Communications in Mathematical Physics

, Volume 359, Issue 3, pp 1079–1089 | Cite as

Repulsion in Low Temperature \({\beta}\)-Ensembles

  • Yacin Ameur
Open Access


We prove a result on separation of particles in a two-dimensional Coulomb plasma, which holds provided that the inverse temperature \({\beta}\) satisfies \({\beta > 1}\). For large \({\beta}\), separation is obtained at the same scale as the conjectural Abrikosov lattice optimal separation.


  1. 1.
    Ameur, Y.: A density theorem for weighted Fekete sets. Int. Math. Res. Not. 16, 5010–5046 (2017)Google Scholar
  2. 2.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Ward identities and random normal matrices. Ann. Probab. 43 (2015), 1157–1201. Cf. arXiv:1109.5941v3 for a different version
  3. 3.
    Ameur, Y., Kang, N.-G., Makarov, N.: Rescaling Ward Identities in the Random Normal Matrix Model. arXiv:1410.4132v4
  4. 4.
    Ameur, Y., Kang, N.-G., Makarov, N., Wennman, A.: Scaling Limits of Random Normal Matrix Processes at Singular Boundary Points, arXiv:1510.08723
  5. 5.
    Ameur Y., Ortega-Cerdà J.: Beurling–Landau densities of weighted Fekete sets and correlation kernel estimates. J. Funct. Anal. 263, 1825–1861 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ameur, Y., Seo, S.-M.: Microscopic densities and Fock-Sobolev spaces. J. d’Analyse Mathématique (to appear). See also arXiv:1610.10052v3
  7. 7.
    Ameur, Y., Seo, S.-M.: On bulk singularities in the random normal matrix model. Constr. Approx. (to appear).
  8. 8.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: The Two-Dimensional Coulomb Plasma: Quasi-free Approximation and Central Limit Theorem, arXiv:1609.08582
  9. 9.
    Bourgade P., Erdős L., Yau H.-T.: Universality of general \({\beta}\)-ensembles. Duke Math. J. 163, 1127–1190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)CrossRefGoogle Scholar
  11. 11.
    Caillol, J.M., Levesque, D, Weiss, J.J., Hansen, J.P.: A Monte-Carlo study of the classical two-dimensional one-component plasma. J. Stat. Phys. 28, 325–349 (1982)Google Scholar
  12. 12.
    Can T., Forrester P.J., Téllez G., Wiegmann P.: Singular behavior at the edge of Laughlin states. Phys. Rev. B 89, 235137 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Can T., Laskin M., Wiegmann P.: Fractional quantum Hall effect in a curved space: gravitational anomaly and electromagnetic response. Phys. Rev. Lett. 113, 046803 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    Carroll, T., Marzo, J., Massaneda, X., Ortega-Cerdà, J.: Equidistribution and \({\beta}\) ensembles. Annales de la Faculté des Sciences de Toulouse (Mathématiques), arXiv:1509.06725
  15. 15.
    Ferrari, F., Klevtsov, S.: FQHE on curved backgrounds, free fields and large \({N}\). JHEP12 (2014) 086Google Scholar
  16. 16.
    Forrester P.J.: Analogies between random matrix ensembles and the one-component plasma in two dimensions. Nucl. Phys. B 904, 253–281 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  18. 18.
    Hedenmalm H., Makarov N.: Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. 106, 859–907 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jancovici B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Jansen S., Lieb E.H., Seiler R.: Symmetry breaking in Laughlin’s state on a cylinder. Commun. Math. Phys. 285, 503–535 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Johansson K.: On fluctuations of eigenvalues of random normal matrices. Duke Math. J. 91, 151–204 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kang, N.-G., Makarov, N.: Gaussian free field and conformal field theory. Astérisque 353, vii+136 (2013)Google Scholar
  23. 23.
    Laskin M., Chiu Y.H., Can T., Wiegmann P.: Emergent conformal symmetry of quantum Hall states on singular surfaces. Phys. Rev. Lett. 117, 266803 (2016)ADSCrossRefGoogle Scholar
  24. 24.
    Nodari, S.R., Serfaty, S.: Renormalized energy equidistribution and local charge balance in 2D Coulomb systems. Int. Math. Res. Not. 11, 3035–3093 (2015)Google Scholar
  25. 25.
    Rougerie N., Yngvason J.: Incompressibility estimates for the Laughlin phase, part II. Comm. Math. Phys. 339, 263–277 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Saff E.B., Totik V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  27. 27.
    Seo, S.-M.: Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge, arXiv:1508.06591

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceLund UniversityLundSweden

Personalised recommendations