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

TL;DR

This work establishes a uniform, N-independent bound on the operator norm of the two-point correlation matrix $\mathbf{M}$ for the Sherrington–Kirkpatrick model in the replica symmetric region with nonzero external field. The authors combine a TAP-inspired expansion with a resolvent-type approximation, showing that $\mathbf{M}$ behaves like the inverse of $\mathbf{D}+\beta^2(1-q)-\beta\mathbf{G}$ up to small errors, and they prove a uniform lower bound on this linear operator to deduce the bound on $\|\mathbf{M}\|_{op}$. A key technical contribution is a detailed decay analysis for $(k,p)$-point correlation functions via Talagrand’s overlap concentration, which feeds into precise error controls (e.g., an $O(N^{-5/2})$ bound) for the TAP representation. The final step uses a Sudakov–Fernique comparison in Bolthausen’s TAP framework to obtain a positive spectral gap, guaranteeing the uniform bound and extending known high-temperature results to positive fields. The results complement prior vanishing-field proofs and deepen the understanding of RS stability through spectral properties of the correlation matrix.

Abstract

We prove that the two point correlation matrix $ \textbf{M}= (\langle σ_i ; σ_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 $ε>0$ there exists $K_ε>0$, 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 $(β,h)$ 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 $ (β,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 $β<1$ in the special case of vanishing external field.

Theorems & Definitions (22)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• proof
• Theorem 2.1: talagrand2010mean-vol2
• Theorem 2.2: talagrand2010mean-vol1
• Lemma 2.3
• proof
• Lemma 2.4