Exceptionally small balls in stable trees

The γ-stable trees are random measured compact metric spaces that appear as the scaling limit of Galton-Watson trees whose offspring distribution lies in a γ-stable domain, γ ∈ (1, 2]. They form a specific class of Lévy trees (introduced by Le Gall and Le Jan in [24]) and the Brownian case γ = 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 γ-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 γ γ−1 (log 1/r) 1 γ−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 γ = 2, we prove that this quantity is of order r 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]. AMS 2000 subject classifications: Primary 60G57, 60J80. Secondary 28A78.