Critical percolation on any quasi-transitive graph of exponential growth\n has no infinite clusters

2016/05/17 by Tom Hutchcroft, Hutchcroft, Tom · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1605.05301

openalex publication_date 2016/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that critical percolation on any quasi-transitive graph of\nexponential volume growth does not have a unique infinite cluster. This allows\nus to deduce from earlier results that critical percolation on any graph in\nthis class does not have any infinite clusters. The result is new when the\ngraph in question is either amenable or nonunimodular.\n

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• Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry

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