Typical behavior of the harmonic measure in critical Galton-Watson trees

2015/02/19 by Shen Lin, Lin, Shen
Mathematics · #60G50 #60J80 #60K37 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G50 #msc:60J80 #msc:60K37

paper · pdf · doi:10.48550/arxiv.1502.05584

38 pages, 8 figures; minor changes with one reference added; to appear in the Annales de l'I.H.P. Probabilités et statistiques

arxiv created 2016/03/03 · arxiv updated 2016/03/04

Abstract

We study the typical behavior of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution has finite variance. The harmonic measure considered here refers to the hitting distribution of height n by simple random walk on a critical Galton-Watson tree conditioned to have height greater than n. We prove that, with high probability, the mass of the harmonic measure carried by a random vertex uniformly chosen from height n is approximately equal to n^-λ, where the constant λ>1 does not depend on the offspring distribution. This universal constant λ is equal to the first moment of the asymptotic distribution of the conductance of size-biased Galton-Watson trees minus 1.

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