Genealogy in critical generations of a diffusive random walk in random environment on trees

Abstract: We consider the range $R^{(n)}$, the tree made up of visited vertices by a diffusive null-recurrent randomly biased walk $\mathbb{X}$ on a Galton-Watson tree $\mathbb{T}$ up to the $n$-th return time to its root and we consider the following genealogy problem: pick two vertices uniformly at random in a generation of order $n$ in the tree $R^{(n)}$. Where does the coalescence occur? it turns out that the coalescence happens either in the recent past or in the remote past.