Rigidity of the three-dimensional hierarchical Coulomb gas

  • Sourav ChatterjeeEmail author


A random set of points in Euclidean space is called ‘rigid’ or ‘hyperuniform’ if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This paper gives the first proof of hyperuniformity in a Coulomb type system in dimension three, known as the hierarchical Coulomb gas. This is a simplified version of the actual 3D Coulomb gas. The interaction potential in this model, inspired by Dyson’s hierarchical model of the Ising ferromagnet, has a hierarchical structure and is locally an approximation of the Coulomb potential. Hyperuniformity is proved at both macroscopic and microscopic scales, with upper and lower bounds for the order of fluctuations that match up to logarithmic factors. The fluctuations have cube-root behavior, in agreement with a well-known prediction for the 3D Coulomb gas. For completeness, analogous results are also proved for the 2D hierarchical Coulomb gas and the 1D hierarchical log gas.


Coulomb gas Interacting particles Rigidity Hyperuniformity 

Mathematics Subject Classification

60K35 82B05 



I thank Erik Bates for carefully checking the proofs, and Paul Bourgade, Persi Diaconis, Subhro Ghosh, Adrien Hardy, Joel Lebowitz, Satya Majumdar, Charles Radin, Sylvia Serfaty and H.-T. Yau for helpful discussions and comments.


