Biased random walks on Galton–Watson trees 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 γ = γ(β) ∈ (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) = ⌊λβ^γk⌋, 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.