Ends of branching random walks on planar hyperbolic Cayley graphs

TL;DR

This paper proves that the trace of a transient branching random walk on a planar hyperbolic Cayley graph almost surely has infinitely many ends and no isolated end, revealing complex boundary behavior.

Contribution

It establishes the almost sure topological structure of the trace of branching random walks on planar hyperbolic Cayley graphs, a novel result in geometric group theory and probability.

Findings

Trace has continuum many ends

Trace has no isolated end

Results apply to transient branching random walks on hyperbolic groups

Abstract

We prove that the trace of a transient branching random walk on a planar hyperbolic Cayley graph has a.s. continuum many ends and no isolated end.