Spread of visited sites of a random walk along the generations of a branching process. (English) Zbl 1312.60118

Andreoletti, Pierre; Debs, Pierre

Electron. J. Probab. 19, Paper No. 42, 22 p. (2014).

Summary: 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 et al. [Probab. Theory Relat. Fields 154, No. 3–4, 621–660 (2012; Zbl 1257.05162)] 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. We also show that the visited sites spread all over the tree until generation \(\ell\).

MSC:

60K37	Processes in random environments
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60G50	Sums of independent random variables; random walks
60J55	Local time and additive functionals

Keywords:

random walks; branching process; random environment

Citations:

Zbl 1257.05162