Every countable infinite group admits a long range percolation with a phase transition

Abstract

Considering G as a countable infinite group and μ a symmetric probability on it whose support generates G, and calling μ a generating measure of G. Here, we prove that for some probability μ, group G admits a long-range percolation phase transition in which the corresponding percolation threshold λc(μ) is finite. Consequently, the group invariant λc(G)=infμλc(μ) is well-defined, where the infimum is taken over all generating measures μ.