Spread of visited sites of a random walk along the generations of a branching process

In this paper we consider a null recurrent random walk in random environment on a super-critical Galton-Watson tree. We consider the case where the log-Laplace transform ψ of the branching process satisfies ψ(1) = ψ′(1) = 0 for which G. Faraud, Y. Hu and Z. Shi have shown that, with probability one, the largest generation visited by the walk, until the instant n, is of the order of (logn). We have already proved that the largest generation entirely visited behaves almost surely like logn up to a constant. Here we study how the walk visits the generations ` = (logn) , with 0 < ζ < 2. We obtain results in probability giving the asymptotic logarithmic behavior of the number of visited sites at a given generation. We prove that there is a phase transition at generation (logn) for the mean of visited sites until n returns to the root. Also we show that the visited sites spread all over the tree until generation `.