Operator Norm Bounds on the Correlation Matrix of the SK Model at High Temperature

TL;DR

This paper proves that the correlation matrix of the SK model at high temperature has bounded operator norm with high probability, extending previous results to a broader parameter range.

Contribution

It establishes operator norm bounds for the SK model's correlation matrix in the entire high-temperature regime with external field, beyond the zero-field case.

Findings

Operator norm of the correlation matrix is bounded with high probability.

Results hold for all  in the high-temperature region with external field.

Extends previous bounds to include non-zero external fields.

Abstract

We prove that the two point correlation matrix $ \textbf{M}= (\langle \sigma_i ; \sigma_j\rangle)_{1\leq i,j\leq N} \in \mathbb{R}^{N\times N}$ of the Sherrington-Kirkpatrick model has the property that for every $\epsilon>0$ there exists $K_\epsilon>0$, that is independent of $N$, such that \[ \mathbb{P}\big( \| \textbf{M} \|_{\text{op}} \leq K_{\epsilon}\big) \geq 1- \epsilon \] for $N$ large enough, for suitable interaction and external field parameters $(\beta,h)$ in the replica symmetric region. In other words, the operator norm of $\textbf{M}$ is of order one with high probability. Our results are in particular valid for all $ (\beta,h)\in (0,1)\times (0,\infty) $ and thus complement recently obtained results in \cite{EAG,BSXY} that imply the operator norm boundedness of $\textbf{M}$ for all $\beta<1$ in the special case of vanishing external field.