Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson Trees

Pierre Rousselin

Published 2017-08-23Version 1

We consider infinite Galton-Watson trees without leaves together with i.i.d. random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, to build explicit invariant measures. We apply this result, together with the ergodic theory on Galton-Watson trees developed by Lyons, Pemantle and Peres, to the computation of Hausdorff dimensions of harmonic measures in two cases. The first one is the harmonic measure of the (transient) $\lambda$-biased random walk on Galton-Watson trees, for which the invariant measure and the Hausdorff dimension were not explicitly known. The second case is a model of random walk on a Galton-Watson trees with random lengths for which we compute the Hausdorff dimensions and show dimension drop phenomenons for the natural metric on the boundary and another metric that depends on the random lengths.