Random coverage from within with variable radii, and Johnson-Mehl cover times

TL;DR

This work analyzes the time required for Johnson-Mehl tessellations to completely cover a compact region A, revealing that boundary effects crucially shape the limiting distribution in dimensions d≥2. By connecting restricted Johnson-Mehl cover times to high-intensity spherical Poisson Boolean models with random radii, the authors derive precise extreme-value limits: in 2D polygonal or smooth-boundary cases the restricted J-M cover time exhibits a boundary-term–driven Gumbel or two-component extreme value distribution, while in higher dimensions the boundary term dominates and yields a Gumbel-type limit. They extend these results to the restricted SPBM, providing explicit limit formulas involving boundary measures and constants c_{d,k} and c_{d,k,Y}, including the special case of uniform radii. The proofs hinge on a boundary/ interior decomposition, half-space SPBM analysis, polytopal approximation of ∂A, and an induced-coverage construction thatorganizes boundary contributions as a product over charts, yielding a robust framework for geometric extreme-value limits in spatial growth models with random radii. These results clarify the role of edge effects, generalize prior deterministic-radius findings, and offer tools for analyzing crystallization-like growth processes in continuum spaces.

Abstract

Given a compact planar region $A$, let $τ_A$ be the (random) time it takes for the Johnson-Mehl tessellation of $A$ to be complete, i.e. the time it takes for $A$ to be fully covered by a spatial birth-growth process in $A$ with seeds arriving as a unit-intensity Poisson point process in $A \times [0,\infty)$, where upon arrival each seed grows at unit rate in all directions. We show that if $\partial A$ is smooth or polygonal then $\Pr [ πτ_{sA}^3 - 6 \log s - 4 \log \log s \leq x]$ tends to $\exp(- (\frac{81}{4π})^{1/3} |A|e^{-x/3} -(\frac{9}{2π^2})^{1/3} |\partial A| e^{-x/6})$ in the large-$s$ limit; the second term in the exponent is due to boundary effects, the importance of which was not recognized in earlier work on this model. We present similar results in higher dimensions (where boundary effects dominate). These results are derived using new results on the asymptotic probability of covering $A$ with a high-intensity spherical Poisson Boolean model restricted to $A$ with grains having iid small random radii, which generalize recent work of the first author that dealt only with grains of deterministic radius.

Figures (10)

• Figure 1: Tessellations of a disc $A$ (diameter $0.9$) by the restricted Johnson--Mehl process with $\rho \approx 125$ (left) and restricted spherical Poisson Boolean model with $Y_i$ exponentially distributed (right). In both cases the seeds are marked inside each cell with a red dot, and $\mathrm{argmax}_{x \in A} \Xi_x$ (the "last location covered") is marked with a blue square. Later-arriving seeds are marked with paler dots (left diagram); larger dots indicate larger $Y_i$ (right diagram). All vertices of both tessellations are of degree 3, including those on the boundary.
• Figure 2: Empirical distributions of the standardised coverage times $T_\rho$ and $\tilde{T}_{\rho}$ as described in Proposition \ref{['Hallthm']} and Theorem \ref{['thmwksq']}, obtained by sampling many independent realisations of the Johnson-Mehl process in $[0,1]^2$. The standardisation function $g(T,\rho) = \pi \rho T^3 - 2 \log \rho - 4 \log\log \rho$ is the same for both $T_\rho$ and $\tilde{T}_\rho$.
• Figure 3: Examples in $d=2$ and $d=3$ showing a $d$-tuple $\{{\bf x}_1,\ldots,{\bf x}_d\}$, satisfying $h_n({\bf x}_1,\ldots,{\bf x}_d)=1$ and showing $p_n(\{{\bf x}_1,\ldots ,{\bf x}_d\})$ (white square) and $q_n(\{{\bf x}_1,\ldots,{\bf x}_d\})$ (black square), where, for $i= 1,\ldots,d$, ${\bf x}_i= (x_i,s_i)$ and the arrow from $x_i$ is of length $r_n s_i$.
• Figure 4: Example in $d=2$ with $k=1$. Here $M_n(\Omega,a) = 11$ since, taking the uncovered regions below height $a$ in descending order, we have a region with 5 corners that all contribute to $M_n(\Omega,a)$, one with 4 corners three of which contribute, and one with 3 corners that all contribute, and a region with one interior corner that does not contribute. The thick blue line within the $x$-axis is the region $\Omega_n$.
• Figure 5: Illustration of the geometric estimate used to derive equation \ref{['e:0212b']}. The thick line at the bottom is $\partial \mathbb{H}$.

Theorems & Definitions (75)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Theorem 2.9: Limiting probability of $k$-coverage of a polygonal domain by a restricted SPBM
• Theorem 2.10: Limiting probability of $k$-coverage of a smoothly bounded region in $d\geq 2$ by a restricted SPBM