  1. 1.
    Aizenman, M., Martin, P.A.: Structure of Gibbs states of one-dimensional Coulomb systems. Commun. Math. Phys. 78(1), 99–116 (1980)MathSciNetGoogle Scholar
  2. 2.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Fluctuations of eigenvalues of random normal matrices. Duke Math. J. 159(1), 31–81 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ameur, Y., Hedenmalm, H., Makarov, N.: Random normal matrices and Ward identities. Ann. Probab. 43(3), 1157–1201 (2011)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  5. 5.
    Bardenet, R., Hardy, A.: Monte Carlo with Determinantal Point Processes. (2016) arXiv preprint arXiv:1605.00361
  6. 6.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: Local Density for Two-Dimensional One-Component Plasma. (2015) arXiv preprint arXiv:1510.02074
  7. 7.
    Bauerschmidt, R., Bourgade, P., Nikula, M., Yau, H.-T.: The Two-Dimensional Coulomb Plasma: Quasi-free Approximation and Central Limit Theorem. (2016) arXiv preprint arXiv:1609.08582
  8. 8.
    Beck, J.: Irregularities of distribution. I. Acta Math. 159(1–2), 1–49 (1987)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bekerman, F., Leblé, T., Serfaty, S.: CLT for Fluctuations of \(\beta \)-Ensembles with General Potential (2013). arXiv preprint arXiv:1706.09663
  10. 10.
    Bekerman, F., Lodhia, A.: Mesoscopic Central Limit Theorem for General \(\beta \)-Ensembles (2016). arXiv preprint arXiv:1605.05206
  11. 11.
    Ben Arous, G., Zeitouni, O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bendikov, A.D., Grigor’yan, A.A., Pittet, C., Woess, W.: Isotropic Markov Semigroups on Ultra-Metric Spaces. (Russian) Uspekhi Mat. Nauk, 69(4), 418, 3–102 (2014); translation in Russian Math. Surv., 69(4), 589–680Google Scholar
  13. 13.
    Bendikov, A.A., Grigor’yan, A.A., Pittet, Ch.: On a class of Markov semigroups on discrete ultra-metric spaces. Potential Anal. 37(2), 125–169 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Benfatto, G., Renn, J.: Nontrivial fixed points and screening in the hierarchical two-dimensional Coulomb gas. J. Stat. Phys. 67(5), 957–980 (1992)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Benfatto, G., Gallavotti, G., Nicolò, F.: The dipole phase in the two-dimensional hierarchical Coulomb gas: analyticity and correlations decay. Commun. Math. Phys. 106(2), 277–288 (1986)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Berman, R.J.: Determinantal point processes and fermions on complex manifolds: large deviations and bosonization. Commun. Math. Phys. 327(1), 1–47 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Bolley, F., Chafaï, D., Fontbona, J.: Dynamics of a Planar Coulomb Gas (2017). arXiv preprint arXiv:1706.08776
  18. 18.
    Borodin, A., Sinclair, C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291(1), 177–224 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Borodin, A., Gorin, V., Guionnet, A.: Gaussian asymptotics of discrete \(\beta \)-ensembles. Publ. Math. Inst. Hautes Études Sci. 125, 1–78 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Borot, G., Guionnet, A.: Asymptotic Expansion of \(\beta \) Matrix Models in the Multi-Cut Regime (2013). arXiv preprint arXiv:1303.1045
  21. 21.
    Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the one-cut regime. Commun. Math. Phys. 317(2), 447–483 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Borot, G., Guionnet, A., Kozlowski, K.K.: Large-\(N\) asymptotic expansion for mean field models with Coulomb gas interaction. Int. Math. Res. Not. IMRN 2015(20), 10451–10524 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Bourgade, P., Erdős, L., Yau, H.-T.: Bulk universality of general \(\beta \)-ensembles with non-convex potential. J. Math. Phys. 53(9), 095221 (2012). 19 ppMathSciNetzbMATHGoogle Scholar
  24. 24.
    Bourgade, P., Erdős, L., Yau, H.-T.: Universality of general \(\beta \)-ensembles. Duke Math. J. 163(6), 1127–1190 (2014)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Bourgade, P., Erdős, L., Yau, H.-T.: Edge universality of beta ensembles. Commun. Math. Phys. 332(1), 261–353 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bourgade, P., Yau, H.-T., Yin, J.: Local circular law for random matrices. Probab. Theory Relat. Fields 159(3–4), 545–595 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Bourgade, P., Yau, H.-T., Yin, J.: The local circular law II: the edge case. Probab. Theory Relat. Fields 159(3–4), 619–660 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Fixed energy universality for generalized Wigner matrices. Commun. Pure Appl. Math. 69(10), 1815–1881 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Brascamp, H.J., Lieb, E.H.: Some Inequalities for Gaussian Measures and the Long-Range Order of the One-Dimensional Plasma. Functional Integration and its Applications, pp. 1–14. Clarendon Press, Oxford (1975)zbMATHGoogle Scholar
