Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^d$

TL;DR

This paper establishes exponential decay of connection probabilities below the critical threshold and a linear lower bound above it for Voronoi percolation in any dimension, providing new insights especially in two dimensions.

Contribution

It proves exponential decay in the subcritical phase and a linear lower bound in the supercritical phase for Voronoi percolation, advancing understanding of phase transition sharpness.

Findings

Exponential decay of connection probabilities for p<p_c

Linear lower bound for connection probability for p>p_c

New proof that p_c(2)=1/2 in two dimensions

Abstract

We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.