Abstract
A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].
Highlights
The main purpose of this work is to answer, for a specific family of continuous random trees (CRT in short), the following general question about measured metric spaces: if m(r) denotes the measure assigned to the ball centered at some fixed distinguished point and with radius r ≥ 0, is the non-decreasing function m absolutely continuous with respect to the Lebesgue measure on [0, ∞) ? When the answer is positive, the density m (r) can be viewed as the measure of the sphere with radius r
When the metric space is a continuum tree, the density m is sometimes known as the profile of the tree
Recall that a conservative self-similar fragmentation describes the evolution of a branching particle system such that at every branching event, the sum of the masses of the children coincides with the mass of the parent, and self-similarity refers to the property that the evolution of a particle with mass x > 0 is a scaling transformation of that of a particle
Summary
The main purpose of this work is to answer, for a specific family of continuous random trees (CRT in short), the following general question about measured metric spaces: if m(r) denotes the measure assigned to the ball centered at some fixed distinguished point and with radius r ≥ 0, is the non-decreasing function m absolutely continuous with respect to the Lebesgue measure on [0 , ∞) ? When the answer is positive, the density m (r) can be viewed as the measure of the sphere with radius r. When the metric space is a continuum tree, the density m is sometimes known as the profile of the tree This question has been answered by Haas [13] for the class of self-similar fragmentation trees, which notably includes Aldous’ CRT. The motivation of the present work is not just getting a formal extension of the results of Haas; it stems from the connection between random surfaces and growthfragmentations as we shall explain informally It was pointed out in [1] and [5] that for certain random surfaces with a boundary, the process obtained by slicing the surface at fixed distances from the boundary and measuring the lengths of the resulting cycles yields a self-similar growth-fragmentation with negative index.
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