Limit theorems for the empirical distribution of supercritical branching random walks on transitive graphs

TL;DR

This paper establishes limit theorems for the empirical distribution of supercritical branching random walks on transitive graphs, including a law of large numbers and a central limit theorem, advancing understanding of their long-term behavior.

Contribution

It provides the first rigorous proof of a law of large numbers and a Stam-type central limit theorem for these processes on transitive graphs.

Findings

Law of large numbers for mean displacement

Central limit theorem for empirical distributions

Answers open questions from prior research

Abstract

We consider supercritical branching random walks on transitive graphs and we prove a law of large numbers for the mean displacement of the ensemble of particles, and a Stam-type central limit theorem for the empirical distributions, thus answering the questions from Kaimanovich-Woess [KW23, Section 6.2].