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


 
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1. Title Title of document Spread of visited sites of a random walk along the generations of a branching process
 
2. Creator Author's name, affiliation, country Pierre Andreoletti; Université d'Orléans; France
 
2. Creator Author's name, affiliation, country Pierre Debs; Université d'Orléans; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walks, random environment, trees, branching random walk
 
3. Subject Subject classification MSC 2010 60J55 ; 60J80 ; 60G50 ; 60K37
 
4. Description 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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7. Date (YYYY-MM-DD) 2014-05-04
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2790
 
10. Identifier Digital Object Identifier 10.1214/EJP.v19-2790
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 19
 
12. Language English=en en
 
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