Self-similar real trees defined as fixed-points and their geometric properties

TL;DR

This paper investigates fixed-point equations for probability measures on compact metric spaces that generate continuum random trees, analyzing their existence, uniqueness, convergence, and geometric fractal properties, including Minkowski and Hausdorff dimensions.

Contribution

It introduces new methods to study fixed-point equations for random trees and derives tight bounds on their fractal dimensions, applicable to classical and novel tree models.

Findings

Existence and uniqueness of fixed-point solutions established.

Derived tight bounds on Minkowski and Hausdorff dimensions.

Applicable to classical continuum random trees and dual trees of random triangulations.

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 dimension.