Coupling and an application to level-set percolation of the Gaussian free field

TL;DR

This paper refines the coupling between the Gaussian free field and random interlacements, applying it to level-set percolation on regular trees to derive bounds on the critical percolation threshold.

Contribution

It provides a refined coupling construction and applies it to establish bounds on the critical level for percolation in Gaussian free fields on regular trees.

Findings

Established bounds on the critical level h_* for percolation.

Derived the inequality 0 < h_* < √(2u_*).

Extended understanding of level-set percolation in Gaussian free fields.

Abstract

We consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in arXiv:1402.0298. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*}$, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.