Level set percolation for random interlacements and the Gaussian free field

We consider continuous-time random interlacements on Z d , d ≥ 3 , and investigate the percolation model where a site x of Z d is occupied if the total amount of time spent at x by all the trajectories of the interlacement at level u ≥ 0 exceeds some constant α ≥ 0 , and empty otherwise. We also examine percolation properties of empty sites. A recent isomorphism theorem (Sznitman, 2012) enables us to “translate” some of the relevant questions into the language of level-set percolation for the Gaussian free field on Z d , d ≥ 3 , about which new insights of independent interest are also gained.