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Transience of the vacant set for near-critical random interlacements in high dimensions. (English. French summary) Zbl 1333.60202
Summary: The model of random interlacements is a one-parameter family \(\mathcal{I}^{u}\), \(u\geq0\), of random subsets of \(\mathbb{Z}^{d}\), which locally describes the trace of a simple random walk on a \(d\)-dimensional torus run up to time \(u\) times its volume. Its complement, the so-called vacant set \(\mathcal{V}^{u}\), has been shown to undergo a non-trivial percolation phase-transition in \(u\), i.e., there exists \(u_{*}(d)\in(0,\infty)\) such that for \(u\in[0,u_{*}(d))\) the vacant set \(\mathcal{V}^{u}\) contains a unique infinite connected component \(\mathcal{V}_{\infty}^{u}\), while for \(u>u_{*}(d)\) it consists of finite connected components. It is known [A.-S, Sznitman, Probab. Theory Relat. Fields 150, No. 3–4, 575–611 (2011; Zbl 1230.60103); Ann. Probab. 39, No. 1, 70–103 (2011; Zbl 1210.60047)] that \(u_{*}(d) \sim \log d\), and in this article we show the existence of \(u(d)>0\) with \(\frac{u(d)}{u_{*}(d)}\to1\) as \(d\to\infty\) such that \(\mathcal{V}_{\infty}^{u}\) is transient for all \(u\in[0,u(d))\).
MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B43 Percolation
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