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Published October 2022 | public
Journal Article

Slightly supercritical percolation on non-amenable graphs I: The distribution of finite clusters


We study the distribution of finite clusters in slightly supercritical (p↓pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc0 such that Pp(n≤|K|<∞)≍n−1/2exp[−Θ(|p−pc|2n)] and Pp(r≤Rad(K)<∞)≍r−1exp[−Θ(|p−pc|r)] for every p∈(pc−δ,pc+δ) and n,r≥1, where all implicit constants depend only on G. We deduce in particular that the critical exponents γ′ and Δ′ describing the rate of growth of the moments of a finite cluster as p↓pc take their mean-field values of 1 and 2 respectively. These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on ℤd even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.

Additional Information

We thank Jonathan Hermon and Asaf Nachmias for many helpful discussions, and thank Remco van der Hofstad for helpful comments on an earlier version of this manuscript. We also thank Antoine Godin for sharing his simplified proof of Proposition 3.1 with us.

Additional details

August 22, 2023
October 24, 2023