Random trees with local catastrophes: the Brownian case

Abstract

We introduce and study a model of plane random trees generalizing the famous Bienaym\'e--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 {\L}ukasiewicz exploration, but rather the stochastic flow point of view introduced by Bertoin and Le Gall.