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Random walks on discrete cylinders and random interlacements. (English) Zbl 1172.60316
Summary: We explore some of the connections between the local picture left by the trace of simple random walk on a cylinder \({(\mathbb {Z} / N\mathbb {Z})^d \times \mathbb {Z}}, d \geq 2\), running for times of order \(N^{2d}\) and the model of random interlacements recently introduced in [A. Sznitman, Vacant set of random interlacements and percolation, preprint, arXiv:0704.2560, to appear in Ann. Math.]. In particular, we show that for large \(N\) in the neighborhood of a point of the cylinder with vertical component of order \(N^{d}\) the complement of the set of points visited by the walk up to times of order \(N^{2d}\) is close in distribution to the law of the vacant set of random interlacements with a level which is determined by an independent Brownian local time. The limit behavior of the joint distribution of the local pictures in the neighborhood of finitely many points is also derived.

60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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