Abstract
We describe an extremal property of the hexagonal lattice \(\Lambda \subset \mathbb {R}^2\). Let p denote the circumcenter of its fundamental triangle (a so-called deep hole) and let \(A_r\) denote the set of lattice points that are at distance r from p
If \(\Gamma \) is a small perturbation of \(\Lambda \) in the space of lattices with fixed density and \(C_r\) denotes the set of points in \(A_r\) shifted to the new lattice, then
where \(d(\Lambda , \Gamma )\) denotes the distance between the lattices: the hexagonal lattice has the property that “far away points are closer than they are for nearby lattices”. This has implications in the calculus of variations: assume
For a certain class of compactly supported functions f, the hexagonal lattice \(\Lambda \) is then a strict local maximizer of
where the maximum runs over all lattices of fixed density.
Similar content being viewed by others
References
Baernstein II, A.: Landau’s constant, and extremal problems involving discrete subsets of \(\mathbb{C}\). In: Havin, V.P., Nikolski, N.K. (eds.) Linear and Complex Analysis Problem Book 3, Part 2. Lecture Notes in Mathematics, Chapter 18, vol. 1574, pp. 404–407. Springer, Berlin (1994)
Baernstein II, A.: A minimum problem for heat kernels of flat tori. In: Extremal Riemann Surfaces (San Francisco. CA, 1995). Contemporary Mathematics, vol. 201, pp. 227–243. American Mathematical Society, Providence, RI (1997)
Baernstein II, A., Vinson, J.P.: Local minimality results related to the Bloch and Landau constants. In: Duren, P., Heinonen, J., Osgood, B., Palka, B. (eds.) Quasiconformal Map**s and Analysis: A Collection of Papers Honoring F.W. Gehring, pp 55–89. Springer, New York (1998)
Betermin, L.: Two-dimensional theta functions and crystallization among Bravais lattices. SIAM J. Math. Anal. 48(5), 3236–3269 (2016)
Betermin, L., Petrache, M.: Dimension reduction techniques for the minimization of theta functions on lattices. J. Math. Phys. 58, 071902 (2017)
Betermin, L.: Minimal soft lattice theta functions, preprint. ar**v:1809.00473
Betermin, L.: Minimizing lattice structures for Morse potential energy in two and three dimensions, preprint. ar**v:1901.08957 (2019)
Blanc, X., Lewin, M.: The crystallization conjecture: a review. EMS Surv. Math. Sci. 2(2), 225–306 (2015)
Borwein, J.M., Borwein, P.B.: A cubic counterpart of Jacobi’s identity and the AGM. Trans. Am. Math. Soc. 332(2), 691–701 (1991)
Cohn, H., Kumar, A.: Universally optimal distribution of points on spheres. J. Am. Math. Soc. 20(1), 99–148 (2007)
Cohn, H., Kumar, A., Miller, S.D., Radchenko, D., Viazovska, M.S.: The sphere packing problem in dimension 24. Ann. Math. 187(3), 1035–1068 (2017)
Cohn, H., Kumar, A., Miller, S.D., Radchenko, D., Viazovska, M.S.: Universal optimality of \(E_8\) and Leech lattices and interpolation formulas. ar**v:1902:05438 (2019)
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, 3rd edn. Springer, New York (1999)
Faulhuber, M.: Extremal Bounds of Gaussian Gabor Frames and Properties of Jacobi’s Theta Functions. Doctoral thesis, University of Vienna (2016)
Faulhuber, M., Steinerberger, S.: Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions. J. Math. Anal. Appl. 445(1), 407–422 (2017)
Faulhuber, M.: Minimal frame operator norms via minimal theta functions. J. Fourier Anal. Appl. 24(2), 545–559 (2018)
Hales, T.: A proof of the Kepler conjecture. Ann. Math. Second Ser 162(3), 1065–1185 (2005)
Montgomery, H.: Minimal theta functions. Glasg. Math. J. 30(1), 75–85 (1988)
Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of laplacians. J. Funct. Anal. 80(1), 148–211 (1988)
Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 365–373 (1981)
Rogers, C.A.: Packing and Covering. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 54. Cambridge University Press, New York (1964)
Serre, J.P.: A Course in Arithmetic. Graduate Texts in Mathematics, vol. 7. Springer, Berlin (1973)
Steinerberger, S.: A geometric uncertainty principle with applications to Pleijel’s estimate. Annales Henri Poincare 15, 2299–2319 (2014)
Strohmer, T., Beaver, S.: Optimal OFDM design for time-frequency dispersive channels. IEEE Trans. Commun. 51(7), 1111–1122 (2003)
Theil, F.: A proof of crystallization in two dimensions. Commun. Math. Phys. 262(1), 209–236 (2006)
Viazovska, M.S.: The sphere packing problem in dimension 8. Ann. Math. 187(3), 991–1015 (2017)
Acknowledgements
The funding was provided by Österreichischen Akademie der Wissenschaften (Grant No. J4100-N32), Division of Mathematical Sciences (Grant No. 1763179) and also by Alfred P. Sloan Foundation (Grant No. #1).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Alessandro Giuliani.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M.F. is supported by the Erwin Schrödinger Program of the Austrian Science Fund (FWF): J4100-N32. S.S. is supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
Rights and permissions
About this article
Cite this article
Faulhuber, M., Steinerberger, S. An Extremal Property of the Hexagonal Lattice. J Stat Phys 177, 285–298 (2019). https://doi.org/10.1007/s10955-019-02368-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-019-02368-3