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

TL;DR

This paper investigates how a null recurrent random walk in a random environment on a super-critical Galton-Watson tree visits generations, revealing phase transitions and spread patterns up to certain logarithmic depths.

Contribution

It provides new probabilistic results on the asymptotic behavior and spread of visited sites across generations in a complex branching environment.

Findings

Largest visited generation is of order ( n)^3

Visited sites spread throughout the tree up to generation ( n)^{1+}

Phase transition occurs at generation ( n)^2 for mean visited sites

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 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 $\ell=(\log n)^{1+ \zeta}$, 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 $\ell$.