A localisation phase transition for the catalytic branching random walk

Abstract

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 $\mathbb Z^d$, and that branches (with binary branching) at rate $\lambda>0$ everywhere, except at the origin, where it branches at rate $\lambda_0>\lambda$. We show that, if $\lambda_0$ 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 $\lambda_0$ is close enough to $\lambda$, 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 $\lambda = 0$ (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 $\lambda=0$. 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 $\lambda = 0$ give convergence of moments instead.