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

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 ${\mathbb{P}}_p[0\text{ connected to distance }n]\le \exp (-c_{p}n)$, - there exists $c>0$ such that for $p>p_c$, ${\mathbb{P}}_p[0\text{ connected to }\infty ]\ge c(p-p_c)$. 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.

8 pages

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