  30. 30.
    Breuer, J., Duits, M.: Universality of mesoscopic fluctuations for orthogonal polynomial ensembles. Commun. Math. Phys. 342(2), 491–531 (2016)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Breuer, J., Duits, M.: Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Am. Math. Soc. 30(1), 27–66 (2017)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Castin, Y.: Basic Theory Tools for Degenerate Fermi Gases (2006). arXiv preprint arXiv:cond-mat/0612613
  33. 33.
    Chafaï, D., Hardy, A., Maïda, M.: Concentration for Coulomb Gases and Coulomb Transport Inequalities (2016). arXiv preprint arXiv:1610.00980
  34. 34.
    Chafaï, D., Gozlan, N., Zitt, P.-A.: First-order global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Costin, O., Lebowitz, J.: Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75, 69–72 (1995)MathSciNetGoogle Scholar
  36. 36.
    Deift, P.A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  37. 37.
    Dhar, A., Kundu, A., Majumdar, S.N., Sabhapandit, S., Schehr, G.: Exact extremal statistics in the classical 1D Coulomb gas. Phys. Rev. Lett. 119, 060601 (2017)Google Scholar
  38. 38.
    Diaconis, P., Evans, S.N.: Linear functionals of eigenvalues of random matrices. Trans. Am. Math. Soc. 353(7), 2615–2633 (2001)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Dimock, J.: The Kosterlitz–Thouless phase in a hierarchical model. J. Phys. A 23(7), 1207–1215 (1990)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Dyson, F.J.: The dynamics of a disordered linear chain. Phys. Rev. 92(6), 1331–1338 (1953)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Dyson, F.J.: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12(2), 91–107 (1969)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  43. 43.
    Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Relat. Fields 163(3–4), 643–665 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Ghosh, S.: Palm measures and rigidity phenomena in point processes. Electron. Commun. Probab. 21(85), 1–14 (2016)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ghosh, S., Lebowitz, J.: Number rigidity in superhomogeneous random point fields. J. Stat. Phys. 166(3–4), 1016–1027 (2017)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Ghosh, S., Lebowitz, J.L.: Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey. Indian J. Pure Appl. Math. 48(4), 609–631 (2017)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. Duke Math. J. 166(10), 1789–1858 (2017)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Ghosh, S., Zeitouni, O.: Large deviations for zeros of random polynomials with i.i.d. exponential coefficients. Int. Math. Res. Not. IMRN 2016(5), 1308–1347 (2016)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Girko, V.L.: The circular law. Teor. Veroyatnost. i Primenen. 29(4), 669–679 (1984)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Guidi, L.F., Marchetti, D.H.U.: Renormalization group flow of the two-dimensional hierarchical Coulomb gas. Commun. Math. Phys. 219(3), 671–702 (2001)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Hardy, A.: A note on large deviations for 2D Coulomb gas with weakly confining potential. Electron. Commun. Probab. 17(19), 12 (2012)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes. Electron. J. Probab. 18(74), 24 (2013)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes. American Mathematical Society, Providence (2009)zbMATHGoogle Scholar
  55. 55.
    Jancovici, B., Lebowitz, J.L., Manificat, G.: Large charge fluctuations in classical Coulomb systems. J. Stat. Phys. 72(3), 773–787 (1993)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Johansson, K., Lambert, G.: Gaussian and Non-Gaussian Fluctuations for Mesoscopic Linear Statistics in Determinantal Processes (2015). arXiv preprint arXiv:1504.06455
  57. 57.
    Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Kappeler, T., Pinn, K., Wieczerkowski, C.: Renormalization group flow of a hierarchical Sine-Gordon model by partial differential equations. Commun. Math. Phys. 136(2), 357–368 (1991)MathSciNetzbMATHGoogle Scholar
  59. 59.
    König, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kunz, H.: The one-dimensional classical electron gas. Ann. Phys. 85, 303–335 (1975)MathSciNetGoogle Scholar
  61. 61.
    Lambert, G., Ledoux, M., Webb, C.: Stein’s Method for Normal Approximation of Linear Statistics of Beta-Ensembles (2017). arXiv preprint arXiv:1706.10251
  62. 62.
    Leblé, T., Serfaty, S.: Fluctuations of Two-Dimensional Coulomb Gases (2016). arXiv preprint arXiv:1609.08088
  63. 63.
    Leblé, T., Serfaty, S.: Large Deviation Principle for Empirical Fields of Log and Riesz Gases. arXiv preprint arXiv:1502.02970. To appear in Invent. Math. (2015)
