We consider a random walk on a Galton–Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable Lévy process, jointly with the convergence of the renormalised trace of the walk towards the real tree coded by the latter continuous-time height process.
Partially supported by the ANR project Liouville (ANR-15-CE40-0013).
I thank my advisor Elie Aïdékon for guiding me throughout the development of this article. I thank Xinxin Chen for her advice on the organization of the proofs and for spotting mistakes in an earlier version. I also thank an anonymous referee for his/her numerous precise comments which greatly improved the quality of this paper.
"Scaling limit of the subdiffusive random walk on a Galton–Watson tree in random environment." Ann. Probab. 50 (1) 339 - 396, January 2022. https://doi.org/10.1214/21-AOP1535