Vacant Set of Random Interlacements and Percolation

Abstract

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the vacant set left by the interlacement at level u, which is an infinite, connected, translation invariant random subset of Z^d. We introduce a critical value such that the vacant set percolates for u below the critical value, and does not percolate for u above the critical value. Our main results show that the critical value is finite when d is bigger or equal to 3, and strictly positive when d is bigger or equal to 7.