Determinantal Point Processes and Fermions on Polarized Complex Manifolds: Bulk Universality

  • Robert J. BermanEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)


We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a high power of a given Hermitian holomorphic line bundle L over X. The empirical measure on X of the process, describing the particle locations, converges in probability towards the pluripotential equilibrium measure, expressed in term of the Monge–Ampère operator. The asymptotics of the corresponding fluctuations in the bulk are shown to be asymptotically normal and described by a Gaussian free field and applies to test functions (linear statistics) which are merely Lipschitz continuous. Moreover, a scaling limit of the correlation functions in the bulk is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This geometric setting applies in particular to normal random matrix ensembles, the two dimensional Coulomb gas, free fermions in a strong magnetic field and multivariate orthogonal polynomials.



It is a pleasure to thank Sébastien Boucksom, David Witt-Nyström, Frédéric Faure and Jeff Steif for stimulating and illuminating discussions. The author is particularly grateful to Bo Berndtsson for helpful discussions concerning Theorem 4.3. Thanks also to the referee for comments that helped to improve the exposition.


  1. 1.
    Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys. 112(3), 503–552 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Berezin transform in polynomial Bergman spaces. Commun. Pure Appl. Math. 63(12) (2010). arXiv:0807.0369MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random normal matrices. Duke Math. J. 159(1), 31–81 (2011). arXiv:0807.0375MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Random normal matrices and Ward identities. Ann. Probab. 43(3), 1157–1201 (2015). arXiv:1109.5941MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ameur, Y., Kang, NG., Makarov, N.: Rescaling Ward identities in the random normal matrix model (2014). arXiv:1410.4132
  6. 6.
    Bardenet, R., Hardy, A.: Monte Carlo with determinantal point processes. arXiv:1605.00361
  7. 7.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H-T.: The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem. arXiv:1609.08582
  8. 8.
    Bedford, E., Taylor, A.: The Dirichlet problem for a complex Monge-Ampere equation. Invent. Math 37(1), 1–44 (1976)Google Scholar
  9. 9.
    Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Corrected reprint of the: original. Grundlehren Text Editions. Springer, Berlin (1992)CrossRefGoogle Scholar
  10. 10.
    Berman, R.J., Berndtsson, B., Sjöstrand, J.: A direct approach to asymptotics of Bergman kernels for positive line bundles. Arkiv för Matematik. 46(2), 197–217 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Berman, R.J., Boucksom, S.: Witt Nyström, D: Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207(1), 1–27 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berman, R.J., Ortega-Cerdà, J.: Sampling of real multivariate polynomials and pluripotential theory. Am. J. Math. arXiv:1509.00956. (to appear)
  13. 13.
    Berman, R.J.: Bergman kernels and equilibrium measures for line bundles over projective manifolds. Am. J. Math. 131(5) (2009)Google Scholar
  14. 14.
    Berman, R.J.: Bergman kernels and equilibrium measures for polarized pseudoconcave domains. Int. J. Math. 21(1), 77–115 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Berman, R.J.: Bergman kernels and local holomorphic Morse inequalities. Math. Z 248(2), 325–344 (2004)Google Scholar
  16. 16.
    Berman, R.J.: Bergman kernels and weighted equilibrium measures of \({\mathbb{C}}^{n}.\) Indiana Univ. Math. J. 58(4) (2009)Google Scholar
  17. 17.
    Berman, R.J.: Boucksom, S; Growth of balls of holomorphic sections and energy at equilibrium. 42 pages. Invent. Math. 181(2), 337–394 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Berman, R.J.: Determinantal point processes and fermions on complex manifolds: large deviations and Bosonization. Commun. Math. Phys. 327(1), 1–47 (2014). arXiv:0812.4224MathSciNetCrossRefGoogle Scholar
  19. 19.
    Berman, R.J.: Kähler-Einstein metrics, canonical random point processes and birational geometry. (to appear in the AMS Proceedings of the 2015 Summer Research Institute on Algebraic Geometry)
  20. 20.
