Electronic Journal of Probability

On the overlap distribution of Branching Random Walks

Aukosh Jagannath

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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

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

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

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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

Branching Random Walk Ghirlanda-Guerra identities spin glasses

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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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