Limiting laws of supercritical branching random walks

TL;DR

This paper explicitly characterizes the limiting behavior of supercritical branching random walks and their Gibbs measures, confirming conjectures about their distribution and the nature of their jumps.

Contribution

It provides explicit laws for the renormalized supercritical branching random walk and its Gibbs measures, validating physicists' conjectures in the discrete setting.

Findings

Law of the renormalized supercritical branching random walk is explicitly derived.

The limiting Gibbs measures follow a Poisson-Dirichlet distribution.

Spatial distribution of jumps in the limit is characterized.

Abstract

In this note, we make explicit the law of the renormalized supercritical branching random walk, giving credit to a conjecture formulated in a previous works of the authors for a continuous analog of the branching random walk. Also, in the case of a branching random walk on a homogeneous tree, we express the law of the corresponding limiting renormalized Gibbs measures, confirming, in this discrete model, conjectures formulated by physicists about the Poisson-Dirichlet nature of the jumps in the limit, and precising the conjecture by giving the spatial distribution of these jumps.