Annealed scaling relations for Voronoi percolation

TL;DR

This paper establishes annealed scaling relations for planar Voronoi percolation, marking the first such result for a continuum percolation model, and extends key properties to the near-critical regime.

Contribution

It proves annealed scaling relations for Voronoi percolation and extends the quenched box-crossing property to near-critical regimes, inspired by Bernoulli percolation methods.

Findings

First annealed scaling relations for continuum Voronoi percolation

Extension of quenched box-crossing property to near-critical regime

Analysis of quenched and annealed pivotal events

Abstract

We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [Kes87]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [AGMT16]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [AGMT16] to the near-critical regime.