A localisation phase transition for the catalytic branching random walk

Abstract

We show the existence of a phase transition between a localisation and a nonlocalisation regime for a branching random walk with a catalyst at the origin. More precisely, we consider a continuous-time branching random walk that jumps at rate one, with simple random walk jumps on Zd, and that branches (with binary branching) at rate λ>0 everywhere, except at the origin, where it branches at rate λ0>λ. We show that, if λ0 is large enough, then the occupation measure of the branching random walk localises (i.e., when normalised by the total number of particles, it converges almost surely without spatial renormalisation), whereas, if λ0 is close enough to λ, then the occupation measure delocalises, in the sense that the proportion of particles in any finite given set converges almost surely to zero. The case λ=0 (when branching only occurs at the origin) has been extensively studied in the literature and a transition between localisation and nonlocalisation was also exhibited in this case. Interestingly, the transition that we observe, conjecture, and partially prove in this paper occurs at the same threshold as in the case λ=0. One of the strengths of our result is that, in the localisation regime, we are able to prove convergence of the occupation measure, while existing results in the case λ=0 give convergence of moments instead.