Gaussian asymptotics of discrete $\beta $-ensembles

Alexei Borodin; Vadim Gorin; Alice Guionnet

doi:10.1007/s10240-016-0085-5

Gaussian asymptotics of discrete

β

-ensembles

Alexei Borodin ¹ ; Vadim Gorin ^1,²; Alice Guionnet ¹

¹ Department of Mathematics, Massachusetts Institute of Technology Cambridge MA USA
² Institute for Information Transmission Problems of Russian Academy of Sciences Moscow Russia

Publications Mathématiques de l'IHÉS, Volume 125 (2017), pp. 1-78

Abstract

We introduce and study stochastic $N$ -particle ensembles which are discretizations for general- $β$ 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.

Received: 2015-06-10
Accepted: 2016-06-06
Online First: 2016-06-14
Published online: 2017-06-07

MR Zbl

DOI: 10.1007/s10240-016-0085-5

Author's affiliations:

Alexei Borodin ¹; Vadim Gorin ^{1, 2}; Alice Guionnet ¹

¹ Department of Mathematics, Massachusetts Institute of Technology Cambridge MA USA
² Institute for Information Transmission Problems of Russian Academy of Sciences Moscow Russia

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     title = {Gaussian asymptotics of discrete $\beta $-ensembles},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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     year = {2017},
     publisher = {Springer Berlin Heidelberg},
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Alexei Borodin; Vadim Gorin; Alice Guionnet. Gaussian asymptotics of discrete $\beta $-ensembles. Publications Mathématiques de l'IHÉS, Volume 125 (2017), pp. 1-78. doi: 10.1007/s10240-016-0085-5

References
Cited by

[AM] J. Ambjørn; Yu. Makeenko Properties of loop equations for the Hermitian matrix model and for two-dimensional gravity, Mod. Phys. Lett. A, Volume 5 (1990), pp. 1753-1763 | MR | DOI | Zbl

[AGZ] G. Anderson; A. Guionnet; O. Zeitouni Introduction to Random Matrices (2009) | DOI | Zbl

[BBDT] J. Baik; A. Borodin; P. Deift; T. Suidan A model for the bus system in Cuernevaca (Mexico), J. Phys. A, Math. Gen., Volume 39 (2006), p. 8965 | DOI | Zbl | arXiv

[BKMM] 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

[BFG] F. Bekerman; A. Figalli; A. Guionnet Transport maps for Beta-matrix models and universality, Commun. Math. Phys., Volume 338 (2015), pp. 589-619 | DOI | Zbl | arXiv

[BeGu] G. Ben Arous; A. Guionnet Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Relat. Fields, Volume 108 (1997), pp. 517-542 | MR | DOI | Zbl

[BDE] G. Bonnet; F. David; B. Eynard Breakdown of universality in multi-cut matrix models, J. Phys. A, Math. Gen., Volume 33 (2000), p. 6739 | MR | DOI | Zbl | arXiv

[B1] A. Borodin Schur dynamics of the Schur processes, Adv. Math., Volume 228 (2011), pp. 2268-2291 | MR | DOI | Zbl | arXiv

[B2] A. Borodin CLT for spectra of submatrices of Wigner random matrices, Mosc. Math. J., Volume 14 (2014), pp. 29-38 | MR | Zbl | arXiv

[BF] A. Borodin; P. Ferrari Anisotropic growth of random surfaces in $2 + 1$ dimensions, Commun. Math. Phys., Volume 325 (2014), pp. 603-684 | MR | DOI | Zbl | arXiv

[BoGo] A. Borodin; V. Gorin Shuffling algorithm for boxed plane partitions, Adv. Math., Volume 220 (2009), pp. 1739-1770 | MR | DOI | Zbl | arXiv

[BoGo2] A. Borodin; V. Gorin General beta Jacobi corners process and the Gaussian Free Field, Commun. Pure Appl. Math., Volume 68 (2015), pp. 1683-1884 | DOI | Zbl | arXiv

[BO1] A. Borodin; G. Olshanski Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. Math., Volume 161 (2005), pp. 1319-1422 | MR | DOI | Zbl | arXiv

[BO2] A. Borodin; G. Olshanski Asymptotics of Plancherel-type random partitions, J. Algebra, Volume 313 (2007), pp. 40-60 | MR | DOI | Zbl | arXiv

