Constructive Approximation

, Volume 48, Issue 1, pp 101–136 | Cite as

Point Processes, Hole Events, and Large Deviations: Random Complex Zeros and Coulomb Gases



We consider particle systems (also known as point processes) on the line and in the plane and are particularly interested in “hole” events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) \(\beta \)-Ginibre ensembles. We also discuss some possible directions for future investigations.


Point processes Particle systems Coulomb gases Random matrices Random polynomials Hole probabilities Large deviations Empirical measures 

Mathematics Subject Classification

Primary 60G55 Secondary 60F10 



We thank the authors of the paper [46] for allowing us to use the picture in Fig. 7. We thank Diego Ayala for allowing us to use the picture in Figure 8. We thank the anonymous referee for numerous helpful suggestions. The work of S.G. was supported in part by the ARO Grant W911NF-14-1-0094, the NSF Grant DMS-1148711 and the NUS Grant R-146-000-250-133.


  1. 1.
    Adhikari, K., Reddy, N.: Hole probabilities for finite and infinite Ginibre ensembles, International Mathematics Research Notices, rnw207,
  2. 2.
    Aizenman, M., Goldstein, S., Lebowitz, J.: Conditional equilibrium and the equivalence of microcanonical and grandcanonical ensembles in the thermodynamic limit. Commun. Math. Phys. 62(3), 279–302 (1978)MathSciNetMATHGoogle Scholar
  3. 3.
    Aizenman, M., Martin, P.: Structure of Gibbs states of one-dimensional Coulomb systems. Comm. Math. Phys. 78(1), 99–116 (1980)MathSciNetGoogle Scholar
  4. 4.
    Antezana, J., Buckley, J., Marzo, J., Olsen, J.F.: Gap probabilities for the cardinal sine. J. Math. Anal. Appl. 396(2), 466–472 (2012)MathSciNetMATHGoogle Scholar
  5. 5.
    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, vol. 118. Cambridge University Press, Cambridge (2010)MATHGoogle Scholar
  6. 6.
    Armstrong, S.N., Serfaty, S., Zeitouni, O.: Remarks on a constrained optimization problem for the Ginibre ensemble. Potential Anal. 41(3), 945–958 (2014)MathSciNetMATHGoogle Scholar
  7. 7.
    Bergweiler, W., Eremenko, A.: Distribution of zeros of polynomials with positive coefficients. Ann. Acad. Sci. Fenn. Math. 40(1), 375–383 (2015)MathSciNetMATHGoogle Scholar
  8. 8.
    Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108(4), 517–542 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Arous, G.B., Zeitouni, O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Bufetov, A.: Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel. Bull. Math. Sci. 6(1), 163–172 (2016)MathSciNetMATHGoogle Scholar
  11. 11.
    Bufetov, A., Dabrowski, Y., Qiu, Y.: Linear rigidity of stationary stochastic processes. Ergod. Theory Dyn. Syst. 1–15.
  12. 12.
    Bufetov, A., Qiu, Y.: Determinantal point processes associated with Hilbert spaces of holomorphic functions. Commun. Math. Phys. 351(1), 1–44 (2017)MathSciNetMATHGoogle Scholar
  13. 13.
    Bogomolny, E., Bohigas, O., Leboeuf, P.: Quantum chaotic dynamics and random polynomials. J. Stat. Phys. 85(5-6), 639–679 (1996)MathSciNetMATHGoogle Scholar
  14. 14.
    Borodin, A., Serfaty, S.: Renormalized energy concentration in random matrices. Commun. Math. Phys. 320(1), 199–244 (2013)MathSciNetMATHGoogle Scholar
  15. 15.
    Butez, R.: Large Deviations for the Empirical Measure of Random Polynomials: Revisit of the Zeitouni–Zelditch Theorem. arXiv preprint arXiv:1509.09136 (2015)
  16. 16.
    Butez, R., Zeitouni, O.: Universal large deviations for Kac polynomials. Electron. Commun. Probab. 22(6), 10 (2017)MathSciNetMATHGoogle Scholar
  17. 17.
    Buckley, J., Nishry, A., Peled, R., Sodin, M.: Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence. Probab. Theory Relat. Fields (2017). MATHGoogle Scholar
  18. 18.
    Chesnokov, A., Deckers, K., Van Barel, M.: A numerical solution of the constrained weighted energy problem. J. Comput. Appl. Math. 235(4), 950–965 (2010)MathSciNetMATHGoogle Scholar
  19. 19.
