The model of random interlacements is a one-parameter family of random subsets of $\Z^d$, which locally describes the trace of a simple random walk on a $d$-dimensional torus running up to time $u$ times its volume. Here, $u$ serves as an intensity parameter.
Its complement, the so-called vacant set, has been show to undergo a non-trivial percolation phase transition in $u$, i.e., there is $u_*(d)\in (0,\infty)$ such that for all $u<u_*(d)$ the vacant set has a unique infinite connected component (supercritical phase), while for $u>u_*(d)$ all connected components are finite.
So far all results regarding geometric properties of this infinite connected component have been proven under the assumption that $u$ is close to zero.
I will discuss a recent result, which states that throughout most of the supercritical phase simple random walk on the infinite connected component is transient, provided that the dimension is high enough.
This is joint work with Alexander Drewitz