Random Walk on the Torus and Random Interlacements

  • Alexander Drewitz
  • Balázs Ráth
  • Artëm Sapozhnikov
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


In this chapter we consider simple random walk on the discrete torus \(\mathbb{T}_{N}^{d}:= {(\mathbb{Z}/N\mathbb{Z})}^{d}\), d ≥ 3. We prove that for every u > 0, the local limit (as N) of the set of vertices in \(\mathbb{T}_{N}^{d}\) visited by the random walk up to time uN d steps is given by random interlacements at level u.


Random Interlacements Lazy Random Walk Local Limit Walking Changes Sznitman 
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  1. [1]
    Aldous, D.: Probability approximations via the Poisson clumping heuristic. In: Applied Mathematical Sciences, vol. 77. Springer, New York (1989)Google Scholar
  2. [2]
    Aldous, D.J., Brown, M.: Inequalities for rare events in time-reversible Markov chains. II. Stoch. Process. Appl. 44(1), 15–25 (1993). DOI 10.1016/0304-4149(93)90035-3. URL
  3. [5]
    Antal, P., Pisztora, A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036–1048 (1996). DOI 10.1214/aop/1039639377. URL
  4. [18]
    Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. In: James, G., Propp, David B. (eds.) Wilson American Mathematical Society, Providence, RI (2009).Google Scholar
  5. [39]
    Sznitman, A.S.: Random walks on discrete cylinders and random interlacements. Probab. Theory Relat. Fields 145(1–2), 143–174 (2009). DOI 10.1007/s00440-008-0164-8. URL
  6. [40]
    Sznitman, A.S.: Upper bound on the disconnection time of discrete cylinders and random interlacements. Ann. Probab. 37(5), 1715–1746 (2009). DOI 10.1214/09-AOP450. URL
  7. [41]
    Sznitman, A.S.: Vacant set of random interlacements and percolation. Ann. Math. 171(3), 2039–2087 (2010). DOI 10.4007/annals.2010.171.2039. URL
  8. [52]
    Teixeira, A., Windisch, D.: On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64(12), 1599–1646 (2011). DOI 10.1002/cpa.20382. URL
  9. [53]
    Windisch, D.: Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13, 140–150 (2008). DOI 10.1214/ECP.v13-1359. URL

Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Alexander Drewitz
    • 1
  • Balázs Ráth
    • 2
  • Artëm Sapozhnikov
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew York CityUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  3. 3.Max-Planck Institute of Mathematics in the SciencesLeipzigGermany

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