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, dBranching random walk Random environment Localization Phase transition