an:05950540
Zbl 1231.60117
Teixeira, Augusto
On the size of a finite vacant cluster of random interlacements with small intensity
EN
Probab. Theory Relat. Fields 150, No. 3-4, 529-574 (2011).
00285623
2011
j
60K35 82C41
percolation; finite vacant cluster; model of random interlacements
The paper deals with some properties of percolation for the vacant set of random interlacements on \(\mathbb Z^d\) for \(d\) greater than or equal to 5 and small intensity \(u\). The main result of the paper is a theorem, which proves a stretched exponential bound on the probability that the interlacement-set separates two macroscopic connected sets in a large cube. By applying this theorem, the author estimates the distribution of the diameter and the volume of the vacant component at level \(u\) containing the origin, given that it is finite. As another application, the author shows that with high probability, the unique infinite connected component of the vacant set is ``ubiquitous'' in large neighbourhoods of the origin.
Anatoliy Pogorui (Zhytomyr)