@article{EJP2790,
author = {Pierre Andreoletti and Pierre Debs},
title = {Spread of visited sites of a random walk along the generations of a branching process},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {random walks, random environment, trees, branching random walk},
abstract = {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 $\ell$.},
pages = {no. 42, 1-22},
issn = {1083-6489},
doi = {10.1214/EJP.v19-2790},
url = {http://ejp.ejpecp.org/article/view/2790}}