Exceptionally small balls in stable trees

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 $r^{\frac{\gamma }{\gamma -1}}(\log 1\mathrm{/}r{)}^{-\frac{1}{\gamma -1}}$. 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 $r^2\log 1\mathrm{/}r$. 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.