On the internal distance in the interlacement set
  We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus. 
Ji\v{r}\'i \v{C}ern\'y, Serguei Popov
math.PR