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Local percolative properties of the vacant set of random interlacements with small intensity. (English. French summary) Zbl 1319.60180
Summary: Random interlacements at level \(u\) is a one parameter family of connected random subsets of \(\mathbb{Z}^{d}\), \(d\geq3\) [A.-S. Sznitman, Ann. Math. (2) 171, No. 3, 2039–2087 (2010; Zbl 1202.60160)]. Its complement, the vacant set at level \(u\), exhibits a non-trivial percolation phase transition in \(u\) (see [loc. cit.; V. Sidoravicius and A.-S. Sznitman, Commun. Pure Appl. Math. 62, No. 6, 831–858 (2009; Zbl 1168.60036)]), and the infinite connected component, when it exists, is almost surely unique [A. Teixeira, Ann. Appl. Probab. 19, No. 1, 454–466 (2009; Zbl 1158.60046)]. {
} In this paper we study local percolative properties of the vacant set of random interlacements at level \(u\) for all dimensions \(d\geq 3\) and small intensity parameter \(u>0\). We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level \(u\). In particular, this implies that finite connected components of the vacant set at level \(u\) are unlikely to be large. These results are new for \(d\in \{3,4\}\). The case of \(d\geq 5\) was treated in [A. Teixeira, Probab. Theory Relat. Fields 150, No. 3–4, 529–574 (2011; Zbl 1231.60117)] by a method that crucially relies on a certain “sausage decomposition” of the trace of a high-dimensional bi-infinite random walk. Our approach is independent from that of Teixeira [Zbl 1231.60117]. It only exploits basic properties of random walks, such as Green function estimates and Markov property, and, as a result, applies also to the more challenging low-dimensional cases. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
82B43 Percolation
Full Text: DOI Euclid
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