On the boundary at infinity for branching random walk

TL;DR

This paper proves that supercritical branching random walk on a transient graph converges to a random measure on the Martin boundary, providing new insights into its asymptotic behavior and posing open problems.

Contribution

It establishes almost sure convergence of the branching random walk to a measure on the Martin boundary, a novel result in this context.

Findings

Convergence of branching random walk to a measure on the Martin boundary

Identification of properties of the limiting measure

Presentation of open problems and conjectures

Abstract

We prove that supercritical branching random walk on a transient graph converges almost surely under rescaling to a random measure on the Martin boundary of the graph. Several open problems and conjectures about this limiting measure are presented.