Gaussian asymptotics of discrete β-ensembles
Borodin, Alexei1 ; Gorin, Vadim1, 2 ; Guionnet, Alice1
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, Tome 125 (2017), pp. 1-78.

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→∞. 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.

DOI : 10.1007/s10240-016-0085-5
Borodin, Alexei 1 ; Gorin, Vadim 1, 2 ; Guionnet, Alice 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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Borodin, Alexei; Gorin, Vadim; Guionnet, Alice. Gaussian asymptotics of discrete $\beta $-ensembles. Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 1-78. doi : 10.1007/s10240-016-0085-5. https://www.numdam.org/articles/10.1007/s10240-016-0085-5/

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

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

[BBDT] Baik, J.; Borodin, A.; Deift, P.; Suidan, T. A model for the bus system in Cuernevaca (Mexico), J. Phys. A, Math. Gen., Volume 39 (2006), p. 8965 (arXiv:math/0510414) | DOI | Zbl

[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] Bekerman, F.; Figalli, A.; Guionnet, A. Transport maps for Beta-matrix models and universality, Commun. Math. Phys., Volume 338 (2015), pp. 589-619 (arXiv:1311.2315) | DOI | Zbl

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

[BDE] Bonnet, G.; David, F.; Eynard, B. Breakdown of universality in multi-cut matrix models, J. Phys. A, Math. Gen., Volume 33 (2000), p. 6739 (arXiv:cond-mat/0003324) | DOI | MR | Zbl

[B1] Borodin, A. Schur dynamics of the Schur processes, Adv. Math., Volume 228 (2011), pp. 2268-2291 (arXiv:1001.3442) | DOI | MR | Zbl

[B2] Borodin, A. CLT for spectra of submatrices of Wigner random matrices, Mosc. Math. J., Volume 14 (2014), pp. 29-38 (arXiv:1010.0898) | MR | Zbl

[BF] Borodin, A.; Ferrari, P. Anisotropic growth of random surfaces in 2+1 dimensions, Commun. Math. Phys., Volume 325 (2014), pp. 603-684 (arXiv:0804.3035) | DOI | MR | Zbl

[BoGo] Borodin, A.; Gorin, V. Shuffling algorithm for boxed plane partitions, Adv. Math., Volume 220 (2009), pp. 1739-1770 (arXiv:0804.3071) | DOI | MR | Zbl

[BoGo2] Borodin, A.; Gorin, V. General beta Jacobi corners process and the Gaussian Free Field, Commun. Pure Appl. Math., Volume 68 (2015), pp. 1683-1884 (arXiv:1305.3627) | DOI | Zbl

[BO1] Borodin, A.; Olshanski, G. Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. Math., Volume 161 (2005), pp. 1319-1422 (arXiv:math/0109194) | DOI | MR | Zbl

[BO2] Borodin, A.; Olshanski, G. Asymptotics of Plancherel-type random partitions, J. Algebra, Volume 313 (2007), pp. 40-60 (arXiv:math/0610240) | DOI | MR | Zbl

[BP] Borodin, A.; Petrov, L. Integrable probability: from representation theory to Macdonald processes, Probab. Surv., Volume 11 (2014), pp. 1-58 (arXiv:1310.8007) | DOI | MR | Zbl

[BoGu1] Borot, G.; Guionnet, A. Asymptotic expansion of beta matrix models in the one-cut regime, Commun. Math. Phys., Volume 317 (2013), pp. 447-483 (arXiv:1107.1167) | DOI | Zbl

[BoGu2] Borot, G.; Guionnet, A. Asymptotic expansion of beta matrix models in the multi-cut regime, Commun. Math. Phys., Volume 317 (2013), pp. 447-483 (arXiv:1303.1045) | DOI | Zbl

[BEY] Bourgade, P.; Erdos, L.; Yau, H.-T. Edge universality of beta ensembles, Commun. Math. Phys., Volume 332 (2014), pp. 261-353 (arXiv:1306.5728) | DOI | MR | Zbl

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

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

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

[BuGo] Bufetov, A.; Gorin, V. Representations of classical Lie groups and quantized free convolution, Geom. Funct. Anal., Volume 25 (2015), pp. 763-814 (arXiv:1311.5780) | DOI | MR | Zbl

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

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

[CE] Chekhov, L. O.; Eynard, B. Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys., Volume 0612 (2006) (arXiv:math-ph/0604014) | DOI | MR | Zbl

[CJY] Chhita, S.; Johansson, K.; Young, B. Asymptotic domino statistics in the Aztec diamond, Ann. Appl. Probab., Volume 25 (2015), pp. 1232-1278 (arXiv:1212.5414) | DOI | MR | Zbl

[CLP] Cohn Larsen, H.; Propp, J. The shape of a typical boxed plane partition, N.Y. J. Math., Volume 4 (1998), pp. 137-165 (arXiv:math/9801059) | MR | Zbl

[CGM] Collins, B.; Guionnet, A.; Maurel-Segala, E. Asymptotics of unitary and orthogonal matrix integrals, Adv. Math., Volume 222 (2009), pp. 172-215 (arXiv:math/0608193) | DOI | MR | Zbl

