Scaling limit of critical random trees in random environment (Q6614486)

The authors consider Bienaymé-Galton-Watson trees 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. The interest is in the strictly critical regime, that is, where the average of \(\mu_k\) is equal to \(1\) almost surely, and the variance of \(\mu_k\) has finite expectation. The main result is that for almost all realizations of the environment, the scaling limit of the tree, conditioned to be large, is the Brownian continuum random tree.\N\NThe authors remark that usual techniques for analyzing BGW trees do not apply to this model, because the offspring distributions are possibly random and, crucially, can be different depending on the generation. More precisely, associated processes used to study scaling limits on usual BGW trees (such as the Łukasiewicz path and the height process) cease to be Markovian, which makes the analysis more challenging. To tackle this issue, the authors use a plethora of different techniques.