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In this joint work with Bruno Schapira, 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 Z^d, 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. converges almost surely without spatial renormalisation), whereas, if λ0 is close enough to λ, then localisation cannot occur, at least not in a strong sense.
