Biased random walks on a Galton-Watson tree with leaves

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 $γ= γ(β) \in (0,1)$, depending on the bias $β$, such that $X_n$ is of order $n^γ$. Denoting $Δ_n$ the hitting time of level $n$, we prove that $Δ_n/n^{1/γ}$ is tight. Moreover we show that $Δ_n/n^{1/γ}$ does not converge in law (at least for large values of $β$). We prove that along the sequences $n_λ(k)=\lfloor λβ^{γk}\rfloor$, $Δ_n/n^{1/γ}$ 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.

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