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

TL;DR

This paper investigates the typical distribution of harmonic measure in large critical Galton-Watson trees, revealing a universal power-law decay characterized by a constant independent of offspring distribution.

Contribution

It establishes that the harmonic measure's mass on a random vertex scales as a universal power law with exponent mbda, linked to conductance distribution in size-biased trees.

Findings

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

The constant mbda is universal, not depending on offspring distribution.

mbda equals the first moment of conductance distribution minus 1.

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.