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

Considering G as a countable infinite group and mu a symmetric probability on it whose support generates G, and calling mu a generating measure of G. Here, we prove that for some probability mu, group G admits a long-range percolation phase transition in which the corresponding percolation threshold lambda(c)(mu) is finite. Consequently, the group invariant lambda(mu)(G) = inf(mu) lambda(c)(mu) is well-defined, where the infimum is taken over all generating measures mu (C) 2020 Elsevier B.V. All rights reserved.