# Differentiability of the speed of biased random walks on Galton-Watson trees

##### schedule le vendredi 21 mai 2021 de 11h00 à 12h00

Organisé par : Quentin Berger, Nathanaël Enriquez, Thierry Lévy et Shen Lin

Intervenant : Yuki Tokushige (Technische Universität München)
Lieu : https://zoom.us/j/98589875776?pwd=Q2JCYU5FN25uNUZLdDJ4Uk9zU2Mxdz09

Sujet : Differentiability of the speed of biased random walks on Galton-Watson trees

Résumé :

We prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree (with/without leaves) is differentiable for$\lambda$ such that the walk is ballistic and obeys a central limit theorem. We also give an expression of the derivative using a certain 2-dimensional Gaussian random variable, which naturally arise as limits of functionals of a biased random walk.

The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres. In particular, an important role is played by moment estimates of regeneration times, which are locally uniform in $\lambda$. This talk is based on a joint work with Adam Bowditch (University College Dublin).