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Boltzmann-Gibbs weights in the branching random walk. (English) Zbl 0866.60074

Athreya, Krishna B. (ed.) et al., Classical and modern branching processes. Proceedings of the IMA workshop, Minneapolis, MN, USA, June 13–17, 1994. New York, NY: Springer. IMA Vol. Math. Appl. 84, 41-50 (1997).
Summary: 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 “\(k\) 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 A. Joffe [Ann. Appl. Probab. 3, No. 4, 1145-1150 (1993; Zbl 0784.60081)] and simplifies a proof of B. Derrida and J. Spohn [J. Stat. Phys. 51, 817-840 (1988)].
For the entire collection see [Zbl 0855.00027].

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics

Citations:

Zbl 0784.60081
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