A localisation phase transition for the catalytic branching random walk
Résumé
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 continuoustime 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 $λ \geq 0$ everywhere, except at the origin, where it branches at rate $λ_0\ge λ$. We show that, if λ_0 is large enough, then the occupation measure of the branching random walk localises (i.e. converges almost surely without spatial renormalisation), whereas, if λ_0 is close enough to λ, then localisation cannot occur, at least not in a strong sense. The case λ = 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. Strikingly, the transition that we observe, conjecture, and partially prove in this paper occurs at the same threshold as in the case λ = 0.
Domaines
Probabilités [math.PR]| Origine | Fichiers produits par l'(les) auteur(s) |
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