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Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. (arXiv:1711.02590v1 [math.PR])
来源于:arXiv
We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs,
and more generally graphs whose automorphism group has a nonunimodular
quasi-transitive subgroup. We prove that percolation on any such graph has a
non-empty phase in which there are infinite light clusters, which implies the
existence of a non-empty phase in which there are infinitely many infinite
clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This
answers a question of Haggstrom, Peres, and Schonmann (1999), and verifies the
nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996).
We also prove that the triangle condition holds at criticality on any such
graph, which is known to imply that the critical exponents governing the
percolation probability and the cluster volume take their mean-field values.
Finally, we also prove that the susceptibility exponent, the gap exponent, and
the cluster radius exponent each take their mean-field values on any such
graph; 查看全文>>