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On the fragmentation of a torus by random walk. (English) Zbl 1235.60143
One considers a random walk on a \(d\)-dimensional torus with large side length \(N\) and fixed dimension \(d\) larger than 3. The main purpose of the paper is to exhibit some properties of the corresponding vacant set, which loosely speaking, refers to the set of vertices which are not visited by the random walk before a given prior selected time proportional to \(N^d\). Various results are stated, and mainly one solves some open problems stated by Benjamini and Sznitman related to small parameter regime in high dimension. Mainly one proves the existence of two distinct phases for the vacant set, and the mathematical background refers to a coupling of the random walk with random interlacement on \(X^d\)

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
05C81 Random walks on graphs
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