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Random walk on a discrete torus and random interlacements. (English) Zbl 1187.60089
Summary: We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus \((\mathbb{Z}/N\mathbb{Z})^{d}\), \(d \geq 3\), until \(u N^{d}\) time steps, \(u > 0\), and the model of random interlacements recently introduced by A.S. Sznitman [Vacant set of random interlacements and percolation, preprint arXiv:0704.2560; cf. (2001; Zbl 1025.60046)]. In particular, we show that for large \(N\), the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time \(u N^{d}\) converges to independent copies of the random interlacement at level \(u\).

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