A localisation phase transition for the catalytic branching random walk
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Mailler, Cécile Schapira, Bruno |
| author_facet | Mailler, Cécile Schapira, Bruno |
| contents | 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 $λ>0$ everywhere, except at the origin, where it branches at rate $λ_0>λ$. We show that, if $λ_0$ 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 $λ_0$ 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 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 $λ=0$. One of the strengths of our result is that, in the localisation regime, we are able to prove convergence of the occupation measure, whilst existing results in the case $λ= 0$ give convergence of moments instead. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11842 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A localisation phase transition for the catalytic branching random walk Mailler, Cécile Schapira, Bruno Probability 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 $λ>0$ everywhere, except at the origin, where it branches at rate $λ_0>λ$. We show that, if $λ_0$ 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 $λ_0$ 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 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 $λ=0$. One of the strengths of our result is that, in the localisation regime, we are able to prove convergence of the occupation measure, whilst existing results in the case $λ= 0$ give convergence of moments instead. |
| title | A localisation phase transition for the catalytic branching random walk |
| topic | Probability |
| url | https://arxiv.org/abs/2412.11842 |