Electronic Communications in Probability

Random walk on a discrete torus and random interlacements

David Windisch

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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 $uN^d$ time steps, $u > 0$, and the model of random interlacements recently introduced by Sznitman. 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 $uN^d$ converges to independent copies of the random interlacement at level $u$.

Article information

Electron. Commun. Probab., Volume 13 (2008), paper no. 14, 140-150.

Accepted: 10 March 2008
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Random walk random interlacements

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Windisch, David. Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 (2008), paper no. 14, 140--150. doi:10.1214/ECP.v13-1359. https://projecteuclid.org/euclid.ecp/1465233441

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  • D.J. Aldous. Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag, 1989.
  • D.J. Aldous and M. Brown. Inequalities for rare events in time-reversible Markov chains I. Stochastic Inequalities. M. Shaked and Y.L. Tong, ed., IMS Lecture Notes in Statistics, volume 22, 1992.
  • D.J. Aldous and J. Fill. Reversible Markov chains and random walks on graphs. http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • I. Benjamini, A.S. Sznitman. Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. (JEMS), 10(1):133-172, 2008.
  • R. Durrett. Probability: Theory and Examples. (third edition) Brooks/Cole, Belmont, 2005.
  • G.F. Lawler. Intersections of random walks. Birkhäuser, Basel, 1991.
  • L. Saloff-Coste. Lectures on finite Markov chains, volume 1665. Ecole d'Eté de Probabilités de Saint Flour, P. Bernard, ed., Lecture Notes in Mathematics, Springer, Berlin, 1997
  • P.M. Soardi. Potential Theory on Infinite Networks. Springer-Verlag, Berlin, Heidelberg, New York, 1994.
  • A.S. Sznitman. Vacant set of random interlacements and percolation, preprint, available at http://www.math.ethz.ch/u/sznitman/preprints and http://arxiv.org/abs/0704.2560.
  • A.S. Sznitman. Random walks on discrete cylinders and random interlacements, preprint, available at http://www.math.ethz.ch/u/sznitman/preprints.