Percolation in the hyperbolic space: non-uniqueness phase and fibrous clusters

TL;DR

This thesis investigates Bernoulli percolation on hyperbolic space graphs, revealing a non-uniqueness phase and analyzing geometric properties of clusters across various dimensions.

Contribution

It proves the existence of a non-trivial non-uniqueness phase for certain hyperbolic graphs and explores geometric cluster properties in hyperbolic percolation.

Findings

Existence of non-uniqueness phase in hyperbolic percolation models

Characterization of cluster geometry in the non-uniqueness phase

Applicability to a broad class of hyperbolic graphs

Abstract

In this thesis, I am going to consider Bernoulli percolation on graphs admitting vertex-transitive actions of groups of isometries of d-dimensional hyperbolic spaces H^d. In the first chapter, I give an overview concerning percolation and its terminology. In the separate introduction to each of the other chapters, I explain its contents more precisely, giving also some preliminaries needed therein. In the second chapter, I prove the existence of a non-trivial non-uniqueness phase of Bernoulli percolation on Cayley graphs for a wide class of Coxeter reflection groups of finite type polyhedra in H^3. In the third chapter, I consider some geometric property of the clusters in Bernoulli bond percolation in the non-uniqueness phase on a class of connected, transitive, locally finite graphs in H^d, for any d.