Robustness in the Poisson Boolean model with convex grains

Abstract

We study the Poisson Boolean model where the grains are random convex bodies with a rotation-invariant distribution. We say that a grain distribution is dense if the union of the grains covers the entire space and robust if the union of the grains has an unbounded connected component irrespective of the intensity of the underlying Poisson process. If the grains are balls of random radius, then density and robustness are equivalent, but in general this is not the case. We show that in any dimension $d\ge2$ there are grain distributions that are robust but not dense, and give general criteria for density, robustness and non-robustness of a grain distribution. We give examples which show that our criteria are sharp in many instances.

Figures (6)

• Figure 1: The various sets and their relationships from the proof of Theorem \ref{['two']}(a): The point $y$ is in pink, $x$ in red. The yellow area is $C_x$. The orthogonal projection of $C_x$ is dark blue and $H_{x-y}$ is the grey plane with the red grid. $B^{*}_{^{f_{_{i-1}}}}(\textbf{x},y)$ is lime green. Drawn in black are the line with orientation $x-y$, the point lying in $H_{x-y} \cap \partial B_{f_{_{i-1}}}(x)$ and the ball $B_{f_{_{i-1}}}(x)$ containing the location $x$ of the vertex $\textbf{x}$ corresponding to $C_x$.
• Figure 2: The recursive construction of the proof of Theorem \ref{['two']}(a). The convex set $C_{x_{i-1}}$ is in yellow. Mint green is used for $B_{^{f_{_{i}}}}(x_{i-1})\setminus B_{^{f_{_{i}}/2}}(x_{i-1})$; i.e. the possible area for $x_i$. The part in which $x_i$ is not permitted to be is in red. Grey is used for $H_{x_{i-1}-x_i}$, with dark blue for the orthogonal projection of $C_{x_{i-1}}$. In pink, the orthogonal projection of $B_{f_{_{i-1}}}(x_{i-1})$. Finally, the black point is a possible position for $x_i$.
• Figure 3: Visualisation of $\mathcal{T}_{\textbf{w},\textbf{v}}$ in 3 dimensions by colouring of the different parts of $\mathcal{T}(\textbf{w},\textbf{v})$ and $\mathcal{T}(\textbf{v},\textbf{w})$. The perspective in the top figure is along $\operatorname{rot}_{\rho_{v,w}}(e_2)$ and along $\operatorname{rot}_{\rho_{v,w}}(e_3)$ in the bottom figure. The turquoise rectangle represents the set $\operatorname{rot}_{\rho_{v,w}}([-\bar{D}_{w}^{_{(d)}}, \bar{D}_{w}^{_{(d)}}]\times\bigtimes_{i=1}^{d-1} [-\bar{D}_{w}^{_{(i)}} , \bar{D}_{w}^{_{(i)}}])$ and the dark green one represents $\operatorname{rot}_{\rho_{v,w}}([-\bar{D}_{v}^{_{(d)}}, \bar{D}_{v}^{_{(d)}}]\times\bigtimes_{i=1}^{d-1} [-\bar{D}_{v}^{_{(i)}} , \bar{D}_{v}^{_{(i)}}])$. The light orange area and the area on the left side of the orange dashed line is $\mathcal{W}^{-}_{w,v}$ and the light blue area and the area on the right side of the blue dashed line $\mathcal{W}^{+}_{w,v}$. The green line is the connection line of $w$ and $v$ given as the red and blue point. The purple lines are $f_{i}(w,v,\lambda)$ and the black lines $f_{i}(v,w,\lambda)$ for $i\in\{2,3\}$ and $\lambda>0$.
• Figure 4: The partition $\bigl(\mathcal{A}_m(\textbf{w},\textbf{v})\bigr)_{m=1,\dots,8}$ of $\mathbb{R}^3$ from various perspectives.
• Figure 5: Visualisation of the role of $\ell$ in the calculation for $S_3$, with the perspective along $\operatorname{rot}_{\rho_{v,w}}(e_2)$ in the top and $\operatorname{rot}_{\rho_{v,w}}(e_3)$ in the bottom figure. Note also that $\mathcal{A}_3'(\mathbf{0},\mathbf{y})$ represents the everything "above" and "below" of the turquoise box including the box itself. Consequently, $\mathcal{A}_3(\mathbf{0},\mathbf{y})\cap \mathcal{A}_3'(\mathbf{0},\mathbf{y})$ represents the two small red areas "above" and "below" the turquoise box.

Theorems & Definitions (8)

• Proposition 1.1
• Theorem 1.2
• Definition 2.1
• proof : Proof of Proposition \ref{['one']}
• Remark 2.1
• proof : Proof of Theorem \ref{['two']} (a)
• proof : Proof of Theorem \ref{['two']} (b)
• Remark 2.2