Abstract
We consider fixed point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to treat cases for which the mass measure on the real tree induced by natural encodings only provides weak estimates on the Hausdorff dimensions.
Highlights
Since the pioneering work of Aldous [3, 5] who introduced the Brownian continuum random tree (Brownian CRT) as a scaling limit for uniformly random labelled trees, similar objects have been shown to play a crucial role in a number of limits of combinatorial problems that relate to computer science, physics or biology
Self-similar real trees defined as fixed points trees, or tree-like compact metric spaces, and they are usually equipped with a probability measure that yields a notion of “mass”
One may think in particular of the Brownian CRT [6], of trees that are dual to recursive triangulations of the disk [19], and of the genealogies of self-similar fragmentations [32]
Summary
Since the pioneering work of Aldous [3, 5] who introduced the Brownian continuum random tree (Brownian CRT) as a scaling limit for uniformly random labelled trees, similar objects have been shown to play a crucial role in a number of limits of combinatorial problems that relate to computer science, physics or biology. This framework allows for instance to deal with certain recalcitrant cases where the natural height function for the tree is not a “good” encoding, in the sense that its optimal Hölder exponent does not yield the fractal dimension of the metric space (we will be more precise shortly) At this point, let us mention that questions (i), (ii) and (iii) have recently been studied by Albenque and Goldschmidt [2] for the specific example where the fixed point equation is the one described by Aldous in [6] and that is satisfied by the Brownian CRT.
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