The Number of Generations Entirely Visited for Recurrent Random Walks in a Random Environment

In this paper we deal with a random walk in a random environment on a super-critical Galton-Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978-1997, 2007; Probab. Theory Relat. Fields 138:521-549, 2007), Faraud et al. (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174-215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins' law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446-459, 1976).