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 λ c ( μ ) is finite. Consequently, the group invariant λ c ( G ) = inf μ λ c ( μ ) is well-defined, where the infimum is taken over all generating measures μ .