Tightness of the recentered maximum of log-correlated Gaussian fields

TL;DR

This paper proves the tightness of the recentered maximum of log-correlated Gaussian fields, providing tail bounds, and applies these results to the two-dimensional continuous Gaussian free field.

Contribution

It establishes tightness and tail bounds for maxima of log-correlated Gaussian fields, extending to the Gaussian free field in two dimensions.

Findings

Tightness of the recentered maximum is proven.

Exponential tail bounds are derived.

Results apply to the 2D Gaussian free field.

Abstract

We consider a family of centered Gaussian fields on the d-dimensional unit box, whose covariance decreases logarithmically in the distance between points. We prove tightness of the recentered maximum of the Gaussian fields and provide exponentially decaying bounds on the right and left tails. We then apply this result to a version of the two-dimensional continuous Gaussian free field.