Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field

TL;DR

This paper characterizes the tail behavior of the maximum of a 2D discrete Gaussian free field, showing exponential decay on the right and double exponential decay on the left, and confirms the maximum's variance is bounded.

Contribution

It establishes precise tail decay rates for the maximum of the 2D discrete Gaussian free field, improving variance bounds and confirming a folklore conjecture.

Findings

Right tail decays exponentially

Left tail decays double exponentially

Variance of maximum is of order 1

Abstract

We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponential decay for the right tail and a double exponential decay for the left tail. In particular, our result implies that the variance of the maximum is of order 1, improving an $o(\log n)$ bound by Chatterjee (2008) and confirming a folklore conjecture. An important ingredient for our proof is a result of Bramson and Zeitouni (2010), who proved the tightness of the centered maximum together with an evaluation of the expectation up to an additive constant.