    Coroian, D., Dragnev, P.: Constrained Leja points and the numerical solution of the constrained energy problem. J. Comput. Appl. Math. 131(1), 427–444 (2001)MathSciNetMATHGoogle Scholar
  20. 20.
    Daley, D., Vere-Jones, D.: An introduction to the theory of point processes, vol. I and II. Springer, Berlin (2007)MATHGoogle Scholar
  21. 21.
    Dembo, A., Mukherjee, S.: No zero-crossings for random polynomials and the heat equation. Ann. Probab. 43(1), 85–118 (2015)MathSciNetMATHGoogle Scholar
  22. 22.
    Dembo, A., Mukherjee, S.: Persistence of Gaussian processes: non-summable correlations. Probab. Theory Relat. Fields (2016). MATHGoogle Scholar
  23. 23.
    Dumitriu, I., Edelman, A.: Matrix models for beta ensembles. J. Math. Phys. 43(11), 5830–5847 (2002)MathSciNetMATHGoogle Scholar
  24. 24.
    Erdos, L.: Universality for random matrices and log-gases. Lecture Notes for the conference Current Developments in Mathematics, 2012. arXiv preprint arXiv:1212.0839 (2012)
  25. 25.
    Feng, R., Zelditch, S.: Large deviations for zeros of \(P(\varphi )_2\) random polynomials. J. Stat. Phys. 143(4), 619–635 (2011)MathSciNetMATHGoogle Scholar
  26. 26.
    Feldheim, N.D., Feldheim, O.N.: Long gaps between sign-changes of gaussian stationary processes. Int. Math. Res. Not. 2015(11), 3021–3034 (2015)MathSciNetMATHGoogle Scholar
  27. 27.
    Feldheim, N.D., Feldheim, O.N., Nitzan, S.: Persistence of Gaussian stationary processes: a spectral perspective. arXiv preprint arXiv:1709.00204 (math.PR)
  28. 28.
    Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields 163(3-4), 643–665 (2015)MathSciNetMATHGoogle Scholar
  29. 29.
    Ghosh, S.: Palm measures and rigidity phenomena in point processes. Electron. Commun. Probab. 21(85), 14 (2016)MathSciNetMATHGoogle Scholar
  30. 30.
    Ghosh, S., Krishnapur, M.: Rigidity hierarchy in random point fields: random polynomials and determinantal processes. arXiv preprint arXiv:1510.08814 (2015)
  31. 31.
    Ghosh, S., Lebowitz, J.L.: Number rigidity in superhomogeneous random point fields. J. Stat. Phys. 166(3–4), 1016–1027 (2017)MathSciNetMATHGoogle Scholar
  32. 32.
    Ghosh, S., Zeitouni, O.: Large deviations for zeros of random polynomials with iid exponential coefficients. Int. Math. Res. Not. 2016(5), 1308–1347 (2016)MATHGoogle Scholar
  33. 33.
    Ghosh, S., Nishry, A.: Gaussian complex zeros on the hole event: the emergence of a forbidden region. arXiv preprint arXiv:1609.00084 (2016)
  34. 34.
    Ghosh, S., Peres, Y.: Rigidity and Tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. Duke Math. J. 166(10), 1789–1858 (2017)MathSciNetMATHGoogle Scholar
  35. 35.
    Guionnet, A.: Large deviations and stochastic calculus for large random matrices. Probab. Surv. 1, 72–172 (2004)MathSciNetMATHGoogle Scholar
  36. 36.
    Helsen, S., Van Barel, M.: A numerical solution of the constrained energy problem. J. Comput. Appl. Math. 189(1), 442–452 (2006)MathSciNetMATHGoogle Scholar
  37. 37.
    Petz, D., Hiai, F.: Logarithmic energy as an entropy functional. Contemp. Math. 217, 205–221 (1998)MathSciNetMATHGoogle Scholar
  38. 38.
    Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables, and Entropy. American Mathematical Society, Providence (2000)MATHGoogle Scholar
  39. 39.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian analytic functions and determinantal point processes, vol. 51. American Mathematical Society, Providence (2009)MATHGoogle Scholar
  40. 40.
    Jancovici, B., Lebowitz, J.L., Manificat, G.: Large charge fluctuations in classical Coulomb systems. J. Stat. Phys. 72(3–4), 773–787 (1993)MathSciNetMATHGoogle Scholar
  41. 41.
    Krishnapur, M.: Overcrowding estimates for zeroes of planar and hyperbolic Gaussian analytic functions. J. Stat. Phys. 124(6), 1399–1423 (2006)MathSciNetMATHGoogle Scholar
  42. 42.
    Kunz, H.: The one-dimensional classical electron gas. Ann. Phys. 85, 303–335 (1974)MathSciNetGoogle Scholar
  43. 43.
    Leblé, T., Serfaty, S.: Large deviation principle for empirical fields of Log and Riesz gases. arXiv preprint arXiv:1502.02970 (2015). To appear in Inventiones Math
  44. 44.