[BP] A. Borodin; L. Petrov Integrable probability: from representation theory to Macdonald processes, Probab. Surv., Volume 11 (2014), pp. 1-58 | MR | DOI | Zbl | arXiv

[BoGu1] G. Borot; A. Guionnet Asymptotic expansion of beta matrix models in the one-cut regime, Commun. Math. Phys., Volume 317 (2013), pp. 447-483 | DOI | Zbl | arXiv

[BoGu2] G. Borot; A. Guionnet Asymptotic expansion of beta matrix models in the multi-cut regime, Commun. Math. Phys., Volume 317 (2013), pp. 447-483 | DOI | Zbl | arXiv

[BEY] P. Bourgade; L. Erdos; H.-T. Yau Edge universality of beta ensembles, Commun. Math. Phys., Volume 332 (2014), pp. 261-353 | MR | DOI | Zbl | arXiv

[BPS] A. Boutet de Monvel; L. Pastur; M. Shcherbina On the statistical mechanics approach in the random matrix theory: integrated density of states, J. Stat. Phys., Volume 79 (1995), pp. 585-611 | MR | DOI | Zbl

[BD] J. Breuer and M. Duits, Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, J. Amer. Math. Soc. | arXiv

[BIPZ] E. Brézin; C. Itzykson; G. Parisi; J. B. Zuber Planar diagrams, Commun. Math. Phys., Volume 59 (1978), pp. 35-51 | MR | DOI | Zbl

[BuGo] A. Bufetov; V. Gorin Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., Volume 25 (2015), pp. 763-814 | MR | DOI | Zbl | arXiv

[BuGo2] A. Bufetov and V. Gorin, Fluctuations of particle systems determined by Schur generating functions, | arXiv

[Cha] S. Chatterjee, Rigorous solution of strongly coupled $S O (N)$ lattice gauge theory in the large $N$ limit, | arXiv

[CE] L. O. Chekhov; B. Eynard Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys., Volume 0612 (2006) | MR | DOI | Zbl | arXiv

[CJY] S. Chhita; K. Johansson; B. Young Asymptotic domino statistics in the Aztec diamond, Ann. Appl. Probab., Volume 25 (2015), pp. 1232-1278 | MR | DOI | Zbl | arXiv

[CLP] H. Cohn Larsen; J. Propp The shape of a typical boxed plane partition, N.Y. J. Math., Volume 4 (1998), pp. 137-165 | MR | Zbl | arXiv

[CGM] B. Collins; A. Guionnet; E. Maurel-Segala Asymptotics of unitary and orthogonal matrix integrals, Adv. Math., Volume 222 (2009), pp. 172-215 | MR | DOI | Zbl | arXiv

[DF] M. Dolega; V. Feray Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J., Volume 7 (2016), pp. 1193-1282 | MR | Zbl | arXiv

[DS] P. D. Dragnev; E. B. Saff Constrained energy problems with applications to orthogonal polynomials of a discrete variable, J. Anal. Math., Volume 72 (1997), pp. 223-259 | MR | DOI | Zbl

[Du] B. A. Dubrovin Theta functions and non-linear equations, Russ. Math. Surv., Volume 36 (1981), pp. 11-92 | DOI | Zbl

[Ey1] B. Eynard All genus correlation functions for the hermitian 1-matrix model, J. High Energy Phys., Volume 0411 (2004) | DOI | arXiv

[Ey2] B. Eynard All order asymptotic expansion of large partitions, J. Stat. Mech. Theory Exp., Volume 2008 (2008) | MR | DOI

[Ey3] B. Eynard A matrix model for plane partitions, J. Stat. Mech. Theory Exp., Volume 0910 (2009) | MR | DOI

[EO] B. Eynard; N. Orantin Topological recursion in enumerative geometry and random matrices, J. Phys. A, Volume 42 (2009) | MR | DOI | Zbl

[Fe] D. Feral On large deviations for the spectral measure of discrete Coulomb gas, Seminaire de Probabilites XLI Lecture Notes in Mathematics (2008), pp. 19-49

[Fo] P. J. Forrester Log-Gases and Random Matrices (2010) | Zbl

[G] V. Gorin Non-intersecting paths and Hahn orthogonal ensemble, Funct. Anal. Appl., Volume 42 (2008), pp. 180-197 | MR | DOI | Zbl | arXiv

