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arXiv · 1809.11112

No percolation at criticality on certain groups of intermediate growth

Jonathan Hermon·Tom Hutchcroft
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Abstract

We prove that critical percolation has no infinite clusters almost surely on any unimodular quasi-transitive graph satisfying a return probability upper bound of the form $p_n(v,v) \leq \exp\left[-\Omega(n^\gamma)\right]$ for some $\gamma>1/2$. The result is new in the case that the graph is of intermediate volume growth.

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Jonathan Hermon, Tom Hutchcroft. 2018-09-28. No percolation at criticality on certain groups of intermediate growth. https://arxiv.org/abs/1809.11112

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