Fractal random sets associated with multitype Galton-Watson trees

TL;DR

This paper studies the geometric properties of random fractal sets generated by iteratively unioning tiles from scaled tessellations, linking their dimensions to spectral properties of associated multitype Galton-Watson trees.

Contribution

It introduces a novel connection between fractal geometry of tessellation-based sets and multitype Galton-Watson processes, providing explicit dimension formulas.

Findings

Box and Hausdorff dimensions coincide for the limiting set.

Dimensions are expressed as functions of the spectral radius of the reproduction matrix.

Explicit calculations are provided for hexagonal, square, and triangular tessellations.

Abstract

In this paper, we consider a regular tessellation of the Euclidean plane and the sequence of its geometric scalings by negative powers of a fixed integer. We generate iteratively random sets as the union of adjacent tiles from these rescaled tessellations. We encode this geometric construction into a combinatorial object, namely a multitype Galton-Watson tree. Our main result concerns the geometric properties of the limiting planar set. In particular, we show that both box and Hausdorff dimensions coincide and we calculate them in function of the spectral radius of the reproduction matrix associated with this branching process. We then make that spectral radius explicit in several concrete examples when the regular tessellation is either hexagonal, square or triangular.