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Energy of the Coulomb Gas on the Sphere at Low Temperature

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Abstract

We consider the Coulomb gas of N particles on the sphere and show that the logarithmic energy of the configurations approaches the minimal energy up to an error of order log N, with exponentially high probability and on average, provided the temperature is \({\mathcal{O}(1/N)}\).

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References

  1. Alishahi K., Zamani M.: The spherical ensemble and uniform distribution of points on the sphere. Electron. J. Probab. 20(23), 1–27 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Armentano D., Beltrán C., Shub M.: Minimizing the discrete logarithmic energy on the sphere: the role of random polynomials. Trans. Am. Math. Soc. 363(6), 2955–2965 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beltrán C.: Harmonic properties of the logarithmic potential and the computability of elliptic fekete points. Constr. Approx. 37, 135–165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bétermin L., Sandier E.: Renormalized energy and symptotic expansion of optimal logarithmic energy on the sphere. Constr. Approx. 47(1), 39–74 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blum L., Cucker F., Shub M., Smale S.: Complexity and real computation: a manifesto. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 6(1), 3–26 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brauchart J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comput. 77(263), 1599–1613 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Brauchart J.S., Hardin D.P., Saff E.B.: The next-order term for optimal riesz and logarithmic energy asymptotics on the sphere. Contemp. Math. 578(2), 31–61 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Butez R.: Large deviations for the empirical measure of random polynomials: revisit of the Zeitouni–Zelditch theorem. Electron. J. Probab. 21, 37 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Butez, R., Zeitouni, O.: Universal large deviations for Kac polynomials. Electron. Commun. Probab. 22(6), 1–10 (2017)

  10. Dragnev, P.D.: On the separation of logarithmic points on the sphere. Approximation Theory, X (St. Louis, MO, 2001), Innov. Appl. Math., pp. 137–144 (2002)

  11. Dubickas A.: On the maximal product of distances between points on a sphere. Liet. Mat. Rink. 36(3), 303–312 (1996)

    MathSciNet  MATH  Google Scholar 

  12. Etayo, U., Beltrán, C.: Work in progress

  13. Forrester, P.J.: Log-Gases and Random Matrices. Number 34 in London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2010)

  14. Hardin D.P., Michaels T., Saff E.B.: A comparison of popular point configurations on \({\mathbb{S}^{2}}\). Dolomit. Res. Notes Approx. 9, 16–49 (2016)

    Article  MATH  Google Scholar 

  15. Hough J.B., Krishnapur M., Peres Y., Virág B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, vol.51. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  16. Jost J.: Postmodern Analysis, Universitext, 3rd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  17. Krishnapur, M.: Zeros of random analytic functions. Ph.D. Thesis, U.C. Berkley (2006)

  18. Rakhmanov E.A., Saff E.B., Zhou Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1(6), 647–662 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sandier E., Serfaty S.: From the Ginzburg–Landau model to vortex lattice problems. Commun. Math. Phys. 313(3), 635–743 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Smale, S.: Mathematical problems for the next century. Mathematics: Frontiers and Perspectives, pp. 271–294 (2000)

  21. Wagner G.: On the product of distances to a point set on a sphere. J. Aust. Math. Soc. Ser. A 47(3), 466–482 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zeitouni O., Zelditch S.: Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not. 20, 3935–3992 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Zelditch, S., Zhong, Q.: Addendum to “Energies of zeros of random sections on Riemann surfaces”. Indiana Univ. Math. J. 57(4), 1753–1780 (2008). Indiana Univ. Math. J. 59(6), 2001–2005 (2010)

  24. Zhong Q.: Energies of zeros of random sections on Riemann surfaces. Indiana Univ. Math. J. 57(4), 1753–1780 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento MATESCO, Universidad de Cantabria, Avda. Los Castros s/n, Santander, Spain

    Carlos Beltrán

  2. Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655, Villeneuve d’Ascq Cedex, France

    Adrien Hardy

Authors
  1. Carlos Beltrán
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  2. Adrien Hardy
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Corresponding author

Correspondence to Adrien Hardy.

Additional information

Communicated by S. Serfaty

C. Beltrán: Partially supported by MICINN Grants MTM2017-83816-P and MTM2017-90682-REDT and by Banco de Santander-Universidad de Cantabria Grant 21.SI01.64658. A. Hardy: Partially supported by ANR JCJC BoB (ANR-16-CE23-0003) and Labex CEMPI (ANR-11-LABX-0007-01).

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Beltrán, C., Hardy, A. Energy of the Coulomb Gas on the Sphere at Low Temperature. Arch Rational Mech Anal 231, 2007–2017 (2019). https://doi.org/10.1007/s00205-018-1316-3

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  • DOI: https://doi.org/10.1007/s00205-018-1316-3

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