    Leblé, T., Serfaty, S.: Fluctuations of Two-Dimensional Coulomb Gases. arXiv preprint arXiv:1609.08088 (2016)
  45. 45.
    Leblé, T., Serfaty, S., Zeitouni, O.: (with an appendix by W. Wu), Large deviations for the 2D two-component plasma. Commun. Math. Phys. 350(1), 301–360 (2017)Google Scholar
  46. 46.
    Majumdar, S.N., Nadal, C., Scardicchio, A., Vivo, P.: How many eigenvalues of a Gaussian random matrix are positive? Phys. Rev. E 83(4), 041105 (2011)Google Scholar
  47. 47.
    Martin, Ph, Yalcin, T.: The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22(4), 435–463 (1980)MathSciNetGoogle Scholar
  48. 48.
    Mehta, M.: Random Matrices. Academic Press, New York (1962)Google Scholar
  49. 49.
    Nazarov, F., Sodin, M., Volberg, A.: The JancoviciLebowitzManificat law for large fluctuations of random complex zeroes. Commun. Math. Phys. 284(3), 833–865 (2008)MATHGoogle Scholar
  50. 50.
    Nishry, A.: Asymptotics of the hole probability for zeros of random entire functions. Int. Math. Res. Not. 15, 2925–2946 (2010)MathSciNetMATHGoogle Scholar
  51. 51.
    Nishry, A.: The hole probability for Gaussian entire functions. Israel J. Math. 186(1), 197–220 (2011)MathSciNetMATHGoogle Scholar
  52. 52.
    Nishry, A.: Hole probability for entire functions represented by Gaussian Taylor series. J. d’Analyse Mathmatique 118(2), 493–507 (2012)MathSciNetMATHGoogle Scholar
  53. 53.
    Nazarov, F., Sodin, M.: Correlation functions for random complex zeroes: strong clustering and local universality. Commun. Math. Phys. 310(1), 75–98 (2012)MathSciNetMATHGoogle Scholar
  54. 54.
    Obrechkoff, N.: Sur un probl‘eme de Laguerre. C. R. Acad. Sci. (Paris) 177, 102–104 (1923)MATHGoogle Scholar
  55. 55.
    Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41(1), 1–49 (2013)MathSciNetMATHGoogle Scholar
  56. 56.
    Peres, Y., Virág, B.: Zeros of the iid Gaussian power series: a conformally invariant determinantal process. Acta Mathematica 194(1), 1–35 (2005)MathSciNetMATHGoogle Scholar
  57. 57.
    Rougerie, N., Serfaty, S.: Higher dimensional coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. 69(3), 519–605 (2016)MathSciNetMATHGoogle Scholar
  58. 58.
    Shirai, T.: Large deviations for the fermion point process associated with the exponential kernel. J. Stat. Phys. 123(3), 615–629 (2006)MathSciNetMATHGoogle Scholar
  59. 59.
    Shirai, T.: Ginibre-type point processes and their asymptotic behavior. J. Math. Soc. Jpn. 67(2), 763–787 (2015)MathSciNetMATHGoogle Scholar
  60. 60.
    Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes. J. Funct. Anal. 205(2), 414–463 (2003)MathSciNetMATHGoogle Scholar
  61. 61.
    Sandier, E., Serfaty, S.: 2D Coulomb gases and the renormalized energy. Ann. Probab. 43(4), 2026–2083 (2015)MathSciNetMATHGoogle Scholar
  62. 62.
    Sandier, E., Serfaty, S.: 1D log gases and the renormalized energy: crystallization at vanishing temperature. Probab. Theory Relat. Fields 162(3–4), 795–846 (2015)MathSciNetMATHGoogle Scholar
  63. 63.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields, vol. 316. Springer, Berlin (2013)MATHGoogle Scholar
  64. 64.
    Sodin, M., Tsirelson, B.: Random complex zeroes, III. Decay of the hole probability. Israel J. Math. 147(1), 371–379 (2005)MathSciNetMATHGoogle Scholar
  65. 65.
    Tao, T., Vu, V.: Random matrices: The universality phenomenon for Wigner ensembles. Mod. Aspects Random Matrix Theory 72, 121–172 (2012)MathSciNetMATHGoogle Scholar
  66. 66.
    Tricomi, F.G.: Integral Equations, vol. 5. Courier Corporation, North Chelmsford (1957)MATHGoogle Scholar
  67. 67.
    Zeitouni, O., Zelditch, S.: Large deviations of empirical measures of zeros of random polynomials. Int. Math. Res. Not. 2010(20), 3935–3992 (2010)MathSciNetMATHGoogle Scholar
  68. 68.
    Zelditch, S.: Large deviations of empirical measures of zeros on Riemann surfaces. Int. Math. Res. Not. 2013(3), 592–664 (2013)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National University of SingaporeSingaporeSingapore
  2. 2.Tel Aviv UniversityTel AvivIsrael

Personalised recommendations