Extrema of the two-dimensional Discrete Gaussian Free Field

TL;DR

This paper introduces new proofs and insights into the behavior of the two-dimensional Discrete Gaussian Free Field, focusing on the scaling limits of its level sets and maximum, with connections to recent joint research.

Contribution

It provides new proofs for the tightness and distributional convergence of the DGFF maximum without relying on the modified Branching Random Walk, and discusses scaling limits of level sets.

Findings

New proofs of maximum tightness and convergence

Analysis of level set scaling limits

Connections to recent joint research

Abstract

These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text is based on recent joint papers with O. Louidor and with J. Ding and S. Goswami. Still, new proofs of the tightness and distributional convergence of the centered DGFF maximum are presented that by-pass the use of the modified Branching Random Walk. The text contains a wealth of instructive exercises and a list of open questions and conjectures for future research.