The sign clusters of the massless Gaussian free field percolate on $\mathbb{Z}^d$, $d \geqslant 3$ (and more)

TL;DR

This paper proves that the sign clusters of the Gaussian free field on high-dimensional integer lattices percolate for all positive levels, establishing the positivity of the critical parameter and connecting it with random interlacements.

Contribution

It establishes the positivity of the critical level for percolation of Gaussian free field sign clusters on $ abla^d$, and links this phenomenon to random interlacements and isomorphism theorems.

Findings

Critical parameter $h_*(d)$ is strictly positive for all $d geq 3$.

Sign clusters of the Gaussian free field percolate on $ abla^d$, for all $d geq 3$.

Construction involves random interlacements and a Dynkin-type isomorphism theorem.

Abstract

We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.