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

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 $h_{\ast }$. 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.