Electronic Journal of Probability

On the overlap distribution of Branching Random Walks

Aukosh Jagannath

Full-text: Open access

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
Received: 12 April 2016
Accepted: 10 July 2016
First available in Project Euclid: 4 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1470316405

Digital Object Identifier
doi:10.1214/16-EJP3

Mathematical Reviews number (MathSciNet)
MR3539644

Zentralblatt MATH identifier
1345.60100

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 82D60: Polymers

Keywords
Branching Random Walk Ghirlanda-Guerra identities spin glasses

Rights
Creative Commons Attribution 4.0 International License.

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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