[DF] Dolega, M.; Feray, V. Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J., Volume 7 (2016), pp. 1193-1282 (arXiv:1402.4615) | MR | Zbl

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

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

[Ey1] Eynard, B. All genus correlation functions for the hermitian 1-matrix model, J. High Energy Phys., Volume 0411 (2004) (arXiv:hep-th/0407261) | DOI

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

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

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

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

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

[G] Gorin, V. Non-intersecting paths and Hahn orthogonal ensemble, Funct. Anal. Appl., Volume 42 (2008), pp. 180-197 (arXiv:0708.2349) | DOI | MR | Zbl

[GS] Gorin, V.; Shkolnikov, M. Multilevel Dyson Brownian motions via Jack polynomials, Probab. Theory Relat. Fields, Volume 163 (2015), p. 413 (arXiv:1401.5595) | DOI | MR | Zbl

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

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

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

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

[J3] Johansson, K. Shape fluctuations and random matrices, Commun. Math. Phys., Volume 209 (2000), pp. 437-476 (arXiv:math/9903134) | DOI | MR | Zbl

[J4] Johansson, K. Non-intersecting paths, random tilings and random matrices, Probab. Theory Relat. Fields, Volume 123 (2002), pp. 225-280 (arXiv:math/0011250) | DOI | MR | Zbl

[JN] Johansson, K.; Nordenstam, E. Eigenvalues of GUE minors, Electron. J. Probab., Volume 11 (2006), p. 50 (arXiv:math/0606760) | DOI | MR | Zbl

[K] Kenyon, R. Height fluctuations in the honeycomb dimer model, Commun. Math. Phys., Volume 281 (2008), pp. 675-709 (arXiv:math-ph/0405052) | DOI | MR | Zbl

[KO] Kenyon, R.; Okounkov, A. Limit shapes and the complex burgers equation, Acta Math., Volume 199 (2007), pp. 263-302 (arXiv:math-ph/0507007) | DOI | MR | Zbl

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

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

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

[1] Maida, M.; Maurel-Segala, E. 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 (arXiv:1204.3208) | DOI | MR | Zbl

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

[Mo] A. Moll, in preparation.

[Mi] Migdal, A. A. 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] Olshanski, G. The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., Volume 205 (2003), pp. 464-524 (arXiv:math/0109193) | DOI | MR | Zbl

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

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

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

[P1] Petrov, L. Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes, Probab. Theory Relat. Fields, Volume 160 (2014), pp. 429-487 (arXiv:1202.3901) | DOI | MR | Zbl

[P2] Petrov, L. Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field, Ann. Probab., Volume 43 (2015), pp. 1-43 (arXiv:1206.5123) | DOI | MR | Zbl

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

[Shch] Shcherbina, M. Fluctuations of linear eigenvalue statistics of β matrix models in the multi-cut regime, J. Stat. Phys., Volume 151 (2013), pp. 1004-1034 (arXiv:1205.7062) | DOI | MR | Zbl

