May 2021 The annealed spectral sample of Voronoi percolation
Hugo Vanneuville
Author Affiliations +
Ann. Probab. 49(3): 1554-1606 (May 2021). DOI: 10.1214/20-AOP1494


In this paper, we introduce and study the annealed spectral sample of Voronoi percolation, which is a continuous and finite point process in R2 whose definition is mostly inspired by the spectral sample of Bernoulli percolation introduced in (Acta Math. 205 (2010) 19–104) 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.


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Hugo Vanneuville. "The annealed spectral sample of Voronoi percolation." Ann. Probab. 49 (3) 1554 - 1606, May 2021.


Received: 1 December 2019; Revised: 1 October 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1494

Primary: 60K35 , 60K37
Secondary: 82C43

Keywords: annealed , Exceptional times , Fourier spectrum , percolation , Voronoi

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 3 • May 2021
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