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

Abstract

In the present article 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 [9]. We apply our results to level-set percolation of the Gaussian free field on a (d+1)-regular tree, when d≥2, and derive bounds on the critical value h∗. In particular, we show that 0<h∗<√2u∗, where u∗ denotes the critical level for the percolation of the vacant set of random interlacements on a (d+1)-regular tree.