JournalsjemsVol. 10, No. 1pp. 133–172

Giant component and vacant set for random walk on a discrete torus

  • Alain-Sol Sznitman

    ETH Zürich, Switzerland
  • Itai Benjamini

    Weizmann Institute of Science, Rehovot, Israel
Giant component and vacant set for random walk on a discrete torus cover
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Abstract

We consider random walk on a discrete torus EE of side-length NN, in sufficiently high dimension dd. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time uNduN^d. We show that when uu is chosen small, as NN tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const logN\log N. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites NdN^d of EE, and any point of EE lies within distance NβN^\beta of this component, with β\beta an arbitrary positive number.

Cite this article

Alain-Sol Sznitman, Itai Benjamini, Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10 (2008), no. 1, pp. 133–172

DOI 10.4171/JEMS/106