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Fractal random sets associated with multitype Galton-Watson trees

Pierre Calka, Yann Demichel

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.

Fractal random sets associated with multitype Galton-Watson trees

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.

Paper Structure

This paper contains 23 sections, 15 theorems, 80 equations, 18 figures, 13 tables.

Table of Contents

  1. A growth model of planar random sets: An introduction
  2. The multitype Galton-Watson tree
  3. A natural covering of $\partial \cal{K}_\infty$
  4. Coding with a multitype branching process
  5. Properties of the process $(Z_n)_{n\ge 0}$
  6. Hausdorff dimension of the boundary set of a multitype Galton-Watson tree
  7. Proof of Theorems \ref{['theo:main']} and \ref{['theo:perimaire']}
  8. Proof of Theorem \ref{['theo:main']}: exact calculation of $\mathop{\mathrm{\dim_B}}\nolimits(\partial \cal{K}_{\infty})$ and $\mathop{\mathrm{\dim_H}}\nolimits(\partial \cal{K}_{\infty})$
  9. Proof of Theorem \ref{['theo:main']}: the $d$-dimensional Hausdorff measure of $\partial\cal{K}_{\infty}$
  10. Proof of Theorem \ref{['theo:perimaire']}: estimates of the perimeter and defect area
  11. Von Koch-type examples and reduction of $\mathbb M$
  12. A classical random version of the original von Koch model
  13. A deterministic von Koch square model
  14. Reducing the matrix $\mathbb M$
  15. The method in action for the three tessellations
  16. ...and 8 more sections

Key Result

Theorem 1.1

The set $\partial \cal{K}_\infty$ has Hausdorff dimension and box-dimension which coincide almost surely and are equal to Moreover, the $d$-dimensional Hausdorff measure $\cal{H}^d(\partial \cal{K}_\infty)$ of $\partial \cal{K}_\infty$ is positive in mean and finite almost surely.

Figures (18)

  • Figure 1: Simulation of the random growth model for the triangular (resp. square, hexagonal) tessellation with the particular choice $p_*=0$, $p=0.5$ and $\lambda=3$ (resp. $\lambda=4$, $\lambda=3$).
  • Figure 2: Iterative construction of $\cal{K}_{n+1}$ as the union of the deterministic set $\cal{K}_n^\bullet$ and the random set $\cal{K}_n^\circ$ for the square tessellation with parameters $\lambda=4$, $p_*=0$ and $p=0.5$.
  • Figure 3: Description of the random rule in the case of the square tessellation (a) $p_*=0$ and (b) $p_*=1$. Each square $\cal{T}_{n+1,\ell}$ from $\lambda^{-(n+1)}\cal{T}$ along $\partial\cal{K}_n$ is added to $\cal{K}_n$ to obtain $\cal{K}_{n+1}$ according to its Bernoulli variable $\cal{B}(p_\ell)$.
  • Figure 4: Example for the square tessellation of the construction (in grey) of $\cal{R}_0$ (a) and $\cal{R}_1$ (b) for the sets $\cal{K}_0$ and $\cal{K}_1$ (in black) associated with the parameters $\lambda=4$, $p_*=0$ and $p=0.5$.
  • Figure 5: Example for the square tessellation of the 13 different possible types labelled according to the intersection (in red) of the grey tile with $\cal{K}_n$ during the iterative construction of $\cal{K}_\infty$.
  • ...and 13 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 17 more