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

Kainan Xiang and Lang Zou

Statistics & Probability Letters, 2020, vol. 165, issue C

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.

Keywords: Percolation; Random graph; Countable infinite group; Phase transition (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spl.2020.108827

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