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

Abstract

We prove that the two point correlation matrix $\text{M}=(⟨{\sigma }_i;{\sigma }_j⟩{)}_{1\le i,j\le N}\in {\mathbb{R}}^{N\times N}$ 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 \[ \mathbb{P}\big( \| \textbf{M} \|_{\text{op}} \leq K_ε\big) \geq 1- ε\] 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 $\text{M}$ 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 $\text{M}$ for all $\beta <1$ in the special case of vanishing external field.

34 pages, revised version with minor corrections