The concept of hyperuniformity has been introduced by Torquato and Stillinger in 2003 as a notion to detect structural behaviour intermediate between crystalline order and amorphous disorder. The present paper studies a generalisation of this concept to the unit sphere. It is shown that several well studied determinantal point processes are hyperuniform.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Alexander, R.: On the sum of distances between \(n\) points on a sphere. Acta Math. Hung. 23(3–4), 443–448 (1972)
Alishahi, K., Zamani, M.: The spherical ensemble and uniform distribution of points on the sphere. Electron. J. Probab. 20(23), 27 (2015)
Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Axel, F., Gratias, D. (eds.): Beyond Quasicrystals. Springer, Berlin (1995)
Beltrán, C., Marzo, J., Ortega-Cerdà, J.: Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres. J. Complex. 37, 76–109 (2016)
Bourgain, J., Lindenstrauss, J.: Distribution of points on spheres and approximation by zonotopes. Israel J. Math. 64(1), 25–31 (1988)
Brauchart, J.S., Grabner, P.J., Kusner, W.: Hyperuniform point sets on the sphere: deterministic aspects. Constr. Approx. 50(1), 45–61 (2019)
de Bruijn, N.G.: Quasicrystals and their Fourier transform. Indag. Math. 48, 123–152 (1986)
Gigante, G., Leopardi, P.: Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces. Discrete Comput. Geom. 57(2), 419–430 (2017)
Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, vol. 51. American Mathematical Society, Providence, RI (2009)
Krishnapur, M.: Zeros of Random Analytic Functions. Ph.D. thesis, University of California, Berkeley (2006). ArXiv:math/0607504
Kuijlaars, A., Saff, E.B.: Asymptotics for minimal discrete energy on the sphere. Trans. Am. Math. Soc. 350(2), 523–538 (1998)
Lee, Y., Kim, W.C.: Concise Formulas for the Surface Area of the Intersection of Two Hyperspherical Caps. Tech. rep., Department of Industrial and Systems Engineering, KAIST (2014). http://ie.kaist.ac.kr/uploads/professor/tech_file/Concise+Formulas+for+the+Surface+Area+of+the+Intersection+of+Two+Hyperspherical+Caps.pdf
Leopardi, P.: A partition of the unit sphere into regions of equal area and small diameter. Electron. Trans. Numer. Anal. 25, 309–327 (2006). (electronic)
Lošdorfer Božič, A., Čopar, S.: Spherical structure factor and classification of hyperuniformity on the sphere. Phys. Rev. E 99, 032601 (2019)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems for the special functions of mathematical physics. In: Grundlehren der mathematischen Wissenschaften, vol. 52, Third enlarged edition. Springer (1966)
Meyra, A.G., Zarragoicoechea, G.J., Maltz, A.L., Lomba, E., Torquato, S.: Hyperuniformity on spherical surfaces. Phys. Rev. E 100, 022107 (2019)
Mhaskar, H., Narcowich, F., Ward, J.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comput. 70(235), 1113–1130 (2001)
Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)
Soshnikov, A.: Determinantal random point fields. Uspekhi Mat. Nauk 55(5(335)), 107–160 (2000)
Stepanyuk, T.A.: Hyperuniform point sets on flat Tori: deterministic and probabilistic aspects. Constr. Approx. (2020). https://arxiv.org/abs/1902.02973 (to appear)
Torquato, S., Stillinger, F.H.: Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68(4), 041113 (2003)
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while the first three authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Spring 2018 semester. The authors are very grateful to two anonymous referees for their valuable remarks and suggestions.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Johann S. Brauchart was supported by the Lise Meitner scholarship M 2030 of the Austrian Science Foundation FWF.
Peter J. Grabner and Wöden Kusner were supported by the Austrian Science Fund FWF Project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”).
Communicated by Karlheinz Gröchenig.
About this article
Cite this article
Brauchart, J.S., Grabner, P.J., Kusner, W. et al. Hyperuniform point sets on the sphere: probabilistic aspects. Monatsh Math (2020). https://doi.org/10.1007/s00605-020-01439-y
- Determinantal point processes
- Jittered sampling
Mathematics Subject Classification