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Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations
About this Title
Alice Guionnet, Université de Lyon, CNRS, ENS de Lyon, Lyon, France
Publication: CBMS Regional Conference Series in Mathematics
Publication Year:
2019; Volume 130
ISBNs: 978-1-4704-5027-4 (print); 978-1-4704-5194-3 (online)
DOI: https://doi.org/10.1090/cbms/130
MathSciNet review: MR3931304
MSC: Primary 60B20; Secondary 46L54, 60F05, 60F10
Table of Contents
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Front/Back Matter
Chapters
- Introduction
- The example of the GUE
- Wigner random matrices
- Beta-ensembles
- Discrete beta-ensembles
- Continuous beta-models: The several cut case
- Several matrix-ensembles
- Universality for beta-models
- S. Albeverio, L. Pastur, and M. Shcherbina, On the $1/n$ expansion for some unitary invariant ensembles of random matrices, Comm. Math. Phys. 224 (2001), no. 1, 271–305. Dedicated to Joel L. Lebowitz. MR 1869000, DOI 10.1007/s002200100531
- J. Ambjørn and Yu. M. Makeenko, Properties of loop equations for the Hermitian matrix model and for two-dimensional quantum gravity, Modern Phys. Lett. A 5 (1990), no. 22, 1753–1763. MR 1075933, DOI 10.1142/S0217732390001992
- Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118, Cambridge University Press, Cambridge, 2010. MR 2760897
- Zhigang Bao, László Erdős, and Kevin Schnelli, Local law of addition of random matrices on optimal scale, Comm. Math. Phys. 349 (2017), no. 3, 947–990. MR 3602820, DOI 10.1007/s00220-016-2805-6
- F. Bekerman, A. Figalli, and A. Guionnet, Transport maps for $\beta$-matrix models and universality, Comm. Math. Phys. 338 (2015), no. 2, 589–619. MR 3351052, DOI 10.1007/s00220-015-2384-y
- G. Ben Arous, A. Dembo, and A. Guionnet, Aging of spherical spin glasses, Probab. Theory Related Fields 120 (2001), no. 1, 1–67. MR 1856194, DOI 10.1007/PL00008774
- G. Ben Arous and A. Guionnet, Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Related Fields 108 (1997), no. 4, 517–542. MR 1465640, DOI 10.1007/s004400050119
- Gérard Ben Arous and Alice Guionnet, The spectrum of heavy tailed random matrices, Comm. Math. Phys. 278 (2008), no. 3, 715–751.
- Florent Benaych-Georges, Alice Guionnet, and Camille Male, Central limit theorems for linear statistics of heavy tailed random matrices, Comm. Math. Phys. 329 (2014), no. 2, 641–686. MR 3210147, DOI 10.1007/s00220-014-1975-3
- P. Biane, M. Capitaine, and A. Guionnet, Large deviation bounds for matrix Brownian motion, Invent. Math. 152 (2003), no. 2, 433–459. MR 1975007, DOI 10.1007/s00222-002-0281-4
- Pavel M. Bleher and Alexander R. Its, Asymptotics of the partition function of a random matrix model, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 1943–2000 (English, with English and French summaries). MR 2187941
- Charles Bordenave, Pietro Caputo, and Djalil Chafaï, Spectrum of large random reversible Markov chains: heavy-tailed weights on the complete graph, Ann. Probab. 39 (2011), no. 4, 1544–1590. MR 2857250, DOI 10.1214/10-AOP587
- Charles Bordenave and Alice Guionnet, Localization and delocalization of eigenvectors for heavy-tailed random matrices, Probab. Theory Related Fields 157 (2013), no. 3-4, 885–953. MR 3129806, DOI 10.1007/s00440-012-0473-9
- Alexei Borodin, Vadim Gorin, and Alice Guionnet, Gaussian asymptotics of discrete $\beta$-ensembles, Publ. Math. Inst. Hautes Études Sci. 125 (2017), 1–78. MR 3668648, DOI 10.1007/s10240-016-0085-5
- G. Borot and A. Guionnet, Asymptotic expansion of $\beta$ matrix models in the one-cut regime, Comm. Math. Phys. 317 (2013), no. 2, 447–483. MR 3010191, DOI 10.1007/s00220-012-1619-4
- G. Borot and A. Guionnet, Asymptotic expansion of $\beta$ matrix models in the several-cut regime, arXiv 1303.1045, 2013.
