Geometric and Functional Analysis

, Volume 28, Issue 2, pp 443–508 | Cite as

Fluctuations of Two Dimensional Coulomb Gases

Article
  • 26 Downloads

Abstract

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the support of the equilibrium measure. This can be stated in terms of convergence of the random electrostatic potential to a Gaussian Free Field.

Our result is the first to be valid at arbitrary temperature and at the mesoscopic scales, and we recover previous results of Ameur-Hendenmalm-Makarov and Rider-Virág concerning the determinantal case, with weaker assumptions near the boundary. We also prove moderate deviations upper bounds, or rigidity estimates, for the linear statistics and a convergence result for those corresponding to energy-minimizers.

The method relies on a change of variables, a perturbative expansion of the energy, and the comparison of partition functions deduced from our previous work. Near the boundary, we use recent quantitative stability estimates on the solutions to the obstacle problem obtained by Serra and the second author.

Mathematics Subject Classification

60F05 60K35 60B10 60B20 82B05 60G15 

Keywords and phrases

Coulomb gas β-ensembles Log gas Central Limit Theorem Gaussian free field Linear statistics Ginibre ensemble 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AHM11.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random normal matrices. Duke Math. J 159(1), 31–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. AHM15.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Random normal matrices and Ward identities. Ann. Probab. 43(3), 1157–1201 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. AJ81.
    Alastuey, A., Jancovici, B.: On the classical two-dimensional one-component Coulomb plasma. Journal de Physique 42(1), 1–12 (1981)MathSciNetCrossRefGoogle Scholar
  4. AOC12.
    Ameur, Y., Ortega-Cerdà, J.: Beurling-Landau densities of weighted Fekete sets and correlation kernel estimates. J. Funct. Anal 263, 1825–1861 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. BBNY16.
    R. Bauerschmidt, P. Bourgade, M. Nikula, and H.-T. Yau.The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem. arXiv:1609.08582, 2016
  6. BBNY17.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: Local density for two-dimensional one-component plasma. Communications in Mathematical Physics 356(1), 189–230 (2017)MathSciNetCrossRefMATHGoogle Scholar
  7. BEY12.
    P. Bourgade, L. Erdös, and H.-T. Yau. Bulk universality of general \(\beta \)-ensembles with non-convex potential. J. Math. Phys., 53(9): (2012), 095221, 19Google Scholar
  8. BEY14.
    Bourgade, P., Erdös, L., Yau, H.-T.: Universality of general \(\beta \)-ensembles. Duke Math. J. 163(6), 1127–1190 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. BFG13.
    F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for Beta-matrix models and universality. arXiv preprint arXiv:1311.2315, 2013
  10. BG13a.
    G. Borot and A. Guionnet. Asymptotic expansion of beta matrix models in the multi-cut regime. arXiv:1303.1045, 032013
  11. BG13b.
    Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the one-cut regime. Comm. Math. Phys. 317(2), 447–483 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. BL16.
    F. Bekerman and A. Lodhia. Mesoscopic central limit theorem for general \(\beta \)-ensembles. arXiv:1605.05206, 2016
  13. BLS17.
    F. Bekerman, T. Leblé, and S. Serfaty. CLT for fluctuations of \(\beta \)-ensembles with general potential. arXiv:1706.09663, 2017
  14. Caf98.
    L. A. Caffarelli. The obstacle problem revisited. J.Fourier Anal. Appl., (4-5)4 (1998), 383–402Google Scholar
