Lecture 23: Random Tessellations

Introduction

Outline for Today:

Random Tessellations
Zero cells and Typical cells
Poisson Hyperplane Tessellations

Random Tessellations

Definition (Random Tessellations). A random tessellation in $\mathbb{R}^d$ is a particle process $\Phi$ such that:

[(i)] Each $K \in \mathrm{supp}(\Phi)$ is compact, convex, and has interior points.
[(ii)] $\displaystyle \bigcup_{K \in \mathrm{supp}(\Phi)} K = \mathbb{R}^d$.
[(iii)] If $K, K' \in \mathrm{supp}(\Phi)$ and $K \neq K'$, then $\mathrm{int}(K) \cap \mathrm{int}(K') = \varnothing$.

Remarks:

All cells of a random tessellation are convex polytopes.
As a particle process, if $\Phi$ is stationary, we can decompose the intensity measure to obtain the cell intensity $\lambda$ and grain distribution $\mathbb{Q}$.

Zero cells and Typical cells

As with stationary Voronoi tessellations, we study the following two random convex polytopes:

Typical cell $Z$ (only for stationary tessellations)

A random convex polytope with grain distribution $\mathbb{Q}$.
depends on the choice of center $C$.

Zero cell $Z_{0}$

A random convex polytope in $\mathrm{supp}(\Phi)$ containing the origin.
In the stationary case, this has the same distribution as the cell of $\Phi$ containing any fixed $x \in \mathbb{R}^d$, translated by $-x$.

Distribution of the Zero Cell

Let $\mathcal{X}'$ be the space of non-empty closed, convex, compact subsets of $\mathbb{R}^d$.

The distribution of the zero cell is, up to translations, the volume–weighted distribution of the typical cell.

Theorem. Let $\Phi$ be a stationary random tessellation in $\mathbb{R}^d$ with cell intensity $\lambda > 0$ and grain distribution $\mathbb{Q}$.
If $f : \mathcal{X}' \to \mathbb{R}_{+}$ is measurable and translation invariant, then \[ \mathbb{E}\!\left[f(Z_{0})\right] = \lambda\,\mathbb{E}\!\left[V_{d}(Z)\,f(Z)\right]. \]

In particular, the distribution of $Z_{0} - c(Z_{0})$ has density $\lambda V_{d}(\cdot)$ with respect to the distribution of $Z$, i.e.\ to $\mathbb{Q}$.

Letting $f \equiv 1$ gives \[ \mathbb{E}\!\left[V_{d}(Z)\right] = \frac{1}{\lambda}. \]
\[ \mathbb{E}\!\left[V_{d}(Z_{0})\right] = \lambda\,\mathbb{E}\!\left[V_{d}(Z)^{2}\right] = \frac{\mathbb{E}\!\left[V_{d}(Z)^{2}\right]}{\mathbb{E}\!\left[V_{d}(Z)\right]}. \]

Proof. By Campbell’s theorem,

\[\begin{align*} \mathbb{E}\!\left[f(Z_{0})\right] &= \mathbb{E}\!\left[ \sum_{K \in \mathrm{supp}(\Phi)} f(K)\,\mathbf{1}_{\{0 \in \mathrm{int}(K)\}} \right] \\[0.5ex] &= \int_{\mathcal{X}'} f(K)\,\mathbf{1}_{\{0 \in \mathrm{int}(K)\}}\,\Lambda(dK) \\ &= \lambda \int_{\mathcal{X}_0} \int_{\mathbb{R}^d} f(K+x)\,\mathbf{1}_{\{0 \in K+x\}}\,dx\,\mathbb{Q}(dK) \\ &= \lambda \int_{\mathcal{X}_0} f(K) \int_{\mathbb{R}^d} \mathbf{1}_{\{0\in K+x\}}\,dx\, \mathbb{Q}(dK) \\ &= \lambda \int_{\mathcal{X}_0} f(K)\,V_{d}(K)\,\mathbb{Q}(dK) \\ &= \lambda\,\mathbb{E}\!\left[f(Z)\,V_{d}(Z)\right]. \end{align*}\]

$\square$

Poisson Hyperplane Tessellations

Let $\widehat{\Phi}$ be a stationary Poisson hyperplane process in $\mathbb{R}^{d}$ with intensity $\gamma$ and spherical directional distribution $\varphi$ on $S^{d-1}$.

$ = {i } {H(u_i, t_i)}, $ where $\{u_i\}_{i \in \mathbb{Z}}$ are i.i.d. random vectors on $S^{d-1}$ with distribution $\varphi$, and $\{t_i\}_{i \in \mathbb{Z}}$ is a Poisson process on $\mathbb{R}$ with intensity $\gamma$. ($H(u,t) := \{x \in \mathbb{R}^{d} : \langle x, u\rangle = t\}.$)

Assume $\mathrm{supp}(\varphi)$ is not concentrated on a great subsphere.
This ensures that there does not exist a line to which all hyperplanes are almost surely parallel.
Then $\widehat{\Phi}$ induces a stationary random tessellation on $\mathbb{R}^{d}$, and the condition above ensures that all cells are bounded.

Associated Zonoid

Many parameters of the cells of a stationary Poisson hyperplane tessellation are given by quantities relating to a deterministic convex body that characterizes its intensity measure.

Definition. Let $\widehat{\Phi}$ be a stationary Poisson hyperplane process with intensity $\gamma$ and directional distribution $\varphi$.
The associated zonoid of $\widehat{\Phi}$ is the convex body $\Pi$ containing the origin with support function \[ h(\Pi, v) = \sup_{x \in \Pi} \langle x, v\rangle = \frac{\gamma}{2} \int_{S^{d-1}} |\langle u, v\rangle|\, \varphi(du). \]

If $\varphi$ has finite support, then $\widehat{\Pi}$ is the Minkowski sum of a finite number of line segments centered at $0$, i.e. a zonotope.
$\mathbb{E}\!\left[\tilde{\Phi}(\mathcal{F}_{[0,V]})\right] = 2 h(\Pi, v).$
$\mathbb{E}\!\left[V_{d}(Z)\right] = \frac{1}{\lambda} = \frac{1}{V_{d}(\Pi)}.$
$\mathbb{E}\!\left[V_{d}(Z_{0})\right] = 2^{-d} d!\,V_{d}(\Pi^{\circ}),$ where $\Pi^{\circ} := \{x \in \mathbb{R}^{d} : \langle x, y\rangle \le 1 \ \forall y \in \Pi\}$.
$f_0(Z_0)$ is the number of vertices of zero cell: \[ \mathbb{E}\!\left[f_{0}(Z_{0})\right] = 2^{-d} d!\,V_{d}(\Pi^{\circ})\,V_{d}(\Pi). \] \[ \mathbb{E}\!\left[f_{0}(Z_{0})\right] = 2^{-d} d!\,V_{d}(\Pi^{\circ})\,V_{d}(\Pi). \]

\[ \mathbb{E}\!\left[f_{0}(Z_{0})\right] = 2^{-d} d!\,V_{d}(\Pi^{\circ})\,V_{d}(\Pi). \]

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