    Berman, R.J.: Sharp asymptotics for toeplitz determinants and convergence towards the gaussian free field on riemann surfaces. Int. Math. Res. Not. 2012(22), 5031–5062 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Berman, R.J.: Super Toeplitz operators on holomorphic line bundles. J. Geom. Anal. 16(1), 1–22 (2006)Google Scholar
  22. 22.
    Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bogaevskiĭ, I.A.: Singularities of convex hulls of three-dimensional hypersurfaces. Proc. Steklov Inst. Math. 221(2), 71–90 (1998)Google Scholar
  24. 24.
    Bonnet, G., David, F., Eynard, B.: Breakdown of universality in multi-cut matrix models. J. Phys. A33, 6739–6768 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Boutet de Monvel., Sjötrand, J.: Sur la singularite des noyaux de Bergman et de Szegö. Asterisque 34–35, 123–164 (1976)Google Scholar
  26. 26.
    Bryc, W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. Probab. Lett. 18(4), 253–256 (1993). ElsevierMathSciNetCrossRefGoogle Scholar
  27. 27.
    Caffarelli, L.A., Rivière, N.M.: Smoothness and analyticity of free boundaries in variational inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci.(4) 3(2), 289–310 (1976)Google Scholar
  28. 28.
    Cooper, F., Khare, A., Sukhatme, U.: Supersymmetry in Quantum Mechanics. World Scientific Publication (2001)Google Scholar
  29. 29.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant lecture notes in mathematics, vol. 3. New York University, Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence (1999)Google Scholar
  31. 31.
    Deift, P.A.: Universality for mathematical and physical systems. Int. Congr. Math. I, 125–152 (2004). Eur. Math. Soc., ZürichGoogle Scholar
  32. 32.
    Delin, H.: Pointwise estimates for the weighted Bergman projection kernel in \({\mathbb{C}}^{n}\) using a weighted \(L^{2}\) estimate for the \(\bar{\partial }\) equation. Ann. Inst. Fourier (Grenoble) 48(4), 967–997 (1998)Google Scholar
  33. 33.
    Demailly, J-P.: Complex analytic and algebraic geometry.
  34. 34.
    Demailly, J-P.: Estimations \(L^{2}\) pour l’opérateur \(\bar{\partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. (French). Ann. Sci. École Norm. Sup. (4) 15(3), 457–511 (1982)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Demailly, J-P.: Potential theory in several complex variables.
  36. 36.
    Dembo, A., Zeitouni O.: Large deviation techniques and applications. Corrected reprint of the 2nd (1988) edition. Stochastic Modelling and Applied Probability, vol. 38, pp. xvi+396. Springer, Berlin (2010)Google Scholar
  37. 37.
    Donaldson, S.K.: Scalar curvature and projective embeddings. II. Q. J. Math. 56(3), 345–356 (2005)Google Scholar
  38. 38.
    Ferrari, F., Klevtsov, S., Zelditch, S.: Random Kähler metrics. Nucl. Phys. B 869(1), 89–110 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Forrester, P.J.: Fluctuation formula for complex random matrices. J. Phys. A 32(13), L159–L163 (1999)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Forrester, P.J.: Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions. J. Phys. A 39(41), 13025–13036 (2006)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Götz, M., Maymeskul, V.V., Saff, E.B.: Asymptotic distribution of nodes for near-optimal polynomial interpolation on certain curves in \({\mathbb{R}}^{2}\). Constr. Approx. 18(2), 255–283 (2002)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Guedj, V., Zeriahi, A.: Intrinsic capacities on compact Kähler manifolds. J. Geom. Anal. 15(4), 607–639 (2005)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Guionnet, A.: Large deviations and stochastic calculus for large random matrices. Probab. Surv. 1, 72–172 (2004). (electronic)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Gurbatov, S.N., Malakhov, A.I., Saichev, A.I.: Non-Linear Random Waves and Turbulence in Non-dispersive Media: Waves, Rays, Particles. Manchester University Press, Manchester (1991). With an appendix (Singularities and bifurcations of potential flows) by Arnold et alGoogle Scholar
  46. 46.
    Hedenmalm, H., Makarov, N.: Quantum Hele-Shaw flow (2004).
  47. 47.
    Hough, J.B., Krishnapur, M., Peres, Y.l., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Johansson, K.: Random matrices and determinantal processes. arXiv:math-ph/0510038
  50. 50.
    Klevtsov, S.: Geometry and large N limits in Laughlin states. arXiv:1608.02928
  51. 51.
    Klimek, M.: Pluripotential Theory. London mathematical society monographs. New Series, 6. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1991)Google Scholar
