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

摘要

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 $psi(1)=psi'(1)=0$ 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 $(log n)^3$. We already proved 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 $ell=(log n)^{1+ zeta}$, 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 $ell$.