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

  • Li, Shi-Hao; Shen, Bo-Jian; Yu, Guo-Fu; Forrester, Peter J. Discrete orthogonal ensemble on the exponential lattices, Advances in Applied Mathematics, Volume 164 (2025), p. 102836 | DOI:10.1016/j.aam.2024.102836
  • Krieger, Andrew; Menz, Georg; Tassy, Martin Homogenization of the variational principle for discrete random maps, Electronic Journal of Probability, Volume 30 (2025) no. none | DOI:10.1214/24-ejp1236
  • Dimitrov, Evgeni; Knizel, Alisa Multi-level loop equations for β-corners processes, Selecta Mathematica, Volume 31 (2025) no. 1 | DOI:10.1007/s00029-024-01006-5
  • Huang, Jiaoyang Edge statistics for lozenge tilings of polygons, I: concentration of height function on strip domains, Probability Theory and Related Fields, Volume 188 (2024) no. 1-2, p. 337 | DOI:10.1007/s00440-023-01238-0
  • Gorin, Vadim; Huang, Jiaoyang Dynamical loop equation, The Annals of Probability, Volume 52 (2024) no. 5 | DOI:10.1214/24-aop1685
  • Chen, Wen Xuan; Gao, Fu Qing Precise Deviations for Discrete Ensembles, Acta Mathematica Sinica, English Series, Volume 39 (2023) no. 8, p. 1580 | DOI:10.1007/s10114-023-1570-7
  • Dimitrov, Evgeni; Zhang, Hengzhi A global large deviation principle for discrete β-ensembles, Electronic Journal of Probability, Volume 28 (2023) no. none | DOI:10.1214/23-ejp977
  • Forrester, Peter J. A review of exact results for fluctuation formulas in random matrix theory, Probability Surveys, Volume 20 (2023) no. none | DOI:10.1214/23-ps15
  • Forrester, Peter J.; Li, Shi-Hao; Shen, Bo-Jian; Yu, Guo-Fu q-Pearson pair and moments in q-deformed ensembles, The Ramanujan Journal, Volume 60 (2023) no. 1, p. 195 | DOI:10.1007/s11139-022-00639-4
  • Chapuy, Guillaume; Dołęga, Maciej Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions, Advances in Mathematics, Volume 409 (2022), p. 108645 | DOI:10.1016/j.aim.2022.108645
  • Das, Sayan; Dimitrov, Evgeni Large deviations for discrete β-ensembles, Journal of Functional Analysis, Volume 283 (2022) no. 1, p. 109487 | DOI:10.1016/j.jfa.2022.109487
  • Nazarov, A. A.; Nikitin, P. P.; Postnova, O. V. Statistics of Irreducible Components in Large Tensor Powers of the Spinor Representation for so2n+1 as n→∞, Journal of Mathematical Sciences, Volume 261 (2022) no. 5, p. 658 | DOI:10.1007/s10958-022-05778-z
  • Ahn, Andrew Fluctuations of β-Jacobi product processes, Probability Theory and Related Fields, Volume 183 (2022) no. 1-2, p. 57 | DOI:10.1007/s00440-022-01109-0
  • Dimitrov, Evgeni; Knizel, Alisa Asymptotics of discrete β-corners processes via two-level discrete loop equations, Probability and Mathematical Physics, Volume 3 (2022) no. 2, p. 247 | DOI:10.2140/pmp.2022.3.247
  • Gao, Fuqing; Mu, Jianyong Moderate deviations for linear eigenvalue statistics of β-ensembles, Random Matrices: Theory and Applications, Volume 11 (2022) no. 02 | DOI:10.1142/s2010326322500174
  • Huang, Jiaoyang Law of large numbers and central limit theorems through Jack generating functions, Advances in Mathematics, Volume 380 (2021), p. 107545 | DOI:10.1016/j.aim.2020.107545
  • Breuer, Jonathan; Grinshpon, Yoel; White, Moshe J. Spectral Fluctuations for Schrödinger Operators with a Random Decaying Potential, Annales Henri Poincaré, Volume 22 (2021) no. 11, p. 3763 | DOI:10.1007/s00023-021-01082-9
  • Olshanski, Grigori Macdonald-Level Extension of Beta Ensembles and Large-N Limit Transition, Communications in Mathematical Physics, Volume 385 (2021) no. 1, p. 595 | DOI:10.1007/s00220-020-03899-7
  • Huang, Jiaoyang β-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems, International Mathematics Research Notices, Volume 2021 (2021) no. 8, p. 5898 | DOI:10.1093/imrn/rnz021
  • Duits, Maurice; Fahs, Benjamin; Kozhan, Rostyslav Global fluctuations for Multiple Orthogonal Polynomial Ensembles, Journal of Functional Analysis, Volume 281 (2021) no. 5, p. 109062 | DOI:10.1016/j.jfa.2021.109062
  • Katsevich, Alexander; Bertola, Marco; Tovbis, Alexander Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L2, Mathematische Nachrichten, Volume 294 (2021) no. 8, p. 1523 | DOI:10.1002/mana.201800567
  • Ahn, Andrew Global universality of Macdonald plane partitions, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 56 (2020) no. 3 | DOI:10.1214/19-aihp1013
  • Charlier, C.; Duits, M.; Kuijlaars, A. B. J.; Lenells, J. A Periodic Hexagon Tiling Model and Non-Hermitian Orthogonal Polynomials, Communications in Mathematical Physics, Volume 378 (2020) no. 1, p. 401 | DOI:10.1007/s00220-020-03779-0
  • Guionnet, Alice; Huang, Jiaoyang Rigidity and Edge Universality of Discrete β‐Ensembles, Communications on Pure and Applied Mathematics, Volume 72 (2019) no. 9, p. 1875 | DOI:10.1002/cpa.21818
  • Bufetov, Alexey; Gorin, Vadim Fourier transform on high-dimensional unitary groups with applications to random tilings, Duke Mathematical Journal, Volume 168 (2019) no. 13 | DOI:10.1215/00127094-2019-0023
  • Dimitrov, Evgeni; Knizel, Alisa Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions, Journal of Functional Analysis, Volume 276 (2019) no. 10, p. 3067 | DOI:10.1016/j.jfa.2018.12.008
  • Dołęga, Maciej; Śniady, Piotr Gaussian fluctuations of Jack-deformed random Young diagrams, Probability Theory and Related Fields, Volume 174 (2019) no. 1-2, p. 133 | DOI:10.1007/s00440-018-0854-9
  • Chatterjee, Sourav Rigidity of the three-dimensional hierarchical Coulomb gas, Probability Theory and Related Fields, Volume 175 (2019) no. 3-4, p. 1123 | DOI:10.1007/s00440-019-00912-6
  • Cuenca, Cesar Pieri integral formula and asymptotics of Jack unitary characters, Selecta Mathematica, Volume 24 (2018) no. 3, p. 2737 | DOI:10.1007/s00029-017-0373-z
  • Duse, Erik; Johansson, Kurt; Metcalfe, Anthony The Cusp-Airy process, Electronic Journal of Probability, Volume 21 (2016) no. none | DOI:10.1214/16-ejp2

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