Gespeichert in:
| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2307.12535 |
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Inhaltsangabe:
- 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.