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Gaussian asymptotics of discrete \(\beta \)-ensembles

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Abstract

We introduce and study stochastic \(N\)-particle ensembles which are discretizations for general-\(\beta \) log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, \((z,w)\)-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as \(N\to \infty \). The covariance is universal and coincides with its counterpart in random matrix theory.

Our main tool is an appropriate discrete version of the Schwinger-Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.

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References

  1. J. Ambjørn and Yu. Makeenko, Properties of loop equations for the Hermitian matrix model and for two-dimensional gravity, Mod. Phys. Lett. A, 5 (1990), 1753–1763.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Anderson, A. Guionnet and O. Zeitouni, Introduction to Random Matrices, Cambridge Studies in Advanced Mathematics, 2009.

    Book  MATH  Google Scholar 

  3. J. Baik, A. Borodin, P. Deift and T. Suidan, A model for the bus system in Cuernevaca (Mexico), J. Phys. A, Math. Gen., 39 (2006), 8965, arXiv:math/0510414.

    Article  MATH  Google Scholar 

  4. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin and P. D. Miller, Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles, arXiv:math/0310278.

  5. F. Bekerman, A. Figalli and A. Guionnet, Transport maps for Beta-matrix models and universality, Commun. Math. Phys., 338 (2015), 589–619, arXiv:1311.2315.

    Article  MATH  Google Scholar 

  6. G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Relat. Fields, 108 (1997), 517–542.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Bonnet, F. David and B. Eynard, Breakdown of universality in multi-cut matrix models, J. Phys. A, Math. Gen., 33 (2000), 6739, arXiv:cond-mat/0003324.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Borodin, Schur dynamics of the Schur processes, Adv. Math., 228 (2011), 2268–2291, arXiv:1001.3442.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Borodin, CLT for spectra of submatrices of Wigner random matrices, Mosc. Math. J., 14 (2014), 29–38, arXiv:1010.0898.

    MathSciNet  MATH  Google Scholar 

  10. A. Borodin and P. Ferrari, Anisotropic growth of random surfaces in \(2 + 1\) dimensions, Commun. Math. Phys., 325 (2014), 603–684, arXiv:0804.3035.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Borodin and V. Gorin, Shuffling algorithm for boxed plane partitions, Adv. Math., 220 (2009), 1739–1770, arXiv:0804.3071.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian Free Field, Commun. Pure Appl. Math., 68 (2015), 1683–1884, arXiv:1305.3627.

    Article  MATH  Google Scholar 

  13. A. Borodin and G. Olshanski, Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. Math., 161 (2005), 1319–1422, arXiv:math/0109194.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Borodin and G. Olshanski, Asymptotics of Plancherel-type random partitions, J. Algebra, 313 (2007), 40–60, arXiv:math/0610240.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Borodin and L. Petrov, Integrable probability: from representation theory to Macdonald processes, Probab. Surv., 11 (2014), 1–58, arXiv:1310.8007.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Borot and A. Guionnet, Asymptotic expansion of beta matrix models in the one-cut regime, Commun. Math. Phys., 317 (2013), 447–483, arXiv:1107.1167.

    Article  MATH  Google Scholar 

  17. G. Borot and A. Guionnet, Asymptotic expansion of beta matrix models in the multi-cut regime, Commun. Math. Phys., 317 (2013), 447–483, arXiv:1303.1045.

    Article  MATH  Google Scholar 

  18. P. Bourgade, L. Erdos and H.-T. Yau, Edge universality of beta ensembles, Commun. Math. Phys., 332 (2014), 261–353, arXiv:1306.5728.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Boutet de Monvel, L. Pastur and M. Shcherbina, On the statistical mechanics approach in the random matrix theory: integrated density of states, J. Stat. Phys., 79 (1995), 585–611.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Breuer and M. Duits, Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, J. Amer. Math. Soc. arXiv:1309.6224.

