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

Abstract

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.

References

  1. 1

    Ameur, Y.: A density theorem for weighted Fekete sets. Int. Math. Res. Not. 16, 5010–5046 (2017)

  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)

    MathSciNet  Article  MATH  Google 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). https://doi.org/10.1007/s00365-017-9368-4

  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)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Brezis H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)

    Book  Google 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)

  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)

    ADS  Article  Google 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)

    ADS  Article  Google 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) 086

  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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  18. 18

    Hedenmalm H., Makarov N.: Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. 106, 859–907 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Jancovici B.: Exact results for the two-dimensional one-component plasma. Phys. Rev. Lett. 46, 386–388 (1981)

    ADS  MathSciNet  Article  Google 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)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Johansson K.: On fluctuations of eigenvalues of random normal matrices. Duke Math. J. 91, 151–204 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22

    Kang, N.-G., Makarov, N.: Gaussian free field and conformal field theory. Astérisque 353, vii+136 (2013)

  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)

    ADS  Article  Google 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)

  25. 25

    Rougerie N., Yngvason J.: Incompressibility estimates for the Laughlin phase, part II. Comm. Math. Phys. 339, 263–277 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Saff E.B., Totik V.: Logarithmic Potentials with External Fields. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  27. 27

    Seo, S.-M.: Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge, arXiv:1508.06591

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Correspondence to Yacin Ameur.

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Communicated by H.-T. Yau

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Ameur, Y. Repulsion in Low Temperature \({\beta}\)-Ensembles. Commun. Math. Phys. 359, 1079–1089 (2018). https://doi.org/10.1007/s00220-017-3027-2

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