Level-set percolation of the Gaussian free field on regular graphs I: Regular trees

TL;DR

This paper investigates the percolation properties of the Gaussian free field on infinite regular trees, establishing critical thresholds, growth behaviors, and continuity of percolation probability, with implications for spectral analysis.

Contribution

It provides new estimates on component sizes, growth rates, and the continuity of percolation probability for the Gaussian free field on regular trees, including spectral bounds.

Findings

Exponential growth of connected components below critical level

Continuity of percolation probability away from critical point

Matching bounds on eigenfunctions related to spectral characterization

Abstract

We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_\star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_\star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_\star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].