Exponential decay of connection probabilities for subcritical Voronoi percolation in \(\mathbb {R}^d\)
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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 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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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 \(p>p_c, \mathbb {P}_p[0\text { connected to infinity}]\ge c(p-p_c)\).
Mathematics Subject Classification
60K35 82B21 82B43Notes
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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