On the overlap distribution of Branching Random Walks
Copyright policy )On the overlap distribution of Branching Random Walks
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 $\{0,1\}$.
Final Version available at Elect. Jour. Probab
- University of Toronto Canada
- New York University United States
82D60, spin glasses, Sums of independent random variables; random walks, 82B44, overlap distribution, 82D30, Ghirlanda-Guerra identities, Probability (math.PR), Interacting random processes; statistical mechanics type models; percolation theory, 60K35, Branching processes (Galton-Watson, birth-and-death, etc.), Statistical mechanics of polymers, FOS: Mathematics, branching random walk, Branching Random Walk, Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics, Mathematics - Probability, Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
82D60, spin glasses, Sums of independent random variables; random walks, 82B44, overlap distribution, 82D30, Ghirlanda-Guerra identities, Probability (math.PR), Interacting random processes; statistical mechanics type models; percolation theory, 60K35, Branching processes (Galton-Watson, birth-and-death, etc.), Statistical mechanics of polymers, FOS: Mathematics, branching random walk, Branching Random Walk, Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics, Mathematics - Probability, Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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