Scaling limit of critical random trees in random environment

Abstract

We consider Bienaymé-Galton-Watson trees in random environment, where each generation k is attributed a random offspring distribution ${\mathrm{\mu }}_k$, and ${({\mathrm{\mu }}_k)}_{k\ge 0}$ 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 ${\mathrm{\mu }}_k$ is assumed to be equal to 1 almost surely, and the variance of ${\mathrm{\mu }}_k$ 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 Bienaymé-Galton-Watson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.