The chemical distance in random interlacements in the low-intensity regime

Abstract

In Z d with d ≥ 5, we consider the time constant ρ u associated to the chemical distance in random interlacements at low intensity u ≪ 1. We prove an upper bound of order u −1/2 and a lower bound of order u −1/2+ε . The upper bound agrees with the conjectured scale in which u 1/2 ρ u converges to a constant multiple of the Euclidean norm, as u → 0. Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as u → 0; these bounds can be relevant in future studies of the low-intensity geometry.

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

“…This problem is, in turn, closely related to the problem of computing the asymptotics of the time constant for supercritical percolation as p ↓ p c . An analogous problem for high-dimensional random interlacements has recently been solved to within subpolynomial factors in [17], and regularity results for the percolation time constant have been established in [6,9,10].…”