Large deviations for occupation time profiles of random interlacements

We derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of $${\mathbb { Z}}^d$$ Z d , $$d \ge 3$$ d ≥ 3 . As an application, we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box. As a step in this program, we obtain a similar large deviation principle for the occupation-time measure of Brownian interlacements at a fixed level in a large box of $${\mathbb { R}}^d$$ R d , and we derive a new identity for the Laplace transform of the occupation-time measure, which is based on the analysis of certain Schrödinger semi-groups