AI 中文总结
研究高斯最大值的两个问题，一是给出特定条件下概率上界，回答相关问题且指数可由等相关高斯场达到；二是从熵角度证明临界 SK 模型自由能涨落方差为\(\frac{1}{6}\log N+O(1)\)，给出不同于他人的证明方法。
AI 中文摘要
我们研究了高斯最大值的两个问题。首先，设\((X_1,\ldots,X_N)\)是一个中心高斯向量，\(\operatorname{Var}(X_i)\leq 1\)。对于固定的\(\alpha\in(0,\sqrt{2})\)和\(\kappa>0\)，若\(\mathbb{E}\max_i X_i\geq\alpha\sqrt{\log N}\)且\(\mathbb{E}\max_i X_i+\kappa\sqrt{\log N}\leq\sqrt{2\log N}\)，则\(\mathbb{P}\left(\max_i X_i\geq \mathbb{E}\max_i X_i+\kappa\sqrt{\log N}\right) \leq N^{-\kappa^2/(2-\alpha^2)+o(1)}\)，回答了 Ding、Eldan 和 Zhai 的问题，该指数由等相关高斯场达到。其次，对于临界逆温度\(\beta_c = 1/\sqrt{2}\)的 Sherrington - Kirkpatrick 模型，我们证明\(\operatorname{Var}\bigl(F_N(\beta_c)\bigr)=\frac{1}{6}\log N+O(1)\)。我们的证明从熵的角度给出了临界温度下方差渐近性的另一种证明，独立于 Du 和 Huang 的方法。对于上界，我们将方差表示为指数倾斜下的熵，并将此熵与高斯同步模型的 Kullback - Leibler 散度等同起来，然后使用 I - MMSE 公式、信息渗流和临界 Erdős - Rényi 随机图的磁化率估计来界定其导数。对于下界，我们将在副本参数处应用的高斯凸性与球面上逆矩的估计以及连续维度中 GOE 特征值密度的恒等式相结合。
英文摘要
In this paper, we establish a moderate deviation bound for Gaussian maxima and the variance asymptotics of the Sherrington-Kirkpatrick free energy at criticality based on Gaussian convexity. First, let $(X_1,\ldots,X_N)$ be centered Gaussian vector with $\operatorname{Var}(X_i)\leq 1$. Suppose that, for fixed $α\in(0,\sqrt 2)$ and $κ>0$, $\mathbb{E}\max_iX_i\geqα\sqrt{\log N}$ and $\mathbb{E}\max_iX_i+κ\sqrt{\log N}\leq\sqrt{2\log N}$. We prove that $$ \mathbb{P}\left(\max_iX_i\geq \mathbb{E}\max_iX_i+κ\sqrt{\log N}\right) \leq N^{-κ^2/(2-α^2)+o(1)}. $$ This answers a question of Ding, Eldan and Zhai. The exponent is sharp, as witnessed by an equicorrelated Gaussian field. Second, for the Sherrington--Kirkpatrick model at the critical inverse temperature $β_c=1/\sqrt2$, we prove $$ \operatorname{Var}\bigl(F_N(β_c)\bigr)=\frac16\log N+O(1). $$ Our argument provides the variance asymptotics at the critical temperature from an entropy perspective, via a route distinct from that of Du and Huang. For the upper bound, we express the variance as an entropy under exponential tilting and identify this entropy with the Kullback--Leibler divergence of a Gaussian synchronization model. Its derivative is then bounded using the I-MMSE formula, information percolation, and estimates for the susceptibility of the critical Erdős--R'enyi random graph. For the lower bound, we combine Gaussian convexity applied at the replica parameter with an estimate for inverse moments on the sphere and an identity relating GOE eigenvalue densities in consecutive dimensions.
