整个de Almeida–Thouless区域内Sherrington–Kirkpatrick模型的定量复制对称界
A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region
- Graduate School of Science, Kyoto University（京都大学大学院理学研究科）
- Department of Mathematics, Keio University（庆应义塾大学数学系）
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AI总结：
本文针对Sherrington–Kirkpatrick模型，在严格de Almeida–Thouless区域内证明了重叠量的集中度，为该区域的复制对称自由能公式提供新证明并得到定量界，取代了前期预印本的相关结果。
AI中文摘要：
我们考虑逆温度β>0且确定性外场h>0的Sherrington–Kirkpatrick模型。设q为复制对称不动点，满足q=E[tanh²(h+β√q Z)]，其中Z为标准正态变量。我们证明，在严格de Almeida–Thouless区域的紧子集上一致成立：β² E[sech⁴(h+β√q Z)] <1，重叠量满足集中度：E⟨(R₁₂−q)²⟩=O(N⁻¹)。本文结果为Lopatto近期在de Almeida–Thouless区域建立的复制对称自由能公式提供了另一种证明，且我们的方法得出了更强的明确定量界结论。据此，我们得到O(N⁻¹)的复制对称自由能修正并确定了有限体积下的replicon susceptibility。我们的证明是自包含的，未使用极限自由能与Parisi变分公式的对应关系。本文主要结果取代了我们近期预印本arXiv:2607.23427中的对应结果，将复制对称界扩展至严格de Almeida–Thouless区域；但我们保留该预印本，因其论证与本文不同且本质上更简单。
英文摘要：
We consider the Sherrington--Kirkpatrick model with inverse temperature $β>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb E{\rm tanh}^2 (h+β\sqrt q\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: $β^2\mathbb E{\rm sech}^4(h+β\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \mathbb E\langle (R_{12}-q)^2\rangle=O(N^{-1}). $$ As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida--Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.
