On the spectral function of the Poisson–Voronoi cells

Abstract

Denote by ϕ(t)=∑ n⩾1 e −λ nt , t>0 , the spectral function related to the Dirichlet Laplacian for the typical cell C of a standard Poisson–Voronoi tessellation in R d, d⩾2 . We show that the expectation E ϕ(t) , t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E ϕ(t) , when t→0 +,+∞. In particular, we prove that the law of the first eigenvalue λ 1 of C satisfies the asymptotic relation ln P {λ 1⩽t}∼−2 dω dj (d−2)/2 d·t −d/2 when t→0 +, where ω d and j ( d−2)/2 are respectively the Lebesgue measure of the unit ball in R d and the first zero of the Bessel function J ( d−2)/2 .