Extremes of the zero-average Gaussian Free Field on random regular graphs

TL;DR

This paper investigates the extreme value behavior of the zero-average Gaussian free field on large random regular graphs and trees, showing it converges to a Poisson point process with exponential intensity.

Contribution

It establishes the asymptotic distribution of the extremal points of the GFF on random regular graphs and trees, using Gaussian comparison and Green function estimates.

Findings

Extremal point process converges to a Poisson process with exponential intensity.

Results hold for diverging size graphs and finite subsets of vertices.

Method relies on Gaussian comparison and Green function estimates.

Abstract

We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random $r$-regular graphs and the Gaussian free field on $r$-regular trees. For random $r$-regular graphs of diverging size, for every fixed $r\ge3$, we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity $e^{-x}\,\mathrm{d}x$. The same limit behaviour is obeyed by the restriction of the GFF on $r$-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.