  21. E. Brézin, C. Itzykson, G. Parisi and J. B. Zuber, Planar diagrams, Commun. Math. Phys., 59 (1978), 35–51.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Bufetov and V. Gorin, Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., 25 (2015), 763–814, arXiv:1311.5780.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Bufetov and V. Gorin, Fluctuations of particle systems determined by Schur generating functions, arXiv:1604.01110.

  24. S. Chatterjee, Rigorous solution of strongly coupled \(SO(N)\) lattice gauge theory in the large \(N\) limit, arXiv:1502.07719.

  25. L. O. Chekhov and B. Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys., 0612, 026 (2006), arXiv:math-ph/0604014.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Chhita, K. Johansson and B. Young, Asymptotic domino statistics in the Aztec diamond, Ann. Appl. Probab., 25 (2015), 1232–1278, arXiv:1212.5414.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Cohn Larsen and J. Propp, The shape of a typical boxed plane partition, N.Y. J. Math., 4 (1998), 137–165, arXiv:math/9801059.

    MathSciNet  MATH  Google Scholar 

  28. B. Collins, A. Guionnet and E. Maurel-Segala, Asymptotics of unitary and orthogonal matrix integrals, Adv. Math., 222 (2009), 172–215, arXiv:math/0608193.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Dolega and V. Feray, Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J., 7 (2016), 1193–1282, arXiv:1402.4615.

    MathSciNet  MATH  Google Scholar 

  30. P. D. Dragnev and E. B. Saff, Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math., 72 (1997), 223–259.

    Article  MathSciNet  MATH  Google Scholar 

  31. B. A. Dubrovin, Theta functions and non-linear equations, Russ. Math. Surv., 36 (1981), 11–92.

    Article  MATH  Google Scholar 

  32. B. Eynard, All genus correlation functions for the hermitian 1-matrix model, J. High Energy Phys., 0411, 031 (2004), arXiv:hep-th/0407261.

    Article  Google Scholar 

  33. B. Eynard, All order asymptotic expansion of large partitions, J. Stat. Mech. Theory Exp., 2008, P07023 (2008).

    Article  MathSciNet  Google Scholar 

  34. B. Eynard, A matrix model for plane partitions, J. Stat. Mech. Theory Exp., 0910, P10011, (2009).

    Article  MathSciNet  Google Scholar 

  35. B. Eynard and N. Orantin, Topological recursion in enumerative geometry and random matrices, J. Phys. A, 42 (2009), 293001.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Feral, On large deviations for the spectral measure of discrete Coulomb gas, in Seminaire de Probabilites XLI Lecture Notes in Mathematics, vol. 1934, pp. 19–49, 2008.

    Google Scholar 

  37. P. J. Forrester, Log-Gases and Random Matrices, Princeton University Press, Princeton, 2010.

    MATH  Google Scholar 

  38. V. Gorin, Non-intersecting paths and Hahn orthogonal ensemble, Funct. Anal. Appl., 42 (2008), 180–197, arXiv:0708.2349.

    Article  MathSciNet  MATH  Google Scholar 

  39. V. Gorin and M. Shkolnikov, Multilevel Dyson Brownian motions via Jack polynomials, Probab. Theory Relat. Fields, 163 (2015), 413, arXiv:1401.5595.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Guionnet and J. Novak, Asymptotics of unitary multimatrix models: the Schwinger–Dyson lattice and topological recursion, J. Funct. Anal., 268 (2015), 2851–2905.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Hora and N. Obata, Quantum Probability and Spectral Analysis of Graphs, Theoretical and Mathematical Physics, Springer, Berlin, 2007.

    MATH  Google Scholar 

  42. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., 91 (1998), 151–204.

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. (2), 153 (2001), 259–296, math.CO/9906120.

