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Journal Articles Electronic Journal of Probability Year : 2014

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

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 $\ell$.
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Dates and versions

hal-00800339 , version 1 (13-03-2013)
hal-00800339 , version 2 (10-05-2013)
hal-00800339 , version 3 (13-02-2014)

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Pierre Andreoletti, Pierre Debs. Spread of visited sites of a random walk along the generations of a branching process. Electronic Journal of Probability, 2014, 19 (42), pp.1-22. ⟨hal-00800339v3⟩
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