- Gaëtan Borot, Bertrand Eynard, and Nicolas Orantin, Abstract loop equations, topological recursion and new applications, Commun. Number Theory Phys. 9 (2015), no. 1, 51–187. MR 3339853, DOI 10.4310/CNTP.2015.v9.n1.a2
- Gaëtan Borot, Alice Guionnet, and Karol K. Kozlowski, Large-$N$ asymptotic expansion for mean field models with Coulomb gas interaction, Int. Math. Res. Not. IMRN 20 (2015), 10451–10524. MR 3455872, DOI 10.1093/imrn/rnu260
- Gaëtan Borot, Alice Guionnet, and Karol K. Kozlowski, Asymptotic expansion of a partition function related to the sinh-model, Mathematical Physics Studies, Springer, [Cham], 2016. MR 3585351, DOI 10.1007/978-3-319-33379-3
- Jean-Philippe Bouchaud and Pierre Cizeau, Theory of Lévy matrices, Phys. Rev. E 3 (1994), 1810–1822.
- Paul Bourgade, László Erdös, and Horng-Tzer Yau, Edge universality of beta ensembles, Comm. Math. Phys. 332 (2014), no. 1, 261–353. MR 3253704, DOI 10.1007/s00220-014-2120-z
- Paul Bourgade, László Erdős, and Horng-Tzer Yau, Universality of general $\beta$-ensembles, Duke Math. J. 163 (2014), no. 6, 1127–1190. MR 3192527, DOI 10.1215/00127094-2649752
- Paul Bourgade, Laszlo Erdős, Horng-Tzer Yau, and Jun Yin, Universality for a class of random band matrices, Adv. Theor. Math. Phys. 21 (2017), no. 3, 739–800. MR 3695802, DOI 10.4310/ATMP.2017.v21.n3.a5
- Mireille Bousquet-Mélou and Arnaud Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, J. Combin. Theory Ser. B 96 (2006), no. 5, 623–672. MR 2236503, DOI 10.1016/j.jctb.2005.12.003
- A. Boutet de Monvel, L. Pastur, and M. Shcherbina, On the statistical mechanics approach in the random matrix theory: integrated density of states, J. Statist. Phys. 79 (1995), no. 3-4, 585–611. MR 1327898, DOI 10.1007/BF02184872
- Jonathan Breuer and Maurice Duits, Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients, J. Amer. Math. Soc. 30 (2017), no. 1, 27–66. MR 3556288, DOI 10.1090/jams/854
- William G. Brown, Enumeration of triangulations of the disk, Proc. London Math. Soc. (3) 14 (1964), 746–768. MR 168485, DOI 10.1112/plms/s3-14.4.746
- S. Chatterjee, Rigorous solution of strongly coupled so(n) lattice gauge theory in the large $n$ limit, To appear in Comm. Math. Phys., arXiv: 1502.07719, 2015.
