Journal of Statistical Physics

, Volume 175, Issue 6, pp 1043–1065 | Cite as

The High Temperature Crossover for General 2D Coulomb Gases

  • Gernot AkemannEmail author
  • Sung-Soo Byun


We consider N particles in the plane, influenced by a general external potential, that are subject to the Coulomb interaction in two dimensions at inverse temperature \(\beta \). At large temperature, when scaling \(\beta =2c/N\) with some fixed constant \(c>0\), in the large-N limit we observe a crossover from Ginibre’s circular law or its generalisation to the density of non-interacting particles at \(\beta =0\). Using Ward identities and saddle point methods we derive a partial differential equation of generalised Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential as well, and present the analytic solution for the density in the case of a Gaussian plus logarithmic potential.


2D Coulomb gases Normal random matrices High temperature crossover 



The authors gratefully acknowledge discussions and helpful suggestions of Trinh Khanh Duy, Adrien Hardy, Nam-Gyu Kang, Mylène Maïda, Seong-Mi Seo, Pierpaolo Vivo and Oleg Zaboronski, as well as detailed comments by Yacin Ameur and Gaultier Lambert on a preliminary version of the paper. We also wish to express our gratitude to Jeongho Kim for several valuable comments concerning the numerical verifications.


  1. 1.
    Akemann, G., Baik, J., Di Francesco, P. (ed.): The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011)Google Scholar
  2. 2.
    Akemann, G., Cikovic, M., Venker, M.: Universality at weak and strong non-hermiticity beyond the elliptic Ginibre ensemble. Commun. Math. Phys. 362(3), 1111–1141 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allez, R., Bouchaud, J.-P., Guionnet, A.: Invariant \(\beta \)-ensembles and the Gauss–Wigner crossover. Phys. Rev. Lett. 109(9), 094102 (2012)CrossRefGoogle Scholar
  4. 4.
    Allez, R., Bouchaud, J.-P., Majumdar, S.N., Vivo, P.: Invariant \(\beta \)-Wishart ensembles, crossover densities and asymptotic corrections to the Marčenko–Pastur law. J. Phys. A 46(1), 015001 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Allez, R., Guionnet, A.: A diffusive matrix model for invariant \(\beta \)-ensembles. Electron. J. Probab. 18(62), 1–30 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ambjørn, J., Chekhov, L., Kristjansen, C.F., Makeenko, Y.: Matrix model calculations beyond the spherical limit. Nucl. Phys. B 404(1–2), 127–172 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ameur, Y.: Repulsion in low temperature \(\beta \)-ensembles. Commun. Math. Phys. 359(3), 1079–1089 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Random normal matrices and Ward identities. Ann. Probab. 43(3), 1157–1201 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ameur, Y., Kang, N.-G., Makarov, N.: Rescaling Ward identities in the random normal matrix model. Constr. Approx. (2018). Google Scholar
  10. 10.
    Ameur, Y., Kang, N.-G., Seo, S.-M.: The random normal matrix model: insertion of a point charge. preprint arXiv:1804.08587 (2018)
  11. 11.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem. preprint arXiv:1609.08582 (2016)
  12. 12.
    Berman, R.J.: Determinantal point processes and fermions on complex manifolds: bulk universality. preprint arXiv:0811.3341 (2008)
  13. 13.
    Bolley, F., Chafaï, D., Fontbona, J., et al.: Dynamics of a planar Coulomb gas. Ann. Appl. Probab. 28(5), 3152–3183 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Borot, G., Guionnet, A., Kozlowski, K.K.: Large-n asymptotic expansion for mean field models with Coulomb gas interaction. Int. Math. Res. Notices 2015(20), 10451–10524 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bourgade, P., Erdös, L., Yau, H.-T.: Edge universality of beta ensembles. Commun. Math. Phys. 332(1), 261–353 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bourgade, P., Erdös, L., Yau, H.-T.: Universality of general \(\beta \)-ensembles. Duke Math. J. 163(6), 1127–1190 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bucklew, J.A.: Large Deviation Techniques in Decision, Simulation, and Estimation, vol. 190. Wiley, New York (1990)Google Scholar
  18. 18.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Part II. Commun. Math. Phys. 174(2), 229–260 (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Chafai, D., Hardy, A., Maïda, M.: Concentration for Coulomb gases and Coulomb transport inequalities. J. Funct. Anal. 275(6), 1447–1483 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chalker, J.T., Mehlig, B.: Eigenvector statistics in non-Hermitian random matrix ensembles. Phys. Rev. Lett. 81(16), 3367 (1998)CrossRefGoogle Scholar
  22. 22.
    Chau, L.-L., Zaboronsky, O.: On the structure of correlation functions in the normal matrix model. Commun. Math. Phys. 196(1), 203–247 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Crowdy, D.G.: General solutions to the 2d Liouville equation. Int. J. Eng. Sci. 35(2), 141–149 (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Cunden, F.D., Mezzadri, F., Vivo, P.: Large deviations of radial statistics in the two-dimensional one-component plasma. J. Stat. Phys. 164(5), 1062–1081 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    David, F., Kupiainen, A., Rhodes, R., Vargas, V.: Renormalizability of Liouville quantum field theory at the Seiberg bound. Electron. J. Probab. 22(93), 26 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin (2010). Corrected reprint of the second edition (1998)CrossRefzbMATHGoogle Scholar
