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

TL;DR

This paper develops explicit invariant measures for certain flow rules on marked Galton-Watson trees and uses them to compute the Hausdorff dimensions of harmonic measures, revealing dimension drop phenomena.

Contribution

It introduces a method to construct explicit invariant measures for flow rules on marked Galton-Watson trees and applies it to compute harmonic measure dimensions in new cases.

Findings

Explicit invariant measures for flow rules on Galton-Watson trees

Computed Hausdorff dimensions of harmonic measures for $ ext{lambda}$-biased random walks

Demonstrated dimension drop phenomena under certain metrics

Abstract

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 in \cite{LPP95}, 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 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 dimensions of the harmonic measure and show dimension drop phenomenon for the natural metric on the boundary and another metric that depends on the random lengths.