Abstract: In this talk, I will present a new proof of the sharpness of the phase transition for Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on , with a parameter controlling the intensity of the Poisson point process. The model defines a percolation on the edges of with long-range correlations.
We consider the time constant associated to the chemical distance in random interlacements at low intensity . For dimension , it is conjectured that converges to the Euclidean norm as u ↓ 0. In this high-dimensional case, we prove a sharp upper bound (of order ) and an almost sharp lower bound (of order ) for the time constant as the intensity decays to zero.