Windisch, David Logarithmic components of the vacant set for random walk on a discrete torus. (English) Zbl 1191.60118 Electron. J. Probab. 13, 880-897 (2008). 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. Cited in 2 Documents 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) Keywords:giant component; vacant set; random walk; discrete torus Citations:Zbl 1141.60057 PDF BibTeX XML Cite \textit{D. Windisch}, Electron. J. Probab. 13, 880--897 (2008; Zbl 1191.60118) Full Text: DOI arXiv EuDML EMIS OpenURL