  15. Cha17.
    S. Chatterjee. Rigidity of the three-dimensional hierarchical Coulomb gas. arXiv:1708.01965, 2017
  16. CHM16.
    D. Chafai, A. Hardy, and M. Maida. Concentration for Coulomb gases and Coulomb transport inequalities. arXiv preprint arXiv:1610.00980, 2016
  17. CMM15.
    Cunden, F.D., Maltsev, A., Mezzadri, F.: Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition. Physical Review E 91(6), 060105 (2015)CrossRefGoogle Scholar
  18. CR76.
    L.A. Caffarelli and N.M. Riviere. Smoothness and analyticity of free boundaries in variational inequalities. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, (2)3 (1976), 289–310Google Scholar
  19. DL12.
    R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology: Volume 1 Physical Origins and Classical Methods. Springer Science & Business Media, (2012)Google Scholar
  20. Fel71.
    Feller, W.: An introduction to probability theory and itsapplications, vol. II. Second edition. John Wiley & Sons Inc, New York-London-Sydney (1971)Google Scholar
  21. For10.
    P.J. Forrester. Log-gases and random matrices, volume 34 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, (2010)Google Scholar
  22. Fro35.
    O. Frostman. Potentiel d’équilibre et capacité desensembles avec quelques applications à la théorie des fonctions.Meddelanden Mat. Sem. Univ. Lund 3, (1935) 115 sGoogle Scholar
  23. Gin65.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Mathematical Phys 6, 440–449 (1965)MathSciNetCrossRefMATHGoogle Scholar
  24. GP12.
    S. Ghosh and Y. Peres. Rigidity and tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. arXiv:1211.2381, 2012
  25. GT15.
    D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Springer, (2015)Google Scholar
  26. JLM93.
    Jancovici, B., Lebowitz, J., Manificat, G.: Largecharge fluctuations in classical Coulomb systems. J. Statist. Phys 72(3–4), 773–7 (1993)MathSciNetCrossRefMATHGoogle Scholar
  27. Joh98.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)MathSciNetCrossRefMATHGoogle Scholar
  28. Kin78.
    Kinderlehrer, D.: Variational inequalities and free boundary problems. Bulletin of the American Mathematical Society 84(1), 7–26 (1978)MathSciNetCrossRefMATHGoogle Scholar
  29. Lau83.
    R. B. Laughlin. Anomalous quantum hall effect: anincompressible quantum fluid with fractionally charged excitations. Physical Review Letters, (18)50(1983), 1395Google Scholar
  30. Leb17.
    T. Leblé. Local microscopic behavior for 2D Coulomb gases. Probability Theory and Related Fields, (3-4) 169 (2017), 931–976Google Scholar
  31. LLW17.
    G. Lambert, M. Ledoux, and C. Webb. Stein’s method for normal approximation of linear statistics of beta-ensembles. arXiv:1706.10251, 062017
  32. LRY16.
    E. H. Lieb, N. Rougerie, and J. Yngvason. Rigidity of the laughlin liquid. arXiv preprint arXiv:1609.03818, 2016
  33. LS17.
    Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of Log and Riesz gases. Inventiones mathematicae 210(3), 645–757 (2017)MathSciNetCrossRefMATHGoogle Scholar
  34. LSZ17.
    Leblé, T., Serfaty, S., Zeitouni, O.: Large deviations for the two-dimensional two-component plasma. Communications in Mathematical Physics 350(1), 301–360 (2017)MathSciNetCrossRefMATHGoogle Scholar
  35. Meh04.
    M. L. Mehta. Random matrices. Third edition. Elsevier Academic Press, (2004)Google Scholar
  36. NS10.
    F. Nazarov and M. Sodin. Fluctuations in random complex zeroes: asymptotic normality revisited. arXiv preprint arXiv:1003.4251, 2010
  37. PRN18.
