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arXiv · 2307.12535

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

Christian Brennecke·Changji Xu·Horng-Tzer Yau
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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.

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Christian Brennecke, Changji Xu, Horng-Tzer Yau. 2023-07-24. Operator Norm Bounds on the Correlation Matrix of the SK Model at High Temperature. https://arxiv.org/abs/2307.12535

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