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

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 $n^{-\lambda }$, where the constant $\lambda >1$ does not depend on the offspring distribution. This universal constant $\lambda$ is equal to the first moment of the asymptotic distribution of the conductance of size-biased Galton-Watson trees minus 1.

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

References in corpus (1)

• The harmonic measure of balls in random trees

Cited by in corpus (2)

• The harmonic measure of balls in random trees
• Typical behavior of the harmonic measure in critical Galton-Watson trees with infinite variance offspring distribution