Random trees with local catastrophes: the Brownian case

TL;DR

The paper studies a generalization of Bienaymé--Galton--Watson processes in which local catastrophes induce births and deaths that are spatially correlated through a brick-wall construction. Under criticality $\mathbb{E}[B]=\mathbb{E}[H]$ and finite third moments, the genealogical forest coded by the brick-wall model converges, in the local Gromov--Hausdorff sense, to the Brownian forest $\mathcal{F}_{\sigma}$ with $\sigma^2=\frac{1}{\mathbb{E}[B]}\mathbb{E}[(B-H)^2]$, via a stochastic-flow approach inspired by Bertoin and Le Gall. A central contribution is showing that, despite dependencies among subfamilies, finite tuples of subpopulations converge to independent Feller diffusions, enabling tightness and a full scaling limit for the trees themselves. The results establish a universality class for critical brick-wall models with finite third moments, connecting discrete correlated structures to continuous Brownian objects, and suggest extensions to stable regimes and continuous-time dynamics with similar limiting behavior.

Abstract

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.

Figures (7)

• Figure 1: Illustration of the construction of the population evolution process from generation $k$ to $k+1$ given i.i.d. bricks sampled according to the law $\rho$. Notice that the last brick overshoot the population at generation $k$: we put crosses for fictitious individuals.
• Figure 2: Left, a brick wall, with the brick borders in black, and the bricks displayed in various colors. The origin edge $[0,1]\times\{k\}$ of each level $k$ is displayed in red, the brick on its top is size-biased by the variable $B$ whereas all other bricks are sampled iid according to the law $\rho$. The integers points $(i,j)$ are displayed with black crosses and the half-integers points $( i+ \frac{1}{2}, j)$ with blue dots -- except the root node $(\frac{1}{2},0)$, corresponding to the red arrow of the root edge. Center, the associated primal forest $F^{\uparrow}$, in blue. Right, the associated dual forest $F^{\downarrow}$, in orange.
• Figure 3: The different forests at play in \ref{['prop: forest duality']}: left, the strip ${F}_{[0,r]}^{\downarrow}[\rho]$ of the dual forest associated to $\rho$. Center, $\widetilde{{F}}_{[0,r]}^{\downarrow}[\rho]$, which is the same forest re-rooted at the random index $J_r$, then rotated and shifted. Right, the strip ${F}^{\uparrow}_{[0,r]}[^t\rho]$ of the primal forest associated to $^t\rho$, and, highlighted in light blue, the set of vertices $T^{\uparrow}_{(I_r,0)}[^t\rho](r)$. For the left and right forest, the root edges of the underlying brick wall are depicted in red at each level, and the root vertex is circled in black. Note that the center and rightmost (sub-)forests in these examples are identical as metric spaces, but they have slightly different embeddings in $\mathbb{R}^2$ due to the rooting conventions of our brick walls.
• Figure 4: In \ref{['prop:martingale-diffusion-cv-box']}, we prove the convergence of $\frac{A_{\lfloor nt \rfloor}^{(n,\varepsilon)}}{n}$, the rescaled discrete process up until it hits $[0,\varepsilon]\cup[1/\varepsilon, +\infty[$, to $(X_t^{\varepsilon})$, the Feller diffusion process up until it hits $[0,\varepsilon]\cup[1/\varepsilon, +\infty[$.
• Figure 5: A depiction of the event $\mathcal{E}_{\varepsilon,n}$ of \ref{['prop:good-behavior-outside-box']}.

Theorems & Definitions (41)

• Theorem 1: Convergence towards the Brownian forest
• Theorem 2: Convergence to a Feller diffusion and Kolmogorov's estimate
• Definition 1: Brick wall
• Proposition 1: Invariance by translation
• proof
• Definition 2: Primal and dual forests
• Remark 1
• Theorem 3: Aldous aldous-crt1, Le Gall lg-random-trees
• Proposition 2
• proof