Scaling limit of critical random trees in random environment

TL;DR

This paper studies critical Bienaymé-Galton-Watson trees in random environments and proves their large conditioned trees converge to the Brownian continuum random tree, using novel techniques beyond standard methods.

Contribution

It establishes the scaling limit of critical GW trees in random environments as the Brownian CRT, providing new tools for analysis in this setting.

Findings

Scaling limit is the Brownian continuum random tree

Results hold for almost all environmental realizations

Introduces alternative proof techniques for non-exchangeable trees

Abstract

We consider Bienaym\'e-Galton-Watson trees in random environment, where each generation $k$ is attributed a random offspring distribution $\mu_k$, and $(\mu_k)_{k\geq 0}$ is a sequence of independent and identically distributed random probability measures. We work in the ``strictly critical'' regime where, for all $k$, the average of $\mu_k$ is assumed to be equal to $1$ almost surely, and the variance of $\mu_k$ has finite expectation. We prove that, for almost all realizations of the environment (more precisely, under some deterministic conditions that the random environment satisfies almost surely), the scaling limit of the tree in that environment, conditioned to be large, is the Brownian continuum random tree. The habitual techniques used for standard Bienaym\'e-Galton-Watson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.