[GS] V. Gorin; M. Shkolnikov Multilevel Dyson Brownian motions via Jack polynomials, Probab. Theory Relat. Fields, Volume 163 (2015), p. 413 | MR | DOI | Zbl | arXiv

[GN] A. Guionnet; J. Novak Asymptotics of unitary multimatrix models: the Schwinger–Dyson lattice and topological recursion, J. Funct. Anal., Volume 268 (2015), pp. 2851-2905 | MR | DOI | Zbl

[HO] A. Hora; N. Obata Quantum Probability and Spectral Analysis of Graphs (2007) | Zbl

[J1] K. Johansson On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., Volume 91 (1998), pp. 151-204 | MR | DOI | Zbl

[J2] K. Johansson Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. (2), Volume 153 (2001), pp. 259-296 (math.CO/9906120) | MR | DOI | Zbl

[J3] K. Johansson Shape fluctuations and random matrices, Commun. Math. Phys., Volume 209 (2000), pp. 437-476 | MR | DOI | Zbl | arXiv

[J4] K. Johansson Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields, Volume 123 (2002), pp. 225-280 | MR | DOI | Zbl | arXiv

[JN] K. Johansson; E. Nordenstam Eigenvalues of GUE minors, Electron. J. Probab., Volume 11 (2006), p. 50 | MR | DOI | Zbl | arXiv

[K] R. Kenyon Height fluctuations in the honeycomb dimer model, Commun. Math. Phys., Volume 281 (2008), pp. 675-709 | MR | DOI | Zbl | arXiv

[KO] R. Kenyon; A. Okounkov Limit shapes and the complex burgers equation, Acta Math., Volume 199 (2007), pp. 263-302 | MR | DOI | Zbl | arXiv

[KOR] W. Konig; N. O’Connel; S. Roch Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles, Electron. J. Probab., Volume 7 (2002), pp. 1-24 | MR | DOI

[KS] T. Kriecherbauer and M. Shcherbina, Fluctuations of eigenvalues of matrix models and their applications, | arXiv

[Ma] I. G. Macdonald Symmetric Functions and Hall Polynomials (1999) | Zbl

[1] M. Maida; E. Maurel-Segala Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices, Probab. Theory Relat. Fields, Volume 159 (2014), pp. 329-356 | MR | DOI | Zbl | arXiv

[Me] M. L. Mehta Random Matrices (2004) | Zbl

[Mo] A. Moll, in preparation.

[Mi] A. A. Migdal Loop equations and $1 / N$ expansion, Phys. Rep., Volume 102 (1983), pp. 199-290 | DOI

[NS] N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, High Energy Physics - Theory, (2013), 1–83, | arXiv

[NP] N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional $N = 2$ quiver gauge theories, | arXiv

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

[O] G. Olshanski The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., Volume 205 (2003), pp. 464-524 | MR | DOI | Zbl | arXiv

[O2] G. Olshanksi Probability measures on dual objects to compact symmetric spaces and hypergeometric identities, Funct. Anal. Appl., Volume 37 (2001), pp. 281-301 | MR | DOI

[Ox] The Oxford Handbook of Random Matrix Theory (2011) | Zbl

[PS] L. Pastur; M. Shcherbina Eigenvalue Distribution of Large Random Matrices (2011) | DOI | Zbl

[P1] L. Petrov Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Relat. Fields, Volume 160 (2014), pp. 429-487 | MR | DOI | Zbl | arXiv

[P2] L. Petrov Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field, Ann. Probab., Volume 43 (2015), pp. 1-43 | MR | DOI | Zbl | arXiv

[ST] E. B. Saff; V. Totik Logarithmic Potentials with External Fields (1997) | DOI | Zbl

[Shch] M. Shcherbina Fluctuations of linear eigenvalue statistics of $β$ matrix models in the multi-cut regime, J. Stat. Phys., Volume 151 (2013), pp. 1004-1034 | MR | DOI | Zbl | arXiv

[Wi] E. P. Wigner On the distribution of the roots of certain symmetric matrices, Ann. Math., Volume 67 (1958), pp. 325-327 | MR | DOI | Zbl

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