Electronic Journal of Probability

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

Pierre Andreoletti and Pierre Debs

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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 $\ell$.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 42, 22 pp.

Accepted: 4 May 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: MSC 2010 60J55
Secondary: 60J80 60G50 60K37: Processes in random environments

random walks random environment trees branching random walk

This work is licensed under a Creative Commons Attribution 3.0 License.


Andreoletti, Pierre; Debs, Pierre. Spread of visited sites of a random walk along the generations of a branching process. Electron. J. Probab. 19 (2014), paper no. 42, 22 pp. doi:10.1214/EJP.v19-2790. https://projecteuclid.org/euclid.ejp/1465065684

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