Range and critical generations of a random walk on Galton-Watson trees

Pierre Andreoletti; Xinxin Chen

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arXiv · 1510.01121

Range and critical generations of a random walk on Galton-Watson trees

Pierre Andreoletti·Xinxin Chen

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Abstract

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R\_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\log n$. Thisresult is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^2$,which turn out to be the most visited generations. Our proof which involves a quenched analysisgives a description of the typical environments responsible for the behavior of $R\_n$.

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Pierre Andreoletti, Xinxin Chen. 2015-10-05. Range and critical generations of a random walk on Galton-Watson trees. https://arxiv.org/abs/1510.01121

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