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 [12], 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.
Highlights
Consider an infinite rooted tree t without leaves
The boundary ∂t of the tree t is the set of all rays on t, that is infinite nearest-neighbour paths on t that start from the root and never backtrack
For any height n ≥ 0, let Ξn be the last vertex at height n that is visited by the random walk
Summary
Consider an infinite rooted tree t without leaves. The boundary ∂t of the tree t is the set of all rays on t, that is infinite nearest-neighbour paths on t that start from the root and never backtrack. In [13], the authors prove the existence of a unique invariant probability measure μHARM absolutely continuous with respect to the law of the Galton-Watson tree, and conclude that there is a dimension drop phenomenon. They do not (except in the case λ = 1 in [12]) give an explicit formula for the EJP 23 (2018), paper 46. We are dealing with a model of random walk on Galton-Watson trees without marks, it is easier to express an explicit invariant measure in the more general setting of Theorem 3.2. We associate to our tree an age-dependent process whose Malthusian parameter turns out to be the Hausdorff dimension of the boundary with respect to this distance, and prove that we still have a dimension drop phenomenon in this context
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