Localization for branching random walks in random environment
Copyright policy )Localization for branching random walks in random environment
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d \le 2$, or the environment is ``random enough", then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of "replica overlap". We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.
17 pages
- University of Paris France
- University of Paris France
- Kyoto University Japan
Statistics and Probability, random environment, FOS: Physical sciences, 60K37 (Primary), localization, Modelling and Simulation, Branching processes (Galton-Watson, birth-and-death, etc.), FOS: Mathematics, Processes in random environments, Mathematical Physics, Phase transition, Random environment, Applied Mathematics, Probability (math.PR), Interacting random processes; statistical mechanics type models; percolation theory, 60K37 (Primary); 60F05, 60J80, 60K35, 82D30 (Secondary), Mathematical Physics (math-ph), phase transition, Localization, branching random walk, Branching random walk, 60F05, 60J80, 60K35, 82D30 (Secondary), Mathematics - Probability, Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
Statistics and Probability, random environment, FOS: Physical sciences, 60K37 (Primary), localization, Modelling and Simulation, Branching processes (Galton-Watson, birth-and-death, etc.), FOS: Mathematics, Processes in random environments, Mathematical Physics, Phase transition, Random environment, Applied Mathematics, Probability (math.PR), Interacting random processes; statistical mechanics type models; percolation theory, 60K37 (Primary); 60F05, 60J80, 60K35, 82D30 (Secondary), Mathematical Physics (math-ph), phase transition, Localization, branching random walk, Branching random walk, 60F05, 60J80, 60K35, 82D30 (Secondary), Mathematics - Probability, Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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