Biased random walks on a Galton-Watson tree with leaves

Abstract

We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\gamma =\gamma (\beta )\in (0,1)$, depending on the bias $\beta$, such that $X_n$ is of order $n^{\gamma }$. Denoting ${\Delta }_n$ the hitting time of level $n$, we prove that ${\Delta }_n/n^{1/\gamma }$ is tight. Moreover we show that ${\Delta }_n/n^{1/\gamma }$ does not converge in law (at least for large values of $\beta$). We prove that along the sequences $n_{\lambda }(k)=\lfloor \lambda {\beta }^{\gamma k}\rfloor$, ${\Delta }_n/n^{1/\gamma }$ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

49 pages, 2 figures. To appear in Ann. Probab

References in corpus (1)

• Limit laws for transient random walks in random environment on $\z$

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