Sherrington--Kirkpatrick模型的临界窗口涨落与无序普适性

中文摘要 AI 辅助

我们建立了Ising Sherrington--Kirkpatrick模型在其临界窗口非零部分的自由能波动极限。对于固定的$b\ne0$和$\beta_N=1+bN^{-1/3}\sqrt{\log N}$，我们的主要高斯正交系综（GOE）结果是\\[ \sqrt{\frac6{\log N}}\left(F_{N,\beta_N}-N\\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\\,b_+TW_1, \\]其中$G$为标准高斯分布，$TW_1$服从实Tracy--Widom律，且$G$与$TW_1$独立，$\mathrm{FE}$表示球面Sherrington--Kirkpatrick自由能的极限。此外，我们证明在中等超临界区域\\[ \frac{2}{N^{1/3}(\beta_N-1)}\left(F_{N,\beta_N}-N\\,\mathrm{FE}(\beta_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \\]我们还证明上述结果对于独立的、不一定同分布的、其前三阶矩匹配高斯分布且第四阶矩满足平均有界条件的无序矩阵仍然成立。

英文摘要

We establish free-energy fluctuation limits for the Ising Sherrington--Kirkpatrick model in the nonzero parts of its critical window. For fixed $b\ne0$ and $β_N=1+bN^{-1/3}\sqrt{\log N}$, our main Gaussian orthogonal ensemble (GOE) result is \[ \sqrt{\frac6{\log N}}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}G+\sqrt{\frac32}\,b_+TW_1, \] where $G$ is standard Gaussian, $TW_1$ has the real Tracy--Widom law, and $G$ is independent of $TW_1$, and $\mathrm{FE}$ denotes the limit of spherical Sherrington--Kirkpatrick free-energy. Additionally, we show that in a moderately supercritical regime \[ \frac{2}{N^{1/3}(β_N-1)}\left(F_{N,β_N}-N\,\mathrm{FE}(β_N)+\frac{\log N}{12}\right)\xrightarrow{d}TW_1. \] We also show that the above results remains valid for independent, not necessarily identically distributed, disorder matrices whose first three moments match the Gaussian law and whose fourth moments satisfy an averaged bound.

发表机构

• Southern University of Science and Technology（南方科技大学）
• Dalian University of Technology（大连理工大学）

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