This paper determines the intensity parameter for random interlacement, suggesting critical capacity thresholds for random walks in high-dimensional spaces.
The random interlacements I(u) at level u has been introduced by Sznitman, as a Poissonian collection of independent simple random walk trajectories on Zᵈ, d≥ 3, with intensity $u>0$. Since then, several works investigated the properties of the random interlacements intersected with large sets of~Zᵈ. In this paper, we study the asymptotic behavior of the capacity of I(u) ∩ DN, where DN is the blow up of a compact set D, with typical size N. We determine the correct window (uN)N≥ 1 of the intensity parameter for which the capacity cap(I(uN)∩ DN) starts to become negligible compared to cap(DN); this roughly means that a random walk starting from far away starts to see through I(uN)∩ DN. In the same spirit, we investigate the capacity of the simple random walk conditioned to stay in a large Euclidean ball up to time tN, and find similar asymptotics by taking tN = uN Nᵈ.
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Nicolas Bouchot (2025) studied this question.
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