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

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.