Boltzmann-Gibbs Weights in the Branching Random Walk
Considering a branching random walk as a tree model for many physical disordered systems the a.s. convergence of the free energy is proved under minimal assumption (finite mean) on the partition function. The overlap of two nodes in the tree is their last common ancestor (or the common part of their branches). Under a “k log k-type” assumption the overlap of two nodes of height n picked up with Boltzmann-Gibbs weights is proved to have an explicit limit distribution. This extends a result of Joffe and simplify a proof of Derrida and Spohn.
AMS(MOS) subject classifications. Primary: 60J80, 60K35. Secondary: 60F10, 82C41.
Key wordsBranching random walk Boltzmann-Gibbs weights disordered trees
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