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 Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584, 2015) 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.