Clusters in middle-phase percolation on hyperbolic plane

TL;DR

This paper investigates the structure of infinite clusters in hyperbolic plane percolation, showing that in the middle phase, all ends of infinite clusters have a one-point boundary at infinity, revealing geometric properties of percolation phases.

Contribution

It proves that in the middle phase of hyperbolic percolation, all infinite cluster ends have a one-point boundary at infinity, extending understanding of cluster geometry.

Findings

Ends of infinite clusters have one-point boundary at infinity in the middle phase.

Results are similar to Lalley's findings on hyperbolic percolation.

Clarifies geometric structure of clusters in nonamenable planar graphs.

Abstract

I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known (Benjamini and Schramm) that in such a graph G we have three essential phases of percolation, i. e. 0 < p_c(G) < p_u(G) < 1, where p_c is the critical probability and p_u - the unification probability. I prove that in the middle phase a. s. all the ends of all the infinite clusters have one-point boundary in the boundary of H^2. This result is similar to some results of Lalley.