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.
We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub-ballistic regime, where at time $n$ the particle is typically at a distance of order $O(n^kappa)$ from the o
rigin, $kappain(0,1)$. We investigate the probabilities of moderate deviations from this behaviour. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time $n$, the particle is at a distance of order $O(n^{ u_0})$ from the origin, $ u_0in (0,kappa)$), and speedup (at time $n$, the particle is at a distance of order $n^{ u_1}$ from the origin, $ u_1in (kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time $n$, the particle is located around $(-n^ u)$, thus making an unusual excursion to the left. For the slowdown, our results are valid in the ballistic case as well.
