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

TL;DR

This paper investigates the typical behavior of harmonic measure in large critical Galton-Watson trees with infinite variance offspring distributions, revealing how the measure's decay rate depends on the stability index and connecting it to local dimensions.

Contribution

It extends previous results to characterize the harmonic measure's typical decay rate in trees with infinite variance offspring distributions, linking it to a new explicit constant.

Findings

Harmonic measure mass scales as n^{-mbda_lpha} with high probability.

The decay constant mbda_lpha decreases as lpha increases from 1 to 2.

mbda_lpha diverges as lpha approaches 1, at the rate of (lpha - 1)^{-2}.

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.