Advertisement

Communications in Mathematical Physics

, Volume 364, Issue 1, pp 1–43 | Cite as

On Percolation of Two-Dimensional Hard Disks

  • Alexander Magazinov
Article
  • 69 Downloads

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 \({\lambda > \lambda_0}\) \({(\epsilon}\)).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.

References

  1. 1.
    Aristoff D.: Percolation of hard disks. J. Appl. Probab. 51(1), 235–246 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balister P.N., Bollobás B.: Counting regions with bounded surface area. Commun. Math. Phys. 273(2), 305–315 (2007)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Blanc, X., Lewin, M.: The crystallization conjecture: a review (2015). arXiv:1504.01153 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bowen L., Lyons R., Radin C., Winkler P.: A Solidification Phenomenon in Random Packings. SIAM J. Math. Anal. 38(4), 1075–1089 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cohn H., Elkies N.: New upper bounds on sphere packings I. Ann. Math. 157, 689–714 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fejes Tóth L.: Lagerungen in der Ebene, auf der Kugel und in Raum. Springer, New York (1953)CrossRefGoogle Scholar
  7. 7.
    Hales T.C.: A proof of the Kepler conjecture. Ann. Math. 162, 1065–1185 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Harris T.E.: A lower bound for the critical probability in a certain percolation process.Math. Proc. Camb. Philos. Soc. 56(1), 13–20 (1960)ADSCrossRefGoogle Scholar
  9. 9.
    Jansen, S.: Continuum percolation for Gibbsian point processes with attractive interactions. Electron. J. Probab. 21, paper no. 47 (2016)Google Scholar
  10. 10.
    Kingman J.F.C.: Poisson Processes. Wiley, Hoboken (1993)zbMATHGoogle Scholar
  11. 11.
    Lebowitz J.L., Mazel A.E.: Improved Peierls argument for high-dimensional Ising models. J. Stat. Phys. 90(3), 1051–1059 (1998)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Mase S., Møller J., Stoyan D., Waagepetersen R.P., Döge G.: Packing densities and simulated tempering for hard core Gibbs point processes. Ann. Inst. Stat. Math. 53(4), 661–680 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Richthammer T.: Translation-invariance of two-dimensional Gibbsian point processes. Commun. Math. Phys. 274, 81–122 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rogers C.A.: The packing of equal spheres. Proc. Lond. Math. Soc. 3(4), 609–620 (1958)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ruelle D.: Statistical Mechanics: Rigorous Results. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  16. 16.
    Stucki, K.: Continuum percolation for Gibbs point processes. Electron. Commun. Probab. 18, paper no. 67 (2013). arXiv:1305.0492
  17. 17.
    Thue, A.: Omnogle geometrisk-taltheoretiske Theoremer. Forhdl. skandinaviske naturforskeres, 352–353 (1892)Google Scholar
  18. 18.
    Viazovska M.: The sphere packing problem in dimension 8. Ann. Math. 185(3), 991–1015 (2017) arXiv:1603.04246 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cohn H., Kumar A., Miller S.D., Radchenko D., Viazovska M.: The sphere packing problem in dimension 24. Ann. Math 185, 1017–1033 (2017) arXiv:1603.06518 MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

Personalised recommendations