## Electronic Journal of Probability

### On the overlap distribution of Branching Random Walks

Aukosh Jagannath

#### 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 $\{0,1\}$.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 50, 16 pp.

Dates
Accepted: 10 July 2016
First available in Project Euclid: 4 August 2016

https://projecteuclid.org/euclid.ejp/1470316405

Digital Object Identifier
doi:10.1214/16-EJP3

Mathematical Reviews number (MathSciNet)
MR3539644

Zentralblatt MATH identifier
1345.60100

#### Citation

Jagannath, Aukosh. On the overlap distribution of Branching Random Walks. Electron. J. Probab. 21 (2016), paper no. 50, 16 pp. doi:10.1214/16-EJP3. https://projecteuclid.org/euclid.ejp/1470316405

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