  64. 64.
    Leblé, T.: Local Microscopic Behavior for 2D Coulomb Gases (2015). arXiv preprint arXiv:1510.01506
  65. 65.
    Lebowitz, J.L.: Charge fluctuations in coulomb systems. Phys. Rev. A 27, 1491–1494 (1983)Google Scholar
  66. 66.
    Lenard, A.: Exact statistical mechanics of a one-dimensional system with Coulomb forces. J. Math. Phys. 2, 682–693 (1961)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Lenard, A.: Exact statistical mechanics of a one-dimensional system with Coulomb forces. III. Statistics of the electric field. J. Math. Phys. 4, 533–543 (1963)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Marchetti, D.H.U., Perez, J.F.: The Kosterlitz–Thouless phase transition in two-dimensional hierarchical Coulomb gases. J. Stat. Phys. 55(1–2), 141–156 (1989)MathSciNetGoogle Scholar
  69. 69.
    Marino, R., Majumdar, S.N., Schehr, G., Vivo, P.: Phase transitions and edge scaling of number variance in Gaussian random matrices. Phys. Rev. Lett. 112, 254101 (2014)Google Scholar
  70. 70.
    Marino, R., Majumdar, S.N., Schehr, G., Vivo, P.: Number statistics for \(\beta \)-ensembles of random matrices: applications to trapped fermions at zero temperature. Phys. Rev. E 94, 032115 (2016)Google Scholar
  71. 71.
    Martin, P.: Sum rules in charged fluids. Rev. Mod. Phys. 60(4), 1075–1127 (1988)MathSciNetGoogle Scholar
  72. 72.
    Martin, P., Yalcin, T.: The charge fluctuations in classical Coulomb systems. J. Stat. Phys. 22(4), 435–463 (1980)MathSciNetGoogle Scholar
  73. 73.
    Nazarov, F., Sodin, M.: Fluctuations in random complex zeroes: asymptotic normality revisited. Int. Math. Res. Not. IMRN 2011(24), 5720–5759 (2011)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Nazarov, F., Sodin, M., Volberg, A.: Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17(3), 887–935 (2007)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Nazarov, F., Sodin, M., Volberg, A.: The Jancovici–Lebowitz–Manificat law for large fluctuations of random complex zeroes. Commun. Math. Phys. 284(3), 833–865 (2008)MathSciNetzbMATHGoogle Scholar
  76. 76.
    Pastur, L.: Limiting laws of linear eigenvalue statistics for Hermitian matrix models. J. Math. Phys. 47(10), 103303 (2006). 22 ppMathSciNetzbMATHGoogle Scholar
  77. 77.
    Peres, Y., Sly, A.: Rigidity and tolerance for perturbed lattices (2014). arXiv preprint arXiv:1409.4490
  78. 78.
    Petz, D., Hiai, F.: Logarithmic energy as an entropy functional. Contemporary Mathematics, vol. 1998. American Mathematical Society, Providence (1998)zbMATHGoogle Scholar
  79. 79.
    Radin, C.: The ground state for soft disks. J. Stat. Phys. 26(2), 365–373 (1981)MathSciNetGoogle Scholar
  80. 80.
    Rider, B., Virág, B.: The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN, 2007 no. 2, Art. ID rnm006 (2007)Google Scholar
  81. 81.
    Rougerie, N., Serfaty, S.: Higher-dimensional Coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. 69(3), 519–605 (2016)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Sandier, E., Serfaty, S.: 2D Coulomb gases and the renormalized energy. Ann. Probab. 43(4), 2026–2083 (2015)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Serfaty, S.: Ginzburg-Landau vortices, Coulomb gases, and renormalized energies. J. Stat. Phys. 154(3), 660–680 (2014)MathSciNetzbMATHGoogle Scholar
  84. 84.
    Serfaty, S.: Coulomb gases and Ginzburg–Landau vortices. European Mathematical Society (EMS), Zürich (2015)zbMATHGoogle Scholar
  85. 85.
    Shcherbina, M.: Fluctuations of linear eigenvalue statistics of \(\beta \) matrix models in the multi-cut regime. J. Stat. Phys. 151(6), 1004–1034 (2013)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Soshnikov, A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30(1), 171–187 (2002)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Tao, T., Vu, V.: Random matrices: the circular law. Commun. Contemp. Math. 10(2), 261–307 (2008)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Tao, T., Vu, V.: Random matrices: sharp concentration of eigenvalues. Random Matrices Theory Appl. 2(3), 1350007 (2013). 31 ppMathSciNetzbMATHGoogle Scholar
  89. 89.
    Tao, T., Vu, V.: Random matrices: universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43(2), 782–874 (2015)MathSciNetzbMATHGoogle Scholar
  90. 90.
    Torquato, S., Scardicchio, A., Zachary, C.E.: Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory. J. Stat. Mech. Theory Exp. 2008(11), P11019 (2008)MathSciNetGoogle Scholar
  91. 91.
    Wieand, K.: Eigenvalue distributions of random unitary matrices. Probab. Theory Relat. Fields 123(2), 202–224 (2002)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of StatisticsStanford UniversityStanfordUSA

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