On multiple peaks and moderate deviations for supremum of Gaussian field

TL;DR

This paper improves understanding of the extreme values of Gaussian fields by establishing bounds on multiple peaks and moderate deviations, with applications to spin glasses and polymers, advancing theoretical insights into Gaussian process behavior.

Contribution

It enhances existing theorems on multiple peaks by providing exponential bounds without non-negative correlation assumptions and offers improved moderate deviation bounds for Gaussian field suprema.

Findings

Number of peaks is exponential in inverse variance parameter

Polynomially many near-orthogonal sites reach near-maximum values in models

Improved moderate deviation bounds under certain expectation conditions

Abstract

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns with the concept of multiple peaks. A theorem of Chatterjee states that when a centered Gaussian field admits the so-called superconcentration property, it typically attains values near its maximum on multiple near-orthogonal sites, known as multiple peaks. We improve his theorem in two aspects: (i) the number of peaks attained by our bound is of the order $\exp(c / \sigma^2)$ (as opposed to Chatterjee's polynomial bound in $1/\sigma$), where $\sigma$ is the standard deviation of the supremum of the Gaussian field, which is assumed to have variance at most $1$ and (ii) our bound need not assume that the correlations are non-negative. We also prove a similar result based on the superconcentration of the free energy. As primary applications, we infer that for the S-K spin glass model on the $n$-hypercube and directed polymers on $\mathbb{Z}_n^2$, there are polynomially (in $n$) many near-orthogonal sites that achieve values near their respective maxima. Our second theorem gives an upper bound on moderate deviation for the supremum of a general Gaussian field. While the Gaussian isoperimetric inequality implies a sub-Gaussian concentration bound for the supremum, we show that the exponent in that bound can be improved under the assumption that the expectation of the supremum is of the same order as that of the independent case.