- L. O. Chekhov, B. Eynard, and O. Marchal, Topological expansion of the $\beta$-ensemble model and quantum algebraic geometry in the sectorwise approach, Theoret. and Math. Phys. 166 (2011), no. 2, 141–185. Russian version appears in Teoret. Mat. Fiz. 166 (2011), no. 2, 163–215. MR 3165804, DOI 10.1007/s11232-011-0012-3
- Leonid Chekhov and Bertrand Eynard, Matrix eigenvalue model: Feynman graph technique for all genera, J. High Energy Phys. 12 (2006), 026, 29. MR 2276715, DOI 10.1088/1126-6708/2006/12/026
- T. Claeys, T. Grava, and K. D. T.-R. McLaughlin, Asymptotics for the partition function in two-cut random matrix models, Comm. Math. Phys. 339 (2015), no. 2, 513–587. MR 3370612, DOI 10.1007/s00220-015-2412-y
- Benoît Collins, Alice Guionnet, and Edouard Maurel-Segala, Asymptotics of unitary and orthogonal matrix integrals, Adv. Math. 222 (2009), no. 1, 172–215. MR 2531371, DOI 10.1016/j.aim.2009.03.019
- Benoît Collins, Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability, Int. Math. Res. Not. 17 (2003), 953–982. MR 1959915, DOI 10.1155/S107379280320917X
- Antoine Dahlqvist, Free energies and fluctuations for the unitary Brownian motion, Comm. Math. Phys. 348 (2016), no. 2, 395–444. MR 3554890, DOI 10.1007/s00220-016-2756-y
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for orthogonal polynomials, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 491–501. MR 1648182
- P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1677884
- Percy Deift and Dimitri Gioev, Random matrix theory: invariant ensembles and universality, Courant Lecture Notes in Mathematics, vol. 18, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2009. MR 2514781, DOI 10.1090/cln/018
- Percy Deift, Alexander Its, and Igor Krasovsky, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, Ann. of Math. (2) 174 (2011), no. 2, 1243–1299. MR 2831118, DOI 10.4007/annals.2011.174.2.12
- D. Féral, On large deviations for the spectral measure of discrete Coulomb gas, Séminaire de probabilités XLI, Lecture Notes in Math., vol. 1934, Springer, Berlin, 2008, pp. 19–49. MR 2483725, DOI 10.1007/978-3-540-77913-1_{2}
- A. L. Dixon, Clt for fluctuations of $\beta$-ensembles with general potential, arxiv 1706.09663.
- B. A. Dubrovin, Theta-functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2(218), 11–80 (Russian). With an appendix by I. M. Krichever. MR 616797
- Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830–5847. MR 1936554, DOI 10.1063/1.1507823
- E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber, Planar diagrams, Comm. Math. Phys. 59 (1978), no. 1, 35–51. MR 471676
- N. M. Ercolani and K. D. T.-R. McLaughlin, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration, Int. Math. Res. Not. 14 (2003), 755–820. MR 1953782, DOI 10.1155/S1073792803211089
- László Erdős, Universality for random matrices and log-gases, Current developments in mathematics 2012, Int. Press, Somerville, MA, 2013, pp. 59–132. MR 3204344
- László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Wegner estimate and level repulsion for Wigner random matrices, Int. Math. Res. Not. IMRN 3 (2010), 436–479. MR 2587574, DOI 10.1093/imrn/rnp136
- László Erdős and Horng-Tzer Yau, Gap universality of generalized Wigner and $\beta$-ensembles, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 8, 1927–2036. MR 3372074, DOI 10.4171/JEMS/548
- B. Eynard, Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices, Nuclear Phys. B 506 (1997), no. 3, 633–664. MR 1488592, DOI 10.1016/S0550-3213(97)00452-5
- B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), no. 2, 347–452. MR 2346575, DOI 10.4310/CNTP.2007.v1.n2.a4
- Bertrand Eynard, Topological expansion for the 1-Hermitian matrix model correlation functions, J. High Energy Phys. 11 (2004), 031, 35 pp. (2005). MR 2118807, DOI 10.1088/1126-6708/2004/11/031
- Bertrand Eynard, Counting surfaces, Progress in Mathematical Physics, vol. 70, Birkhäuser/Springer, [Cham], 2016. CRM Aisenstadt chair lectures. MR 3468847, DOI 10.1007/978-3-7643-8797-6
- C. Fan, A. Guionnet, Y. Song, and A. Wang, Convergence of eigenvalues to the support of the limiting measure in critical $\beta$ matrix models, Random Matrices Theory Appl. 4 (2015), no. 3, 1550013, 22. MR 3385707, DOI 10.1142/S2010326315500136
- Alessio Figalli and Alice Guionnet, Universality in several-matrix models via approximate transport maps, Acta Math. 217 (2016), no. 1, 81–176. MR 3646880, DOI 10.1007/s11511-016-0142-4
- Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painlevé transcendents, Mathematical Surveys and Monographs, vol. 128, American Mathematical Society, Providence, RI, 2006. The Riemann-Hilbert approach. MR 2264522, DOI 10.1090/surv/128
- F. Benaych Georges, Rectangular random matrices, related convolution, Preprint (2006).
