Some percolations involving the Gaussian free fields

Nathalie Eisenbaum

doi:10.1214/21-ecp379

Abstract

Consider an infinite, connected, locally finite graph with vertex set V. Intuitively a simple point process on V with attractive properties, should percolate more easily than a Bernoulli point process with the same marginales. Although it seems wrong to imagine that it could be true in general, we confirm this intuition on several examples involving Gaussian free fields and permanental free fields.

Highlights

Consider a non-oriented, infinite, connected, locally finite, graph G, with vertex set V and edge set E
It seems wrong to imagine that it could be true in general, we confirm this intuition on several examples involving Gaussian free fields and permanental free fields
Given a family of Bernoulli variables (Yx, x ∈ V), one may ask whether the random subgraph of G with vertex set {x ∈ V : Yx = 1} and edge set {[x, y] ∈ E : Yx = 1 and Yy = 1}, contains an infinite connected component

Summary

Introduction

Consider a non-oriented, infinite, connected, locally finite, graph G, with vertex set V and edge set E. - for h > h∗, the set {x ∈ Zd : ηx > h} a.s. has only finite connected components It has been shown by Bricmont, Lebowitz and Maes [5], that h∗ ≥ 0 and that in dimension 3: h∗ < ∞. This extension has been already noticed by Abächerli and Sznitman (Proposition A2 in [1]).

Extension to all Gaussian and Permanental free fields

Critical level sets for the Gaussian free fields