    Petrache, M., Rota-Nodari, S.: Equidistribution of jellium energy for coulomb and riesz interactions. Constructive Approximation 47(1), 163–210 (2018)MathSciNetCrossRefMATHGoogle Scholar
  38. PS15.
    Petrache, M., Serfaty, S.: Next order asymptotics and renormalized energy for Riesz interactions. Journal of the Institute of Mathematics of Jussieu, FirstView 1–69, 5 (2015)MATHGoogle Scholar
  39. PSU12.
    A. Petrosyan, H. Shahgholian, and N.N. Uraltseva. Regularity of free boundaries in obstacle-type problems, volume136. American Mathematical Society Providence (RI), (2012)Google Scholar
  40. PZ17.
    E. Paquette and O. Zeitouni. The maximum of the cuefield. International Mathematics Research Notices, page rnx033, (2017)Google Scholar
  41. RNS14.
    S. Rota-Nodari and S. Serfaty. Renormalized energy equidistribution and local charge balance in 2d coulomb systems. International Mathematics Research Notices, (2014)Google Scholar
  42. RS15.
    N. Rougerie and S. Serfaty. Higher-dimensional Coulomb gases and renormalized energy functionals. Communications on Pure and Applied Mathematics, (2015)Google Scholar
  43. RSY14.
    Rougerie, N., Serfaty, S., Yngvason, J.: Quantum hallphases and plasma analogy in rotating trapped bose gases. Journal of Statistical Physics 154(1–2), 2–50 (2014)MathSciNetCrossRefMATHGoogle Scholar
  44. RV07.
    B. Rider and B. Virag. The noise in the circular law and the Gaussian free field. Int. Math. Res. Not, (2007), 2Google Scholar
  45. RY15.
    Rougerie, N., Yngvason, J.: Incompressibility estimates for the laughlin phase. Communications in Mathematical Physics 336(3), 1109–1140 (2015)MathSciNetCrossRefMATHGoogle Scholar
  46. Ser15.
    Serfaty, S.: Coulomb Gases and Ginzburg-Landau Vortices. Eur. Math. Soc, Zurich Lectures in Advanced Mathematics (2015)CrossRefMATHGoogle Scholar
  47. Ser17.
    S. Serfaty. Microscopic description of Log and Coulomb gases.arXiv preprint arXiv:1709.04089, (2017)
  48. Shc13.
    Shcherbina, M.: Fluctuations of linear eigenvalue statistics of\(\beta \) matrix models in the multi-cut regime. J. Stat. Phys. 151(6), 1004–1034 (2013)MathSciNetCrossRefMATHGoogle Scholar
  49. Shc14.
    M. Shcherbina. Change of variables as a method to study general \(\beta \)-models: Bulk universality. J. Math. Phys., 55(4): (2014), 043504, 23Google Scholar
  50. She07.
    Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Related Fields 139(3–4), 521–541 (2007)MathSciNetCrossRefMATHGoogle Scholar
  51. SM76.
    Sari, R., Merlini, D.: On the \(\nu \)-dimension alone-component classical plasma: the thermodynamic limit problem revisited. J. Statist. Phys. 14(2), 91–100 (1976)MathSciNetCrossRefGoogle Scholar
  52. Sod04.
    M. Sodin. Zeroes of gaussian analytic functions. arXiv:math/0410343, (2004)
  53. SS15a.
    Sandier, E., Serfaty, S.: 1D Log gases and there normalized energy: Crystallization at vanishing temperature. Prob. Theor. Rel. Fields 162, 795–846 (2015)CrossRefMATHGoogle Scholar
  54. SS15b.
    Sandier, E., Serfaty, S.: 2D Coulomb gases and there normalized energy. Annals Probab. 43, 2026–2083 (2015)MathSciNetCrossRefMATHGoogle Scholar
  55. SS17.
    S. Serfaty and J. Serra. Quantitative stability of the free boundary in the obstacle problem. arXiv:1708.01490, (2017)
  56. ST97.
    E. B. Saff and V. Totik. Logarithmic Potentials with External Fields. Grundlehren der mathematischen Wissenchaften 316, Springer-Verlag, Berlin, (1997)Google Scholar
  57. STG99.
    Stormer, H., Tsui, D., Gossard, A.: The fractional quantum hall effect. Reviews of Modern Physics 71(2), S298 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Courant Institute, New York UniversityNew YorkUSA

Personalised recommendations