BIASED RANDOM WALKS ON GALTON-WATSON TREES 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 gamma = gamma(beta) is an element of (0, 1), depending on the bias beta, such that vertical bar X-n vertical bar 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) = left perpendicular lambda beta(gamma k)right perpendicular, 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.