Extremal process of the zero-average Gaussian Free Field for $d\ge 3$

TL;DR

This paper proves that the extremal points of the zero-average Gaussian Free Field in dimensions three and higher form a Poisson process, with the maximum following a Gumbel distribution after proper normalization.

Contribution

It establishes the weak convergence of the extremal process of the zero-average Gaussian Free Field to a Poisson random measure in dimensions d≥3.

Findings

Extremal point process converges to a Poisson random measure.

Maxima of the field follow a Gumbel distribution after normalization.

Results hold for dimensions d≥3.

Abstract

We consider the Gaussian free field on the torus whose covariance kernel is given by the zero-average Green's function. We show that for dimension $d\ge 3$, the extremal point process associated with this field converges weakly to a Poisson random measure. As an immediate corollary, the maxima of the field converges after appropriate centering and scaling to the Gumbel distribution.