gt can be thought of as a noisy version of g. We are interested in the question of whether there exist a sequence ε N → 0 such that
1 ∗ lim E ξ σ (g), σ ∗ (gε N ) = 0. (1) N →∞ N A Gaussian process which satisfies (1) is said to exhibit the chaos property. While above question was essentially posed in the physics literature, the precise definition of chaos in the broader context of Gaussian fields was made in the seminal paper of Chatterjee [4], where it is also shown that chaos is related to several other natural properties of Gaussian fields which have a disordered nature, some of which we discuss below. Remark 1 In Eq. (1) it may be natural to
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