# The annealed spectral sample of Voronoi percolation

## TL;DR

This paper introduces the annealed spectral sample for Voronoi percolation, analyzing its properties and applying it to study dynamical models where exceptional times with infinite monochromatic clusters occur.

## Contribution

It defines and studies the annealed spectral sample of Voronoi percolation, providing new insights into its clustering behavior and tail estimates, and applies these to dynamical models.

## Key findings

- Clustering effect of the spectral sample identified.
- Lower tail estimates established for the spectral object.
- Existence of exceptional times with unbounded monochromatic components proven.

## 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 L\'evy 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.