Communications in Mathematical Physics

, Volume 226, Issue 3, pp 531–558

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

  • Noam Berger

DOI: 10.1007/s002200200617

Cite this article as:
Berger, N. Commun. Math. Phys. (2002) 226: 531. doi:10.1007/s002200200617

Abstract:

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability \(\). We show that if d<s<2d, then the walk is transient, and if s≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d≥ 1, if d>s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Noam Berger
    • 1
  1. 1.Department of Statistics, The University of California, Berkeley, CA 94720-3860, USA.¶E-mail: noam@stat.berkeley.eduUS