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Logarithmic components of the vacant set for random walk on a discrete torus. (English) Zbl 1191.60118
Summary: This work continues the investigation, initiated in a recent work by I. Benjamini and A.-S. Sznitman [J. Eur. Math. Soc. (JEMS) 10, No.  1, 133–172 (2008; Zbl 1141.60057)], of percolative properties of the set of points not visited by a random walk on the discrete torus \((\mathbb Z/N\mathbb Z)^d\) up to time \(uN^d\) in high dimension \(d\). If \(u>0\) is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length \(c_0\log N\) for some constant \(c_0> 0\), and this component occupies a non-degenerate fraction of the total volume as \(N\) tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant \(c_0> 0\) is crucial in the definition of the giant component.

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
05C80 Random graphs (graph-theoretic aspects)
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