Fluctuations of recentered maxima of discrete Gaussian Free Fields on a class of recurrent graphs

TL;DR

This paper investigates the fluctuations of the recentered maximum of Gaussian free fields on certain recurrent graphs, revealing non-tightness and fluctuation behavior similar to Z, and covering a broad class of fractal graphs.

Contribution

It establishes conditions under which the recentered maximum of Gaussian free fields on recurrent graphs fluctuates similarly to Z, extending understanding to fractal graph classes.

Findings

Recentered maximum fluctuates at the same order as the maximum at the point of highest variance.

On these graphs, the recentered maximum is not tight, unlike in Z^2.

Conditions apply to a large class of fractal graphs.

Abstract

We provide conditions that ensure that the recentered maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal variance. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in Z but in contrast with the situation in Z^2. We show that our conditions cover a large class of "fractal" graphs.