    Article  MathSciNet  MATH  Google Scholar 

  44. K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys., 209 (2000), 437–476, arXiv:math/9903134.

    Article  MathSciNet  MATH  Google Scholar 

  45. K. Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields, 123 (2002), 225–280, arXiv:math/0011250.

    Article  MathSciNet  MATH  Google Scholar 

  46. K. Johansson and E. Nordenstam, Eigenvalues of GUE minors, Electron. J. Probab., 11 (2006), 50, arXiv:math/0606760.

    Article  MathSciNet  MATH  Google Scholar 

  47. R. Kenyon, Height fluctuations in the honeycomb dimer model, Commun. Math. Phys., 281 (2008), 675–709, arXiv:math-ph/0405052.

    Article  MathSciNet  MATH  Google Scholar 

  48. R. Kenyon and A. Okounkov, Limit shapes and the complex burgers equation, Acta Math., 199 (2007), 263–302, arXiv:math-ph/0507007.

    Article  MathSciNet  MATH  Google Scholar 

  49. W. Konig, N. O’Connel and S. Roch, Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles, Electron. J. Probab., 7 (2002), 1–24.

    Article  MathSciNet  Google Scholar 

  50. T. Kriecherbauer and M. Shcherbina, Fluctuations of eigenvalues of matrix models and their applications, arXiv:1003.6121.

  51. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, London, 1999.

    MATH  Google Scholar 

  52. M. Maida and E. Maurel-Segala, Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices, Probab. Theory Relat. Fields, 159 (2014), 329–356, arXiv:1204.3208.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. L. Mehta, Random Matrices, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004.

    MATH  Google Scholar 

  54. A. Moll, in preparation.

  55. A. A. Migdal, Loop equations and \(1/N\) expansion, Phys. Rep., 102 (1983), 199–290.

    Article  Google Scholar 

  56. N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, High Energy Physics - Theory, (2013), 1–83, arXiv:1312.6689.

  57. N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional \(N=2\) quiver gauge theories, arXiv:1211.2240.

  58. N. Nekrasov, Non-perturbative Dyson–Schwinger equations and BPS/CFT correspondence, in preparation.

  59. G. Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., 205 (2003), 464–524, arXiv:math/0109193.

    Article  MathSciNet  MATH  Google Scholar 

  60. G. Olshanksi, Probability measures on dual objects to compact symmetric spaces and hypergeometric identities, Funct. Anal. Appl., 37 (2001), 281–301.

    Article  MathSciNet  Google Scholar 

  61. G. Akemann, J. Baik and P. Di Francesco (eds.), The Oxford Handbook of Random Matrix Theory, Oxford University Press, London, 2011.

    MATH  Google Scholar 

  62. L. Pastur and M. Shcherbina, Eigenvalue Distribution of Large Random Matrices, AMS, Providence, 2011.

    Book  MATH  Google Scholar 

  63. L. Petrov, Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Relat. Fields, 160 (2014), 429–487, arXiv:1202.3901.

    Article  MathSciNet  MATH  Google Scholar 

  64. L. Petrov, Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field, Ann. Probab., 43 (2015), 1–43, arXiv:1206.5123.

    Article  MathSciNet  MATH  Google Scholar 

  65. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, Berlin, 1997.

    Book  MATH  Google Scholar 

  66. M. Shcherbina, Fluctuations of linear eigenvalue statistics of \(\beta \) matrix models in the multi-cut regime, J. Stat. Phys., 151 (2013), 1004–1034, arXiv:1205.7062.

    Article  MathSciNet  MATH  Google Scholar 

  67. E. P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math., 67 (1958), 325–327.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Alexei Borodin, Vadim Gorin & Alice Guionnet

  2. Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia

    Vadim Gorin

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  1. Alexei Borodin
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  2. Vadim Gorin
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  3. Alice Guionnet
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Correspondence to Vadim Gorin.

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Borodin, A., Gorin, V. & Guionnet, A. Gaussian asymptotics of discrete \(\beta \)-ensembles. Publ.math.IHES 125, 1–78 (2017). https://doi.org/10.1007/s10240-016-0085-5

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