- I. P. Goulden and D. M. Jackson, Combinatorial enumeration, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. With a foreword by Gian-Carlo Rota. MR 702512
- Alice Guionnet and Edouard Maurel-Segala, Combinatorial aspects of matrix models, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 241–279. MR 2249657
- Alice Guionnet and Edouard Maurel-Segala, Second order asymptotics for matrix models, Ann. Probab. 35 (2007), no. 6, 2160–2212. MR 2353386, DOI 10.1214/009117907000000141
- A. Guionnet and J. Novak, Asymptotics of unitary multimatrix models: The Schwinger-Dyson lattice and topological recursion, arXiv:1401.2703, 2014.
- A. Guionnet and O. Zeitouni, Concentration of the spectral measure for large matrices, Electron. Comm. Probab. 5 (2000), 119–136. MR 1781846, DOI 10.1214/ECP.v5-1026
- Alice Guionnet and Ofer Zeitouni, Large deviations asymptotics for spherical integrals, J. Funct. Anal. 188 (2002), no. 2, 461–515. MR 1883414, DOI 10.1006/jfan.2001.3833
- Alice Guionnet, First order asymptotics of matrix integrals; a rigorous approach towards the understanding of matrix models, Comm. Math. Phys. 244 (2004), no. 3, 527–569. MR 2034487, DOI 10.1007/s00220-003-0992-4
- J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), no. 3, 457–485. MR 848681, DOI 10.1007/BF01390325
- Jiaoyang Huang and Alice Guionnet, Rigidity and edge universality in discrete beta-ensemble, arXiv+1705.05527, 2017.
- A. Its and I. Krasovsky, Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump, Integrable systems and random matrices, Contemp. Math., vol. 458, Amer. Math. Soc., Providence, RI, 2008, pp. 215–247. MR 2411909, DOI 10.1090/conm/458/08938
- Kurt Johansson, On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), no. 1, 151–204. MR 1487983, DOI 10.1215/S0012-7094-98-09108-6
- Maxim Kontsevich, Vassiliev’s knot invariants, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 137–150. MR 1237836
- Manjunath Krishnapur, Brian Rider, and Bálint Virág, Universality of the stochastic Airy operator, Comm. Pure Appl. Math. 69 (2016), no. 1, 145–199. MR 3433632, DOI 10.1002/cpa.21573
- A. B. J. Kuijlaars and K. T-R McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), no. 6, 736–785. MR 1744002, DOI 10.1002/(SICI)1097-0312(200006)53:6<736::AID-CPA2>3.0.CO;2-5
- M. Ledoux, G. Lambert, and C. Webb, Stein’s method for normal approximation of linear statistics of beta-ensembles, to appear in Ann. Probab., arXiv:1706.10251, 2017.
- Thomas Leblé and Sylvia Serfaty, Fluctuations of two dimensional Coulomb gases, Geom. Funct. Anal. 28 (2018), no. 2, 443–508. MR 3788208, DOI 10.1007/s00039-018-0443-1
- A. Lytova and L. Pastur, Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, Ann. Probab. 37 (2009), no. 5, 1778–1840. MR 2561434, DOI 10.1214/09-AOP452
- Mylène Maïda and Édouard Maurel-Segala, Free transport-entropy inequalities for non-convex potentials and application to concentration for random matrices, Probab. Theory Related Fields 159 (2014), no. 1-2, 329–356. MR 3201924, DOI 10.1007/s00440-013-0508-x
- A. Matytsin, On the large-$N$ limit of the Itzykson-Zuber integral, Nuclear Phys. B 411 (1994), no. 2-3, 805–820. MR 1257846, DOI 10.1016/0550-3213(94)90471-5
- E. Maurel Segala, High order asymptotics of matrix models and enumeration of maps, arXiv:math/0608192v1 [math.PR], 2006.
