Regularity results of the speed of biased random walks on Galton-Watson trees

Abstract: We prove that the speed of $\lambda$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $\lambda\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.