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Abstract
We consider BienayméGaltonWatson trees in random environment, where each generation k is attributed a random offspring distribution μk , and (μk)k≥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 μk is assumed to be equal to 1 almost surely, and the variance of μk 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 BienayméGaltonWatson trees, or trees with exchangeable vertices, do not apply to this case. Our proof therefore provides alternative tools.
Original language  English 

Article number  112 
Pages (fromto)  153 
Number of pages  54 
Journal  Electronic Journal of Probability 
Volume  29 
Early online date  30 Jul 2024 
DOIs  
Publication status  Published  22 Sept 2024 
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 1 Finished

Random walks in dynamic random environment
Kious, D. (PI)
Engineering and Physical Sciences Research Council
1/07/21 → 1/02/24
Project: Research council