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

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 $h_ > 0$.