Self-similar random structures defined as fixed points of distributional equations

TL;DR

This paper studies fixed-point equations for probability distributions on measured metric spaces, focusing on recursive, tree-like structures with loops, and provides new characterizations and bounds on their fractal dimensions.

Contribution

It introduces a recursive framework for fixed points of distributional equations on metric spaces with loops, extending previous models like stable trees and looptrees.

Findings

Established existence and uniqueness of solutions.

Derived bounds on Hausdorff and Minkowski dimensions.

Provided new characterizations of stable trees and looptrees.

Abstract

We consider fixed-point equations for probability distributions on isometry classes of measured metric spaces. The construction is required to be recursive and tree-like, but we allow loops for the geodesics between points in the support of the measure: one can think of a $\beta$-stable looptree decomposed around one loop. We study existence and uniqueness of solutions together with the attractiveness of the fixed-points by iterating. We obtain bounds on the Hausdorff and upper Minkowski dimension, which appear to be tight for the studied models. This setup applies to formerly studied structures as the $\beta$-stable trees and looptrees, of which we give a new characterization and recover the fractal dimensions.