2026/07/23 by Yiming Chen
Mathematics · #math.PR
arxiv created 2026/07/29 · arxiv updated 2026/07/30
In this paper, we study two problems concerning Gaussian maxima. First, let (X1,…,XN) be a centered Gaussian vector with Var(Xi)≤ 1. Suppose that, for fixed α∈(0,√ 2) and κ>0, 𝔼maxiXi≥α√(log N) and 𝔼maxiXi+κ√(log N)≤√(2log N). We prove that ℙ(maxiXi≥ 𝔼maxiXi+κ√(log N)) ≤ N-κ2/(2-α2)+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 βc=1/√2, we prove Var(FN(βc))=\frac16log N+O(1). Our argument establishes 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--Rényi 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.