2017/08/23 by Pierre Rousselin, Rousselin, Pierre · 1 citation
Mathematics · #37A50 (Primary) 60J05 #60G50 #60J10 (Secondary) #60J80 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1708.06965
openalex publication_date 2017/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider infinite Galton-Watson trees without leaves together with\ni.i.d.~random variables called marks on each of their vertices. We define a\nclass of flow rules on marked Galton-Watson trees for which we are able, under\nsome algebraic assumptions, to build explicit invariant measures. We apply this\nresult, together with the ergodic theory on Galton-Watson trees developed in\n citeLPP95, to the computation of Hausdorff dimensions of harmonic measures\nin two cases. The first one is the harmonic measure of the (transient)\n\λ-biased random walk on Galton-Watson trees, for which the invariant\nmeasure and the dimension were not explicitly known. The second case is a model\nof random walk on a Galton-Watson trees with random lengths for which we\ncompute the dimensions of the harmonic measure and show dimension drop\nphenomenon for the natural metric on the boundary and another metric that\ndepends on the random lengths.\n