Non-uniqueness Phase of Percolation on Reflection Groups in $${\mathbb {H}^3}$$

Abstract

We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space \({\mathbb {H}^3}\) corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that \(p_c < p_u\). This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.