- M. L. Mehta, A method of integration over matrix variables, Comm. Math. Phys. 79 (1981), no. 3, 327–340. MR 627056
- Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
- N. I. Muskhelishvili, Singular integral equations, Dover Publications, Inc., New York, 1992. Boundary problems of function theory and their application to mathematical physics; Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok; Corrected reprint of the 1953 English translation. MR 1215485
- Leonid Pastur and Mariya Shcherbina, Eigenvalue distribution of large random matrices, Mathematical Surveys and Monographs, vol. 171, American Mathematical Society, Providence, RI, 2011. MR 2808038, DOI 10.1090/surv/171
- José A. Ramírez, Brian Rider, and Bálint Virág, Beta ensembles, stochastic Airy spectrum, and a diffusion, J. Amer. Math. Soc. 24 (2011), no. 4, 919–944. MR 2813333, DOI 10.1090/S0894-0347-2011-00703-0
- Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
- M. Shcherbina, Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime, J. Stat. Phys. 151 (2013), no. 6, 1004–1034. MR 3063494, DOI 10.1007/s10955-013-0740-x
- M. Shcherbina, Change of variables as a method to study general $\beta$-models: bulk universality, J. Math. Phys. 55 (2014), no. 4, 043504, 23. MR 3390602, DOI 10.1063/1.4870603
- Mariya Shcherbina, Asymptotic expansions for $beta$ matrix models and their applications to the universality conjecture, Random matrix theory, interacting particle systems, and integrable systems, Math. Sci. Res. Inst. Publ., vol. 65, Cambridge Univ. Press, New York, 2014, pp. 463–482. MR 3380697
- Jack W. Silverstein and Z. D. Bai, On the empirical distribution of eigenvalues of a class of large-dimensional random matrices, J. Multivariate Anal. 54 (1995), no. 2, 175–192. MR 1345534, DOI 10.1006/jmva.1995.1051
- Terence Tao and Van Vu, Random matrices: universality of local eigenvalue statistics up to the edge, Comm. Math. Phys. 298 (2010), no. 2, 549–572. MR 2669449, DOI 10.1007/s00220-010-1044-5
- Terence Tao and Van Vu, Random matrices: universality of local eigenvalue statistics up to the edge, Comm. Math. Phys. 298 (2010), no. 2, 549–572. MR 2669449, DOI 10.1007/s00220-010-1044-5
- G. ’t Hooft, Magnetic monopoles in unified gauge theories, Nuclear Phys. B79 (1974), 276–284. MR 0413809, DOI 10.1016/0550-3213(74)90486-6
- Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174. MR 1257246
- Craig A. Tracy and Harold Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), no. 3, 727–754. MR 1385083
- F. G. Tricomi, Integral equations, Dover Publications, New York, NY 1985, Reprint of the 1957 original.
- W. T. Tutte, A census of planar maps, Canadian J. Math. 15 (1963), 249–271. MR 146823, DOI 10.4153/CJM-1963-029-x
- Benedek Valkó and Bálint Virág, Continuum limits of random matrices and the Brownian carousel, Invent. Math. 177 (2009), no. 3, 463–508. MR 2534097, DOI 10.1007/s00222-009-0180-z
- Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201–220. MR 1094052, DOI 10.1007/BF01245072
- Dan Voiculescu, Lectures on probability theory and statistics: Ecole D’Été de Probabilités de Saint-Flour XXVIII - 1998, Lecture Notes in Mathematics, vol. 1738, pp. 283–349, Springer, New York, NY, 2000.
- Dan Voiculescu, Free entropy, Bull. London Math. Soc. 34 (2002), no. 3, 257–278. MR 1887698, DOI 10.1112/S0024609301008992
- Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. of Math. (2) 62 (1955), 548–564. MR 77805, DOI 10.2307/1970079