Extremes of the two-dimensional Gaussian free field with scale-dependent variance

TL;DR

This paper investigates a modified two-dimensional Gaussian free field where the variance varies with scale, deriving key properties like the maximum and high points, and introduces a streamlined proof technique with potential for broader applications.

Contribution

It introduces a new construction of the 2D GFF with scale-dependent variance and computes the maximum and high points, extending previous constant-variance results.

Findings

Computed the first order of the maximum of the field

Determined the log-number of high points

Streamlined the proof using a truncated second moment method

Abstract

In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.