The harmonic measure of balls in random trees

TL;DR

This paper investigates the harmonic measure of large discrete trees, revealing that most of the measure concentrates on a boundary subset of size approximately n^0.78, with results derived from both discrete and continuous tree models.

Contribution

It introduces a universal constant beta describing the harmonic measure distribution in large trees and connects discrete models with the Aldous continuum random tree.

Findings

Most harmonic measure is supported on a boundary set of size n^0.78.

The dimension of harmonic measure equals beta, approximately 0.78.

The constant beta relates to the conductance distribution of critical Galton-Watson trees.

Abstract

We study properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^{\beta}$, where $\beta\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton-Watson trees conditioned to have height greater than $n$. An important ingredient of our approach is the analogous continuous model (related to Aldous' continuum random tree), where the dimension of harmonic measure of a level set of the tree is equal to $\beta$, whereas the dimension of the level set itself is equal to $1$. The constant $\beta$ is expressed in terms of the asymptotic distribution of the conductance of large critical Galton-Watson trees.