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

Area of Science:

• Stochastic Geometry
• Spatial Statistics
• Computational Geometry

Background:

• The Johnson-Mehl tessellation models spatial birth-growth processes.
• Understanding the completion time of such processes within a compact region is crucial.
• Previous studies overlooked the impact of boundary effects on completion time.

Purpose of the Study:

• To analyze the completion time of the Johnson-Mehl tessellation within a compact planar region A.
• To investigate the asymptotic behavior of the completion time in the large-scale limit.
• To quantify the influence of boundary effects on the tessellation process.

Main Methods:

• Analysis of a spatial birth-growth process with seeds arriving as a unit-intensity Poisson point process.
• Derivation of asymptotic probabilities for covering a region A using a spherical Poisson Boolean model.
• Generalization of existing work to include iid small random radii for grains.

Main Results:

• Derived an asymptotic formula for the completion time of the Johnson-Mehl tessellation in 2D.
• Quantified the contribution of boundary effects to the completion time, showing their significant impact.
• Extended results to higher dimensions, where boundary effects become dominant.

Conclusions:

• Boundary effects play a critical role in the Johnson-Mehl tessellation completion time, contrary to previous assumptions.
• The derived formulas provide accurate predictions for tessellation completion times in various dimensions.
• The study introduces new methods for analyzing Poisson Boolean models with random radii.