Sur la fonction spectrale des cellules de Poisson–Voronoi

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 in the two-dimensional case (d=2) the law of the first eigenvalue λ 1 of C satisfies the asymptotic relation ln E e −tλ 1 ∼−t 1/24 π j 0 , when t→+∞, where j 0 is the first zero of the Bessel function J 0 .