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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.06770 |
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| _version_ | 1866911276994658304 |
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| author | Carrance, Ariane Casse, Jérôme Curien, Nicolas |
| author_facet | Carrance, Ariane Casse, Jérôme Curien, Nicolas |
| contents | We introduce and study a model of plane random trees generalizing the famous Bienaymé--Galton--Watson model but where births and deaths are locally correlated. More precisely, given a random variable $(B,H)$ with values in $\{1,2,3, \dots\}^2$, given the state of the tree at some generation, the next generation is obtained (informally) by successively deleting $B$ individuals side-by-side and replacing them with $H$ new particles where the samplings are i.i.d. We prove that, in the critical case $\mathbb{E}[B]=\mathbb{E}[H]$, and under a third moment condition on $B$ and $H$, the random trees coding the genealogy of the population model converges towards the Brownian Continuum Random Tree. Interestingly, our proof does not use the classical height process or the Łukasiewicz exploration, but rather the stochastic flow point of view introduced by Bertoin and Le Gall. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random trees with local catastrophes: the Brownian case Carrance, Ariane Casse, Jérôme Curien, Nicolas Probability Combinatorics 60B05, 60J80 We introduce and study a model of plane random trees generalizing the famous Bienaymé--Galton--Watson model but where births and deaths are locally correlated. More precisely, given a random variable $(B,H)$ with values in $\{1,2,3, \dots\}^2$, given the state of the tree at some generation, the next generation is obtained (informally) by successively deleting $B$ individuals side-by-side and replacing them with $H$ new particles where the samplings are i.i.d. We prove that, in the critical case $\mathbb{E}[B]=\mathbb{E}[H]$, and under a third moment condition on $B$ and $H$, the random trees coding the genealogy of the population model converges towards the Brownian Continuum Random Tree. Interestingly, our proof does not use the classical height process or the Łukasiewicz exploration, but rather the stochastic flow point of view introduced by Bertoin and Le Gall. |
| title | Random trees with local catastrophes: the Brownian case |
| topic | Probability Combinatorics 60B05, 60J80 |
| url | https://arxiv.org/abs/2401.06770 |