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 $r^{\frac{\gamma}{\gamma-1}} (\log1/r)^{-\frac{1}{\gamma-1}}$. 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. 
Thomas Duquesne (LPMA), Guanying Wang (LPMA)
math.PR