On the spectral function of the Poisson-Voronoi cells (Q2738908)

The Poisson-Voronoi tessellation is a partition of \({\mathbb R}^d\) into the following polyedral, convex, bounded cells: NEWLINE\[NEWLINEC(x) = \{y\in {\mathbb R}^d;\;|y-x'|\leq |y-x|,\;x'\in\Phi\},\qquad x\in\Phi,NEWLINE\]NEWLINE where \(\Phi\) is a Poisson point process over \({\mathbb R}^d\) whose intensity is the Lebesgue measure. This paper studies the spectral function \(\varphi(t) = \sum_{n\geq 1} e^{-\lambda_n t}\), \(t > 0\), of the random Laplacian with Dirichlet boundary conditions in the typical partitioning cell (which is defined by some averaging). It is shown that \({\mathbb E} [\varphi(t)]\) is a functional of the convex hull of a \(d\)-dimensional Brownian bridge. In dimension 2, this makes it possible to get precise asymptotics at infinity of the Laplace transform of the law of the first random eigenvalue \(\lambda_1\).