Exceptionally small balls in stable trees

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.