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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 2007–2017 | Cite as

Energy of the Coulomb Gas on the Sphere at Low Temperature

  • Carlos Beltrán
  • Adrien HardyEmail author
Article

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. 1.
    Alishahi K., Zamani M.: The spherical ensemble and uniform distribution of points on the sphere. Electron. J. Probab. 20(23), 1–27 (2015)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beltrán C.: Harmonic properties of the logarithmic potential and the computability of elliptic fekete points. Constr. Approx. 37, 135–165 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bétermin L., Sandier E.: Renormalized energy and symptotic expansion of optimal logarithmic energy on the sphere. Constr. Approx. 47(1), 39–74 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brauchart J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comput. 77(263), 1599–1613 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Butez R.: Large deviations for the empirical measure of random polynomials: revisit of the Zeitouni–Zelditch theorem. Electron. J. Probab. 21, 37 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Butez, R., Zeitouni, O.: Universal large deviations for Kac polynomials. Electron. Commun. Probab. 22(6), 1–10 (2017)Google Scholar
  10. 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)Google Scholar
  11. 11.
    Dubickas A.: On the maximal product of distances between points on a sphere. Liet. Mat. Rink. 36(3), 303–312 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Etayo, U., Beltrán, C.: Work in progressGoogle Scholar
  13. 13.
    Forrester, P.J.: Log-Gases and Random Matrices. Number 34 in London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2010)Google Scholar
  14. 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)CrossRefzbMATHGoogle Scholar
  15. 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)zbMATHGoogle Scholar
  16. 16.
    Jost J.: Postmodern Analysis, Universitext, 3rd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  17. 17.
    Krishnapur, M.: Zeros of random analytic functions. Ph.D. Thesis, U.C. Berkley (2006)Google Scholar
  18. 18.
    Rakhmanov E.A., Saff E.B., Zhou Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1(6), 647–662 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sandier E., Serfaty S.: From the Ginzburg–Landau model to vortex lattice problems. Commun. Math. Phys. 313(3), 635–743 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Smale, S.: Mathematical problems for the next century. Mathematics: Frontiers and Perspectives, pp. 271–294 (2000)Google Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zeitouni O., Zelditch S.: Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not. 20, 3935–3992 (2010)MathSciNetzbMATHGoogle Scholar
  23. 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)Google Scholar
  24. 24.
    Zhong Q.: Energies of zeros of random sections on Riemann surfaces. Indiana Univ. Math. J. 57(4), 1753–1780 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Departamento MATESCOUniversidad de CantabriaSantanderSpain
  2. 2.Laboratoire Paul PainlevéUniversité de LilleVilleneuve d’Ascq CedexFrance

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