Exponential decay of connection probabilities for subcritical Voronoi percolation in ℝ^d

2017/05/22 by Hugo Duminil‐Copin, Hugo Duminil-Copin, Aran Raoufi +4 · 2 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #math.CO #math.PR

paper · pdf · doi:10.48550/arxiv.1705.07978

8 pages

arxiv created 2017/05/22 · openalex publication_date 2017/05/22 · arxiv updated 2017/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for Voronoi percolation on ℝ^d, there exists p_c∈[0,1] such that - for p<p_c, there exists c_p>0 such that ℙ_p[0 connected to distance n]≤ exp(-c_p n), - there exists c>0 such that for p>p_c, ℙ_p[0 connected to ∞]≥ 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.

Cited by

• Criticality and Covered Area Fraction in Confetti and Voronoi Percolation
• The OSSS Method in Percolation Theory

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