Convergence of the centered maximum of log-correlated Gaussian fields

TL;DR

This paper proves the convergence in distribution of the centered maximum of log-correlated Gaussian fields to a randomly shifted Gumbel distribution, under certain covariance convergence conditions, with implications for related probabilistic models.

Contribution

It establishes the convergence of the maximum of log-correlated Gaussian fields to a universal limit distribution, extending understanding of extreme values in such correlated systems.

Findings

Centered maximum converges to a randomly shifted Gumbel distribution.

Limit characterized by a derivative martingale-like sequence.

Additional structural assumptions are necessary for convergence in certain cases.

Abstract

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.