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 ¸ 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 · 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. Key words and phrases: branching random walk, random environment, localization, phase transition. MSC 2000 subject classifications. Primary 60K37; secondary 60J80, 60K35, 82D30.