  52. 52.
    Laughlin, R.B.: Elementary theory: the incompressible quantum fluid. In: The Quantum Hall Effect. Springer, Berlin (1987)CrossRefGoogle Scholar
  53. 53.
    Lazarsfeld, R.: Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. II. Positivity for vector bundles, and multiplier ideals. A series of modern surveys in mathematics, vol. 48 and 49. Springer, Berlin (2004)Google Scholar
  54. 54.
    Leblé, T., Serfaty, S.: Fluctuations of two-dimensional coulomb gases. arXiv:1609.08088
  55. 55.
    Lindholm, N.: Sampling in weighted \(L^{p}\) spaces of entire functions in \({\mathbb{C}}^{n}\) and estimates of the Bergman kernel. J. Funct. Anal. 182, 390–426 (2001)Google Scholar
  56. 56.
    Macchi, O.: The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7, 83–122 (1975)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Pastur, L., Shcherbina, M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130(2), 205–250 (2008)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Pastur, L.: A simple approach to the global regime of Gaussian ensembles of random matrices. Ukraïn. Mat. Zh. 57(6), 790–817 (2005), Translation in Ukrainian Math. J. 57(6), 936–966 (2005)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Pastur, L.: Limiting laws of linear eigenvalue statistics for Hermitian matrix models. J. Math. Phys. 47(10) (2006)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley classics library. Wiley, New York (1994)CrossRefGoogle Scholar
  61. 61.
    Pokorny, F.T., Singer, M.: Toric partial density functions and stability of toric varieties. Math. Ann. 358(3–4), 879–923 (2014). SpringerGoogle Scholar
  62. 62.
    Rider, B., Virag, B.: Complex determinantal processes and H1 noise. Electron. J. Probab. 12 (2007)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN, (2) (2007)Google Scholar
  64. 64.
    Ross, J., Singer, M.: Asymptotics of Partial Density Functions for Divisors. arXiv:1312.1145
  65. 65.
    Ross, J., Witt Nyström, D.: Homogeneous Monge-Ampère Equations and Canonical Tubular Neighbourhoods in Kähler Geometry. arXiv:1403.3282
  66. 66.
    Saff.E., Totik.V.: Logarithmic Potentials with Exteriour Fields. Springer, Berlin (1997) (with an appendix by Bloom, T)Google Scholar
  67. 67.
    Scardicchio, A., Torquato, S., Zachary, C.E.: Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory . J. Stat. Mech. Theory Exp. (1) (2008)Google Scholar
  68. 68.
    Scardicchio, A., Torquato, S., Zachary, C.E.: Statistical properties of determinantal point processes in high-dimensional Euclidean spaces. Phys. Rev. E (3) 79(4) (2009)Google Scholar
  69. 69.
    Schaeffer, D.: Some examples of singularities in a free boundary. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4(1), 133–144 (1977)Google Scholar
  70. 70.
    Sheffield, Scott: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3–4), 521–541 (2007)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Commun. Math. Phys. 200(3), 661–683 (1999)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Shiffman, B., Zelditch S.: Number variance of random zeros on complex manifolds, II: smooth statistics. Pure Appl. Math. Q. 6(4) (2010). Special Issue: In honor of Joseph J. Kohn. Part 2MathSciNetCrossRefGoogle Scholar
  73. 73.
    Shigekawa, I.: Spectral properties of Schrodinger operators with magnetic fields for a spin 1/2 particle. 101(2), 255–285 (1991)Google Scholar
  74. 74.
    Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21(1–2), 107–125 (2004)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Soshnikov, A.: Determinantal random point fields. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160; translation. Russian Math. Surv. 55(5), 923–975 (2000)MathSciNetCrossRefGoogle Scholar
  76. 76.
    Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30(1), 171–187 (2002)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Zabrodin, A.: Matrix models and growth processes: from viscous flows to the quantum Hall effect. NATO Sci. Ser. II Math. Phys. Chem. 221 (2006). Springer, Dordrecht
  78. 78.
    Zelditch, S., Zhou, P.: Interface asymptotics of partial Bergman kernels on S1-symmetric Kaehler manifoldsGoogle Scholar
  79. 79.
    Zelditch, S.: Szegö kernels and a theorem of Tian. Internat. Math. Res. Not. (6), 317–331 (1998)Google Scholar

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Authors and Affiliations

  1. 1.Chalmers University of TechnologyGothenburgSweden

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