Level set percolation for random interlacements and the Gaussian free field

We consider continuous-time random interlacements on ZdZd, d≥3d≥3, and investigate the percolation model where a site xx of ZdZd is occupied if the total amount of time spent at xx by all the trajectories of the interlacement at level u≥0u≥0 exceeds some constant α≥0α≥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 ZdZd, d≥3d≥3, about which new insights of independent interest are also gained.