Exceptionally small balls in stable trees

TL;DR

This paper investigates the detailed measure properties of $oldsymbol{ ext{γ-stable trees,}}$ revealing the asymptotic behavior of the smallest and largest mass measure balls and providing exact constants for local densities.

Contribution

It provides new asymptotic results for the minimum and maximum mass measure of balls in $ ext{γ-stable trees}$, including exact constants for local densities, extending previous knowledge.

Findings

Minimum mass measure of balls scales as $r^{rac{ ext{γ}}{ ext{γ}-1}} ( ext{log}1/r)^{-rac{1}{ ext{γ}-1}}$

Maximum mass measure of balls in the Brownian case scales as $r^2 ext{log} 1/r$

Exact constants for the lower local density of the mass measure are computed.

Abstract

The $\gamma$-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a $\gamma$-stable domain, $\gamma \in (1, 2]$. They form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in1998) and the Brownian case $\gamma= 2$ corresponds to Aldous Continuum Random Tree (CRT). In this paper, we study fine properties of the mass measure, that is the natural measure on $\gamma$-stable trees. We first discuss the minimum of the mass measure of balls with radius $r$ and we show that this quantity is of order $r^{\frac{\gamma}{\gamma-1}} (\log1/r)^{-\frac{1}{\gamma-1}}$. We think that no similar result holds true for the maximum of the mass measure of balls with radius $r$, except in the Brownian case: when $\gamma = 2$, we prove that this quantity is of order $r^2 \log 1/r$. In addition, we compute the exact constant for the lower local density of the mass measure (and the upper one for the CRT), which continues previous results.