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

Abstract

We study the typical behavior of the harmonic measure in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\alpha \in (1,2]$. Let ${\mu }_n$ denote the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. We extend the results of arxiv:1502.05584 to prove that, with high probability, the mass of the harmonic measure ${\mu }_n$ carried by a random vertex uniformly chosen from height $n$ is approximately equal to $n^{-{\lambda }_{\alpha }}$, where the constant ${\lambda }_{\alpha }>\frac{1}{\alpha -1}$ depends only on the index $\alpha$. In the analogous continuous model, this constant ${\lambda }_{\alpha }$ turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for ${\lambda }_{\alpha }$, we are able to show that ${\lambda }_{\alpha }$ decreases with respect to $\alpha \in (1,2]$, and it goes to infinity at the same speed as $(\alpha -1{)}^{-2}$ when $\alpha$ approaches 1.

35 pages, 4 figures; revised version with Proposition 4 corrected

References in corpus (2)

• The harmonic measure of balls in random trees
• Typical behavior of the harmonic measure in critical Galton-Watson trees