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2014 Spread of visited sites of a random walk along the generations of a branching process
Pierre Andreoletti, Pierre Debs
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Electron. J. Probab. 19: 1-22 (2014). DOI: 10.1214/EJP.v19-2790

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 $\ell$.

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Pierre Andreoletti. Pierre Debs. "Spread of visited sites of a random walk along the generations of a branching process." Electron. J. Probab. 19 1 - 22, 2014. https://doi.org/10.1214/EJP.v19-2790

Information

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

zbMATH: 1312.60118
MathSciNet: MR3210543
Digital Object Identifier: 10.1214/EJP.v19-2790

Subjects:
Primary: MSC 2010 60J55
Secondary: 60G50 , 60J80 , 60K37

Keywords: Branching random walk , random environment , Random walks , trees

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