Scaling limit of critical random trees in random environment

Guillaume Conchon--Kerjan, Daniel Kious, Cécile Mailler

Published 2022-09-22Version 1

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 realisations 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. Standard techniques used for standard Bienaym\'e-Galton-Watson trees do not apply to this case, and our proof therefore provides an alternative approach for showing scaling limits of random trees. In particular, we make a (to our knowledge) novel connection between the Lukasiewicz path and the height process of the tree, by combining a discrete version of the L\'evy snake introduced by Le Gall and the spine decomposition.