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On Percolation of Two-Dimensional Hard Disks

  • Alexander Magazinov
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Abstract

Let QL =  [−L, L]2 be a square in the plane \({\mathbb{R}^{2}}\) . We consider the hard-core model with arbitrary boundary conditions in which a random set of non-intersecting unit disks (i.e., a packing) with centers in QL is sampled. The density of the packing is controlled by the an intensity parameter \({\lambda}\) similarly to the Poisson point process. Given \({\epsilon}\) > 0, we consider the random graph \({{G}_{\epsilon}}\) in which disks (the vertices) are connected by an edge if they are at distance ≤ \({\epsilon}\) from each other.We prove that G is highly connected when \({\lambda}\) is greater than a certain threshold λ0 =  λ0(\({\epsilon}\)). Namely, given a square annulus with inner radius L1 and outer radius L2 (L1 < L2 < L), the probability that the annulus is crossed by \({{G}_{\epsilon}}\) is at least 1 − C exp(−cL1). We also extend our results to random packings of disks in the entire plane using the well-known notion of a Gibbs state. We show that a random graph \({{G}_{\epsilon}}\) corresponding to any Gibbs state almost surely has an infinite connected component whenever the intensity parameter \({\lambda}\) satisfies \({\lambda > \lambda_0}\) \({(\epsilon}\)).

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Notes

Acknowledgments

The author is thankful to A. Sodin, R. Peled and N. Chandgotia for discussion that helped improving the earlier versions of the paper. The research is supported in part by ERC Starting Grant 678520.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

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