The uniform measure on a Galton-Watson tree without the XlogX condition

TL;DR

This paper investigates the Hausdorff dimension of the uniform measure on the boundary of a Galton-Watson tree, showing it drops to zero when the expected offspring log is infinite, thus answering a longstanding open question.

Contribution

It extends the understanding of the boundary measure's Hausdorff dimension to cases where the XlogX condition fails, specifically when the expectation is infinite.

Findings

Dimension is zero when E[ν ln(ν)] = ∞

Confirmed the dimension equals ln(m) when E[ν ln(ν)] < ∞

Answered a question posed by Lyons, Pemantle, and Peres

Abstract

We consider a Galton--Watson tree with offspring distribution $\nu$ of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass $1$ on each vertex of the $n$-th generation and taking the limit $n\to \infty$. In the case $E[\nu\ln(\nu)]<\infty$, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to $\ln(m)$ (\cite{hawkes}, \cite{lpp95}). When $E[\nu \ln(\nu)]=\infty$, we show that the dimension drops to $0$. This answers a question of Lyons, Pemantle and Peres \cite{LyPemPer97}.