Level-set percolation for the Gaussian free field on a transient tree

TL;DR

This paper studies the percolation properties of the Gaussian free field on transient trees, comparing it with random interlacements, and establishes inequalities and conditions for critical values of percolation thresholds.

Contribution

It provides new inequalities relating critical percolation thresholds of the Gaussian free field and random interlacements on transient trees, and offers conditions for positivity of these thresholds.

Findings

Proves that h_* < sqrt(2u_*) in a broad set-up.

Provides an example where h_* = u_* = 0.

Identifies conditions under which h_* > 0.

Abstract

We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* < \sqrt{2u}_*$ in a broad enough set-up, but provide an example where $0 = h_* = u_*$ occurs. We also obtain some sufficient conditions ensuring that $h_* > 0$.