  27. 27.
    Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Duy, K.T., Shirai, T.: The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles. Electron. Commun. Probab. 20, 1–13 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3(6), 1191–1198 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Edelman, A.: The probability that a random real gaussian matrix haskreal eigenvalues, related distributions, and the circular law. J. Multivar. Anal. 60(2), 203–232 (1997)CrossRefzbMATHGoogle Scholar
  31. 31.
    Forrester, P.: Exact results for two-dimensional Coulomb systems. Phys. Rep. 301(1–3), 235–270 (1998)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Forrester, P.J.: Log-gases and Random Matrices (LMS-34). Princeton University Press, Princeton (2010)CrossRefzbMATHGoogle Scholar
  33. 33.
    Forrester, P.J.: Analogies between random matrix ensembles and the one-component plasma in two-dimensions. Nucl. Phys. B 904, 253–281 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    García-Zelada, D.: A large deviation principle for empirical measures on polish spaces: Application to singular Gibbs measures on manifolds. arXiv preprint arXiv:1703.02680 (2017)
  35. 35.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6(3), 440–449 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Götze, F., Tikhomirov, A.: The circular law for random matrices. Ann. Probab. 38(4), 1444–1491 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hastings, M.: Eigenvalue distribution in the self-dual non-Hermitian ensemble. J. Stat. Phys. 103(5–6), 903–913 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Hedenmalm, H., Makarov, N.: Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. 106(4), 859–907 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Hedenmalm, H., Wennman, A.: Planar orthogogonal polynomials and boundary universality in the random normal matrix model. preprint arXiv:1710.06493, (2017)
  40. 40.
    Itoi, C.: Universal wide correlators in non-Gaussian orthogonal, unitary and symplectic random matrix ensembles. Nucl. Phys. B 493(3), 651–659 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kang, N.G., Makarov, N.G.: Gaussian free field and conformal field theory. Astérisque 353, viii–136 (2013)Google Scholar
  43. 43.
    Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of Log and Riesz gases. Invent. Math. 210(3), 645–757 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lehmann, N., Sommers, H.-J.: Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67(8), 941–944 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Livan, G., Novaes, M., Vivo, P.: Introduction to Random Matrices: Theory and Practice. Springer, Cham (2017)zbMATHGoogle Scholar
  46. 46.
    Mehta, M.L.: Random Matrices. Pure and Applied Mathematics (Amsterdam), vol. 142, 3rd edn. Elsevier, Amsterdam (2004)Google Scholar
  47. 47.
    Olver, F.W., Lozier, D.W., Boisvert, R.F., Clark, C.W. (ed.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)Google Scholar
  48. 48.
    Petz, D., Hiai, F.: Logarithmic energy as an entropy functional. Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997). Contemporary Mathematics, vol. 217, pp. 205–221. American Mathematical Society, Providence (1998)CrossRefGoogle Scholar
  49. 49.
    Ramirez, J., Rider, B., Virág, B.: Beta ensembles, stochastic Airy spectrum, and a diffusion. J. Am. Math. Soc. 24(4), 919–944 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Rogers, L., Shi, Z.: Interacting Brownian particles and the Wigner law. Probab. Theory Relat. Fields 95(4), 555–570 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316. Springer, Berlin (1997). Appendix B by Thomas BloomCrossRefGoogle Scholar
  52. 52.
    Serfaty, S.: Microscopic description of log and Coulomb gases. preprint arXiv:1709.04089 (2017)
  53. 53.
    Tao, T., Vu, V.: Random matrices: universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43(2), 782–874 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Valkó, B., Virág, B.: Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177(3), 463–508 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Valkó, B., Virág, B.: The sine-\(\beta \) operator. Invent. Math. 209(1), 275–327 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Venker, M.: Particle systems with repulsion exponent \(\beta \) and random matrices. Electron. Commun. Probab 18(83), 1–12 (2013)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Zabrodin, A., Wiegmann, P.: Large-N expansion for the 2D Dyson gas. J. Phys. A 39(28), 8933–8964 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of PhysicsBielefeld UniversityBielefeldGermany
  2. 2.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea

Personalised recommendations