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Localization and selection in a mean field branching random walk in a random environment. (English) Zbl 0771.60095

Authors’ abstract: We consider a continuous time branching random walk on the finite set \(\{1,2,\dots,N\}\) with totally symmetric diffusion jumps and some site-dependent i.i.d. random birth rates which are unbounded. We study this process as the time \(t\) and the space size \(N\) tend to infinity simultaneously. In the classical law of large numbers setup for spatial branching models, the growth of the population obeys an exponential limit law due to the localization of the overwhelming portion of particles in the record point of the medium. This phenomenon is analyzed further: The historical path (in space) of a typical particle picked at time \(t\) (selection) is of a rather simple and special nature and becomes in the limit singular (in distribution) to the path of the underlying mean field random walk. In general, the properties of the typical path depend on the relation in which \(t\) and \(N\) tend to infinity.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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