Central limit theorems for cavity and local fields of the Sherrington-Kirkpatrick model

Wei-Kuo Chen

doi:10.1214/ejp.v18-1763

Abstract

One of the remarkable applications of the cavity method is to prove the Thouless-Anderson-Palmer (TAP) system of equations in the high temperature regime of the Sherrington-Kirkpatrick (SK) model. This naturally leads us to the important study of the limit laws for cavity and local fields. The first quantitative results for both fields based on Stein's method were studied by Chatterjee. Although Stein's method provides us an efficient approach for obtaining the limiting distributions, the nature of this method restricts the derivation of optimal and general results. In this paper, our study based on Gaussian interpolation obtains the CLT for the cavity field. With the help of this result, we conclude the CLT for local fields. In both cases, more refined moment estimates are given.

Highlights

Introduction and Main Results1.1 The Sherrington-Kirkpatrick Model and TAP EquationsLet N be a positive integer
The key observation to establish the limit law for the cavity field is that from the definition of li, {gij}j≤N,j=i is independent of the randomness of · −, which motivates our approach by using Gaussian interpolation on the cavity field
In our study we prove that the limit of l is still concentrative and under the Gibbs measure, l is approximately Gaussian with mean r and variance 1 − q

Summary

The Sherrington-Kirkpatrick Model and TAP Equations

To denote configurations chosen independently from the Gibbs measure (with the same given disorder) This observation provides us another main approach to studying SK model in the high temperature regime via the ThoulessAnderson-Palmer (TAP) system of equations as outlined in [5]: σi ≈ tanh gij j≤N,j=i σj. The key observation to establish the limit law for the cavity field is that from the definition of li, {gij}j≤N,j=i is independent of the randomness of · − , which motivates our approach by using Gaussian interpolation on the cavity field. We deduce the limit law for the local field In both cases, the quantitative results for the moment estimates are given and will be stated

Limit Law for the Cavity Field

Limit Law for the Local Fields

Proofs

Proof of Theorem 2

Proof of Theorem 3