The Two Point Function of the SK Model without External Field at High Temperature

TL;DR

This paper characterizes the asymptotic behavior of the two-point correlation matrix in the SK model at high temperature, showing it converges to a specific inverse matrix involving the GOE matrix.

Contribution

It provides a precise limit description of the two-point correlation matrix in the SK model without external field at high temperature.

Findings

The correlation matrix converges in operator norm to a specific inverse matrix.

The result holds in the full high temperature regime where eta < 1.

The limit involves the GOE interaction matrix G.

Abstract

We show that the two point correlation matrix $ \textbf{M}= (\langle \sigma_i \sigma_j\rangle)_{1\leq i,j\leq N} $ of the Sherrington-Kirkpatrick model with zero external field satisfies \[ \lim_{N\to\infty} \| \textbf{M} - ( 1+\beta^2 - \beta \textbf{G})^{-1} \|_{\text{op}} =0 \] in probability, in the full high temperature regime $\beta < 1$. Here, $\textbf{G}$ denotes the GOE interaction matrix of the model.