New York Journal of Mathematics
Volume 12 (2006) 1-18


Yuval Peres, Gábor Pete, and Ariel Scolnicov

Critical percolation on certain nonunimodular graphs

Published: February 25, 2006
Keywords: Critical percolation, nonunimodular, nonamenable, Diestel-Leader graphs, grandmother graph, lamplighter group, decay of connection probability
Subject: 60B, 60K, 82B, 20F

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z2 and Zd for large d, as well as for all nonamenable graphs with unimodular automorphism groups. We show that the conjecture holds for the basic classes of nonamenable graphs with nonunimodular automorphism groups: for decorated trees and the nonunimodular Diestel-Leader graphs. We also show that the connection probability between two vertices decays exponentially in their distance. Finally, we prove that critical percolation on the positive part of the lamplighter group has no infinite clusters.


Our work was partially supported by the NSF grant DMS-0244479 (Peres, Pete), and OTKA (Hungarian National Foundation for Scientific Research) grants T30074 and T049398 (Pete).

Author information

Yuval Peres:
Departments of Statistics and Mathematics, 367 Evans Hall, University of California, Berkeley, CA 94720

Gábor Pete:
Department of Statistics, 367 Evans Hall, University of California, Berkeley, CA 94720

Ariel Scolnicov:
VPN Products, Check Point Software Technologies, 44 Betzal'el Ramat Gan, ISRAEL 52521