On the overlap distribution of branching random walks

TL;DR

This paper analyzes the overlap distribution and Gibbs measure of Gaussian branching random walks, confirming replica symmetry breaking and Poisson-Dirichlet statistics, with a key result that overlaps are supported only on 0 and 1.

Contribution

It provides a rigorous proof that the branching random walk exhibits 1-step replica symmetry breaking and characterizes its overlap distribution precisely.

Findings

Overlap distribution supported on {0,1}

Gibbs measure satisfies Ghirlanda-Guerra identities

Limiting Gibbs measure has Poisson-Dirichlet statistics

Abstract

In this paper, we study the overlap distribution and Gibbs measure of the Branching Random Walk with Gaussian increments on a binary tree. We first prove that the Branching Random Walk is 1 step Replica Symmetry Breaking and give a precise form for its overlap distribution, verifying a prediction of Derrida and Spohn. We then prove that the Gibbs measure of this system satisfies the Ghirlanda-Guerra identities. As a consequence, the limiting Gibbs measure has Poisson-Dirichlet statistics. The main technical result is a proof that the overlap distribution for the Branching Random Walk is supported on the set $\{0,1\}$.