The maximum of log-correlated Gaussian fields in random environments

TL;DR

This paper investigates the maximum distribution of a broad class of Gaussian fields with logarithmic correlations in random environments, revealing a universal asymptotic Gumbel distribution with applications to percolation clusters and conductance models.

Contribution

It generalizes previous assumptions to include fields with local defects and demonstrates the universality of the maximum's distribution in various random environments.

Findings

Centered maximum follows a randomly-shifted Gumbel distribution

Results apply to Gaussian free fields on super-critical percolation clusters

Applicable to Gaussian free fields in i.i.d. bounded conductance models

Abstract

We study the distribution of the maximum of a large class of Gaussian fields indexed by a box $V_N\subset Z^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding, Roy and Zeitouni (Annals Probab. (45) 2017, 3886-3928), we show that asymptotically, the centered maximum of the field has a randomly-shifted Gumbel distribution. We prove that the two dimensional Gaussian free field on a super-critical bond percolation cluster with $p$ close enough to $1$, as well as the Gaussian free field in i.i.d. bounded conductances, fall under the assumptions of our general theorem.