Random interlacements and stochastic dimension
We consider the random interlacements process on $Z^d$, introduced by Alain-Sol Sznitman. This process is a Poisson point process on the space of doubly infinite nearest neighbor trajectories modulo time shift on $Z^d$. The intensity measure of the Poisson process essentially make the trajectories look like infinite double sided simple random walk paths. We use ideas from stochastic dimension theory developed by Benjamini, Kesten, Peres and Schramm to prove the following: Given that two vertices $x,y$ belong to the interlacement set, it is possible to find a path between $x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most $\lceil d/2 \rceil-1$ trajectories. The talk is based on joint work with E. Procaccia.