Limits of Poisson-Laguerre tessellations

For sequences of Poisson-Laguerre tessellations and their duals in $\mathbb{R}^d$, generated by Poisson point processes $(η_n){n\in\mathbb{N}}$ in $\mathbb{R}^d \times \mathbb{R}$, we prove limit theorems as $n\to \infty$. The intensity measure of $η_n$ has density of the form $(v,h)\mapsto f_n(h)$ with respect to the Lebesgue measure, where $v\in \mathbb{R}^d$ and $h\in \mathbb{R}$. Identifying a tessellation with its skeleton (the union of the boundaries of all its cells) we provide verifiable conditions on $(f_n){n\in\mathbb{N}}$ that ensure convergence either to the classical Poisson-Voronoi/Poisson-Delaunay tessellation or to another Poisson-Laguerre tessellation. We also discuss convergence of the corresponding typical cells. As a corollary, we show that the Poisson-Voronoi and the Poisson-Delaunay tessellations arise as limits of the $β$-Voronoi and the $β$-Delaunay tessellations, respectively, as $β\to -1$.

💡 Research Summary

The paper investigates the asymptotic behavior of Poisson‑Laguerre tessellations and their duals when the underlying Poisson point processes are defined on the product space (\mathbb{R}^d\times\mathbb{R}) with intensity measures of the form ((v,h)\mapsto f_n(h)). Here (v) denotes the spatial coordinate and (h) a weight (interpreted as time, height, or another auxiliary variable). For each point ((v,h)) the associated Laguerre cell is defined by
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