The annealed spectral sample of Voronoi percolation

Abstract

In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in ${\mathbb{R}}^2$ whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in [GPS10] by Garban, Pete and Schramm. We show a clustering effect as well as estimates on the full lower tail of this spectral object. Our main motivation is the study of two models of dynamical critical Voronoi percolation in the plane. In the first model, the Voronoi tiling does not evolve in time while the colors of the cells are resampled at rate $1$. In the second model, the centers of the cells move according to (independent) long range stable Lévy processes but the colors do not evolve in time. We prove that for these two dynamical processes there exist almost surely exceptional times with an unbounded monochromatic component.

51 pages, 4 figures. Minor changes introduced (see in particular Lemma 2.4), an appendix (about measurability issues) added, two figures added in Section 3.3, some local changes/corrections introduced in Section 3.3