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Rectangular Gilbert Tessellation

Emily Ewers, Tatyana Turova

TL;DR

This work analyzes a rectangular Gilbert tessellation—the axis-aligned ray-growth quadrangulation spawned from a Poisson point process with random horizontal/vertical marks. It establishes exponential tails for the total ray length and exponential decay of correlations with seed-distance, with explicit bounds scaling as $\\sqrt{\\lambda}$, and shows that escaped rays in a bounded square scale linearly with the side length $N$. The approach hinges on geometric constructs (cones, stripes, tops) and probabilistic tools (independence in nonoverlapping areas, Chernoff bounds) to quantify stabilization and boundary effects. Collectively, the results provide rigorous quantitative insight into segment-length distributions, dependence structure, and boundary behavior in Gilbert tessellations and related planar growth models.

Abstract

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the lines grow on both sides of vertical and horizontal directions until they meet another line. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segments are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box $[0,N]^2$, it is proved that the average number of rays reaching the border of the box has a linear order in $N$.

Rectangular Gilbert Tessellation

TL;DR

This work analyzes a rectangular Gilbert tessellation—the axis-aligned ray-growth quadrangulation spawned from a Poisson point process with random horizontal/vertical marks. It establishes exponential tails for the total ray length and exponential decay of correlations with seed-distance, with explicit bounds scaling as , and shows that escaped rays in a bounded square scale linearly with the side length . The approach hinges on geometric constructs (cones, stripes, tops) and probabilistic tools (independence in nonoverlapping areas, Chernoff bounds) to quantify stabilization and boundary effects. Collectively, the results provide rigorous quantitative insight into segment-length distributions, dependence structure, and boundary behavior in Gilbert tessellations and related planar growth models.

Abstract

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the lines grow on both sides of vertical and horizontal directions until they meet another line. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segments are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box , it is proved that the average number of rays reaching the border of the box has a linear order in .

Paper Structure

This paper contains 19 sections, 9 theorems, 159 equations, 9 figures.

Table of Contents

  1. Introduction
  2. From cellular automata to a planar quadrangulation
  3. Model of planar quadrangulation
  4. Gilbert tessellation
  5. Results
  6. Rectangular Gilbert tessellation model
  7. Bounds for distribution of the rays' length
  8. Correlations
  9. Example of negative correlations
  10. Example of positive correlations
  11. Exponential decay of correlations
  12. Bounded initial state
  13. Quadrangulation of a square
  14. Percolation
  15. Proofs
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $G_{\mathcal{V}}(t)$ be a rectangular Gilbert tessellation model with random set $\mathcal{V}$ generated by a Poisson process in $\mathbb{R}^2$ with intensity $\lambda$. There are positive constants $c_1, c_2$ and $\alpha_1 \geq \alpha_2$ such that for any $\lambda >0$, for any arbitrarily fixed for any fixed $t>0$ uniformly in $T>t$.

Figures (9)

  • Figure 1: Example of a process with four points from the left to the right: (1) at $t=0$, (2) at some moment before any collision, (3) at the time of the first collision, (4) at $t=N$.
  • Figure 2: Example of a quadrangulation with 200 initial points.
  • Figure 3: The shaded area is the cone$C_{v,d_v}^+(t)$, the entire square is $D_{v,d_v}^+(t)$, including examples of other lines growing and influencing a potential ray ${\cal L}_{v,d_v}^+(t)$ marked by the right arrow pointed to the centre of the square.
  • Figure 4: Illustration of tops. The horizontal arrow of length $t$ represents $\mathcal{L}_{v, d_v}^+(t)$. Blue points with small lines are the tops. Points that are crossed out in red are not tops, because the growth of lines from these points is blocked by the lines of other points.
  • Figure 5: Illustration of the stripe $S_v^-(t)$ of width $\frac{1}{\sqrt{\lambda}}$, shaded in red. The (blue) dots represent points of $\mathcal{V}$.
  • ...and 4 more figures

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Definition 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Definition 2.2
  • Corollary 2.2
  • Proposition 2.2
  • Theorem 2.2
  • ...and 6 more