A localisation phase transition for the catalytic branching random walk

We show the existence of a phase transition between a localisation and a non-localisation 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 > 0 everywhere, except at the origin, where it branches at rate 0 > . We show that, if 0 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 0 is close enough to , 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 =0(when branching only occurs at the origin) has been extensively studied in the literature and a transition between localisation and non-localisation 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 =0. One of the strengths of our result is that, in the localisation regime, we are able to prove convergence of the occupation measure, whilst existing results in the case = 0 give convergence of moments instead.