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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2307.12535 |
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| _version_ | 1866929676114460672 |
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| author | Brennecke, Christian Xu, Changji Yau, Horng-Tzer |
| author_facet | Brennecke, Christian Xu, Changji Yau, Horng-Tzer |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12535 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Operator Norm Bounds on the Correlation Matrix of the SK Model at High Temperature Brennecke, Christian Xu, Changji Yau, Horng-Tzer Mathematical Physics Probability 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. |
| title | Operator Norm Bounds on the Correlation Matrix of the SK Model at High Temperature |
| topic | Mathematical Physics Probability |
| url | https://arxiv.org/abs/2307.12535 |