Exceptionally small balls in stable trees
Résumé
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 is an element of (1,2]. They form a specific class of Levy trees (introduced by Le Gall and Le Jan in [24]) 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 (gamma/gamma-1) (log 1/r)-(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 from [9, 10, 13].