Abstract
We prove that for Voronoi percolation on \(\mathbb {R}^d\)\((d\ge 2)\), there exists \(p_c=p_c(d)\in (0,1)\) such that
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for \(p<p_c\), there exists \(c_p>0\) such that \(\mathbb {P}_p[0\text { connected to distance }n]\le \exp (-c_pn)\),
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there exists \(c>0\) such that for \(p>p_c, \mathbb {P}_p[0\text { connected to infinity}]\ge c(p-p_c)\).
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; see [10, 11].
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Acknowledgements
The authors are thankful to Asaf Nachmias for reading the manuscript and his helpful comments. This research was supported by the IDEX grant from Paris-Saclay, a grant from the Swiss FNS, and the NCCR SwissMAP.
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Duminil-Copin, H., Raoufi, A. & Tassion, V. Exponential decay of connection probabilities for subcritical Voronoi percolation in \(\mathbb {R}^d\). Probab. Theory Relat. Fields 173, 479–490 (2019). https://doi.org/10.1007/s00440-018-0838-9
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DOI: https://doi.org/10.1007/s00440-018-0838-9