Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field

TL;DR

This paper investigates the maximum of the 2D discrete Gaussian free field, establishing tightness along subsequences and advancing understanding of its probabilistic behavior using branching random walk techniques.

Contribution

It proves the existence of a dense subsequence where the maximum is tight, contributing new insights into Gaussian free field extremal behavior.

Findings

Existence of a dense subsequence with tight maximum

Application of branching random walk methods to Gaussian fields

Progress towards proving tightness without subsequences

Abstract

We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight; this still leaves open the conjecture that tightness holds without the need for subsequences. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.