Martin boundaries and asymptotic behavior of branching random walks

Let G be an infinite, locally finite graph. We investigate the relation between super- critical, transient branching random walk and the Martin boundary of its underlying random walk. We show results regarding the typical asymptotic directions taken by the particles, and as a consequence we find a new connection between t-Martin boundaries and standard Martin boundaries. Moreover, given a subgraph U we study two aspects of branching random walks on U : when the trajectories visit U infinitely often (survival) and when they stay inside U forever (persistence). We show that there are cases, when U is not connected, where the branching random walk almost surely does not survive in U , but the random walk on G converges to the boundary of U with positive probability. In contrast, the branching random walk can survive with positive probability in U even though the random walk eventually exits U almost surely. We provide several examples and counterexamples.