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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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G50 Sums of independent random variables; random walks 82B43 Percolation
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##### References:
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