Transience of the vacant set for near-critical random interlacements in high dimensions

Original · EN

The model of random interlacements is a one-parameter family Iᵘ, u ≥ 0, of random subsets of Zᵈ, which locally describes the trace of simple random walk on a d-dimensional torus run up to time u times its volume. Its complement, the so-called vacant set Vᵘ, has been shown to undergo a non-trivial percolation phase-transition in u; i.e., there exists u*(d) ∈ (0, ∞) such that for u ∈ [0, u*(d)) the vacant set Vᵘ contains a unique infinite connected component V∞ᵘ, while for u > u*(d) it consists of finite connected components. Sznitman SZ11,SZ11B showed that u*(d) d, and in this article we show the existence of u(d) > 0 with u(d)/u*(d) → 1 as d → ∞ such that V∞ᵘ is transient for all u ∈ [0, u(d)).