On the overlap distribution of Branching Random Walks

Aukosh Jagannath

doi:10.1214/16-ejp3

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 $\{0,1\}$.

Highlights

We study the Branching Random Walk (BRW), or directed polymer, on a binary tree
Let TN be the binary tree of depth N and let {gv}v∈TN \∅ be a collection of i.i.d. standard Gaussian random variables indexed by this tree without its root
This implies a mode of convergence of Gibbs measures and the convergence of the weights of balls in support a PoissonDirichlet process, which was first proved by Barral, Rhodes, and Vargas in greater generality by different methods [8]

Summary

Introduction

We study the Branching Random Walk (BRW), or directed polymer, on a binary tree. An immediate consequence of this is that the overlap array distribution for these systems converges to a Ruelle Probability Cascade, see Corollary 3.6 This implies a mode of convergence of Gibbs measures and the convergence of the weights of balls in support a PoissonDirichlet process, which was first proved by Barral, Rhodes, and Vargas in greater generality by different methods [8]. Just as with the REM, both of these predictions can be shown to follow from standard concentration and integration-by-parts arguments provided one can show that the model is at most 1RSB and that the top of the support is at 1 when it is 1RSB To our knowledge this result is thought of as folklore in the Branching Random Walk community. As a study of the extendability of these results are not within the scope of this paper we do not examine these questions further

The support of the overlap distribution

Tilted barrier estimates

Applications of concentration and tilted barrier estimates

The Derrida-Spohn conjecture and the Ghirlanda-Guerra identities

Ghirlanda-Guerra identities