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.