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 $\psi$ of the branching process satisfies $\psi (1)={\psi }^{\mathrm{\prime }}(1)=0$ for which G. Faraud, Y. Hu and Z. Shi in \cite{HuShi10b} show that, with probability one, the largest generation visited by the walk, until the instant $n$, is of the order of $(\log n{)}^3$. In \cite{AndreolettiDebs1} we prove that the largest generation entirely visited behaves almost surely like $\log n$ up to a constant. Here we study how the walk visits the generations $\mathrm{\ell }=(\log n{)}^{1+\zeta }$, with $0<\zeta <2$. 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 $(\log n{)}^2$ 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 $\mathrm{\ell }$.