Non-uniqueness phase of Bernoulli percolation on reflection groups for some polyhedra in H^3

TL;DR

This paper demonstrates the existence of a non-uniqueness phase in Bernoulli percolation on Cayley graphs of reflection groups derived from certain hyperbolic polyhedra, expanding understanding of percolation phenomena in hyperbolic geometry.

Contribution

It proves the non-trivial non-uniqueness phase (p_c < p_u) for specific classes of hyperbolic Coxeter polyhedra Cayley graphs, a novel result in hyperbolic percolation theory.

Findings

Non-uniqueness phase exists for polyhedra with at least 13 faces.

Non-uniqueness phase exists for compact right-angled polyhedra.

Established bounds on growth rates of Cayley graphs and cycle sequences.

Abstract

In the present paper I consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in 3-dimensional hyperbolic space H^3, with standard sets of generators. As the main result, I prove the existence of non-trivial non-uniqueness phase of bond and site Bernoulli percolation on such graphs, i.e. that p_c < p_u, for two classes of such polyhedra: * for any k-hedra as above with k at least 13; * for any compact right-angled polyhedra as above. I also establish a natural lower bound for the growth rate of such Cayley graphs (when the number of faces of the polyhedron is at least 6; see thm. 5.2) and an upper bound for the growth rate of the sequence (#{simple cycles of length n through o})_n for a regular graph of degree at least 2 with a distinguished vertex o, depending on its spectral radius (see thm. 5.1 and rem. 2.3), both used to prove the main result.