Tightness of the recentered maximum of the two-dimensional discrete Gaussian Free Field

TL;DR

This paper proves that the centered maximum of the two-dimensional discrete Gaussian free field is tight, confirming a long-standing conjecture and providing precise estimates of its expected maximum.

Contribution

It establishes the tightness of the maximum of the 2D GFF and offers a detailed evaluation of its expected maximum, advancing understanding of Gaussian field extrema.

Findings

Maximum of 2D GFF is tight when centered at its mean

Expected maximum of the GFF is precisely estimated up to an order of 1

Results extend to related Gaussian fields like the GFF on a torus

Abstract

We consider the maximum of the discrete two dimensional Gaussian free field (GFF) in a box, and prove that its maximum, centered at its mean, is tight, settling a long-standing conjecture. The proof combines a recent observation of Bolthausen, Deuschel and Zeitouni with elements from (Bramson 1978) and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed.