Probability Seminar: Balazs Rath

The random interlacements (at level u) is a one parameter family of random subsets of Z^d (d>=3), introduced recently by A.-S. Sznitman, which arises as the local limit of the trace of a simple random walk on a d-dimensional torus, when the size of the torus goes to infinity. The parameter u controls the density of the interlacement. The vacant set at level u (i.e. the complement set of the random interlacement at level u) undergoes non-trivial percolation phase transition as u varies. We study the supercritical phase and show that finite connected components of the vacant set are "small" for all d>=3 if u is small enough. Our method is markedly different from that of A. Teixeira (2011), which gives the analogous result for d>=5.