On the size of a finite vacant cluster of random interlacements with small intensity

By Augusto Teixeira. 2010

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In this paper, we establish some properties of percolation for the vacant set of random interlacements, for d ≥ 5 and small intensity u. The model of random interlacements was first introduced by Sznitman in (Ann Math, arXiv:0704.2560, 2010). It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see Sidoravicius and Sznitman (Commun Pure Appl Math 62(6):831–858, 2009) and Teixeira (Ann Appl Probab 19(1):454–466, 2009). We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is “ubiquitous” in large neighborhoods of the origin.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/35ea451c9508eff4d92c6df896dec1bb7f98cda8

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Citations in the corpus

Soft local times and decoupling of random interlacements (2012, background): “Recently, great effort has been devoted to studying this model [18], [19], [31], [24], [25], [14], [3] as well as to establishing rigorously the relation between random interlacements and the trace left by random walks on large graphs [20], [34], [32], [4].”
On the fragmentation of a torus by random walk (2010, background): “Applying the simple Markov property at the times τl+1 and τl+1 + t∗, the probability on the right-hand side becomes ∑ x′∈T\B Ex [ Yτl = z, τl < H̃A ∧ U, PYτl+1 [HB > t∗]PYτl+1 [Yt∗ = x ′|HB > t∗] ] Px′ [YHA = y], hence by Lemma 3.9, ∣ ∣ ∣ Px[U H̃A, YH̃A = y]− Px[U H̃A]Pσ[YHA = y] ∣ ∣ ∣ ≤ e−cǫ lo”