# 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 Lévy 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-00640841
Contributor : Thomas Duquesne <>
Submitted on : Monday, November 14, 2011 - 12:40:58 PM
Last modification on : Tuesday, May 14, 2019 - 11:04:15 AM
Long-term archiving on: Wednesday, February 15, 2012 - 2:23:00 AM

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### Identifiers

• HAL Id : hal-00640841, version 1
• ARXIV : 1111.3465

### Citation

Thomas Duquesne, Guanying Wang. Exceptionally small balls in stable trees. 2011. ⟨hal-00640841⟩

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