Abstract
In this paper we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J. Brauchart, P. Grabner, W. Kusner and J. Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls, or balls of threshold order on the flat tori are uniformly distributed. Moreover, it is also shown that QMC-designs sequences for Sobolev classes, probabilistic point sets (obtained from jittered samplings), and some determinantal point process are hyperuniform.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications, (1964)
Brandolini, L., Choirat, Ch., Colzani, L., Gigante, G., Seri, R., Travaglini, G.: Quadrature rules and distribution of points on manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13(4), 889–923 (2014)
Brauchart, J.S., Grabner, P.J., Kusner, W.: Hyperuniform point sets on the sphere: Deterministic aspects. Constructive Approximation 1–17 (2018)
Brauchart, J.S., Grabner, P.J., Kusner, W.B., Ziefle, J.: Hyperuniform point sets on the sphere: probabilistic aspects, arXiv:1809.02645
Brauchart, J.S., Hesse, K.: Numerical integration over spheres of arbitrary dimension. Constr. Approx. 25(1), 41–71 (2007)
Brauchart, J.S., Saff, E.B., Sloan, I.H., Womersley, R.S.: QMC designs: optimal order quasi Monte Carlo integration schemes on the sphere. Math. Comput. 83(290), 2821–2851 (2014)
Gariboldi, B., Gigante, G.: Optimal asymptotic bounds for designs on manifolds, arXiv:1811.12676
Gigante, G., Leopardi, P.: Diameter bounded equal measure partitions of ahlfors regular metric measure spaces. Discrete Comput. Geom. 57(2), 419–430 (2017)
Hardin, D., Saff, E., Simanek, B., Su, Y.: Next Order Energy Asymptotics for Riesz Potentials on Flat Tori, IMRN (2018)
Hardin, D.P., Saff, E.B., Simanek, B.: Periodic discrete energy for long-range potentials. J. Math. Phys. 55, 123509 (2014)
Hesse, K.: A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numer. Math. 103(3), 413–433 (2006)
Hesse, K., Sloan, I.H.: Optimal lower bounds for cubature error on the sphere \(S^{2}\). J. Complex. 21(6), 790–803 (2005)
Hesse, K., Sloan, I.H.: Cubature over the sphere \(S^{2}\) in Sobolev spaces of arbitrary order. J. Approx. Theory 141(2), 118–133 (2006)
Jiao, Y., Lau, T., Hatzikirou, H., Meyer-Hermann, M., Corbo, J.C., Torquato, S.: Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem. Phys. Rev. E 89, 022721 (2014)
Krishnapur, M., Peres, Y., Ben Hough, J., Virag, B.: Zeros of Gaussian analytic functions and determinantal point processes, Providence, RI, vol. 51, University Lecture Series, American Mathematical Society (2009)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)
Lei, Q.-L., Ciamarra, M.P., Ni, R.: Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations, Sci. Adv. 5, eaau7423 (2019)
Marzo, J., Ortega-Cerda, J.: Expected Riesz energy of some determinantal processes on flat tori. Constr. Approx. 47, 75–78 (2018)
Oğuz, E.C., Socolar, J.E.S., Steinhardt, P.J., Torquato, S.: Hyperuniformity of quasicrystals. Phys. Rev. B 95, 054119 (2017)
Torquato, S., Stillinger, F.H.: Local density fluctuations, hyperuniformity, and order metrics. Phys. Rev. E 68(4), 041–113 (2003)
Torquato, S.: Hyperuniform states of matter. Phys. Rep. 745, 1–95 (2018)
Zachary, C.E., Torquato, S.: Hyperuniformity in point patterns and two-phase heterogeneous media. J. Stat. Mech. Theory Exp., P12015 (2009)
Acknowledgements
I would like to express my huge gratitude to Peter Grabner, who posed this problem, gave much valuable advice and ideas, and also for his support and fruitful discussions. Also I would like to thank Dmitriy Bilyk for his very useful comments and help in the proof of Lemma 3.5 and two anonymous referees for their many valuable remarks.
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Communicated by Edward B. Saff.
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The author is supported by the Austrian Science Fund FWF Projects F5503 and F5506-N26 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”) and partially is supported by grant of NASU for groups of young scientists (Project No. 16-10/2018) .
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Stepanyuk, T.A. Hyperuniform Point Sets on Flat Tori: Deterministic and Probabilistic Aspects. Constr Approx 52, 313–339 (2020). https://doi.org/10.1007/s00365-020-09512-3
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DOI: https://doi.org/10.1007/s00365-020-09512-3