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...

We consider a one-dimensional random walk which is conditioned to stay non-negative and is "weakly pinned" to zero. This model is known to exhibit a phase transition as the strength of the weak pinning varies. We prove path space limit theorems which describe the macroscopic shape of the path...