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Jacobi polynomial moments and products of random matrices


Authors: Wolfgang Gawronski, Thorsten Neuschel and Dries Stivigny
Journal: Proc. Amer. Math. Soc. 144 (2016), 5251-5263
MSC (2010): Primary 30E05; Secondary 15B52, 30F10, 46L54
DOI: https://doi.org/10.1090/proc/13153
Published electronically: June 10, 2016
MathSciNet review: 3556269
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Abstract: Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class of measures and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters. Solving this moment problem requires a study of the Riemann surfaces associated to a class of algebraic equations. The connection to random matrix theory is then established using methods from free probability.


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  • [1] Gernot Akemann, Zdzislaw Burda, Mario Kieburg, and Taro Nagao, Universal microscopic correlation functions for products of truncated unitary matrices, J. Phys. A 47 (2014), no. 25, 255202, 26. MR 3224113, https://doi.org/10.1088/1751-8113/47/25/255202
  • [2] Gernot Akemann, Mario Kieburg, and Lu Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A 46 (2013), no. 27, 275205, 22. MR 3081917, https://doi.org/10.1088/1751-8113/46/27/275205
  • [3] 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
  • [4] Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674
  • [5] Z. Burda, R. A. Janik, and B. Waclaw, Spectrum of the product of independent random Gaussian matrices, Phys. Rev. E (3) 81 (2010), no. 4, 041132, 12. MR 2736204, https://doi.org/10.1103/PhysRevE.81.041132
  • [6] Zdzislaw Burda, Maciej A. Nowak, Andrzej Jarosz, Giacomo Livan, and Artur Swiech, Eigenvalues and singular values of products of rectangular Gaussian random matrices--the extended version, Acta Phys. Polon. B 42 (2011), no. 5, 939-985. MR 2806772, https://doi.org/10.5506/APhysPolB.42.939
  • [7] Romain Couillet and Mérouane Debbah, Random matrix methods for wireless communications, Cambridge University Press, Cambridge, 2011. MR 2884783
  • [8] T. Dupic and I. Isaac Pérez Castillo, Spectral density of products of Wishart dilute random matrices. Part I: the dense case, preprint arXiv:1401.7802.
  • [9] Peter J. Forrester, Eigenvalue statistics for product complex Wishart matrices, J. Phys. A 47 (2014), no. 34, 345202, 22. MR 3251989, https://doi.org/10.1088/1751-8113/47/34/345202
  • [10] Peter J. Forrester and Dang-Zheng Liu, Raney distributions and random matrix theory, J. Stat. Phys. 158 (2015), no. 5, 1051-1082. MR 3313617, https://doi.org/10.1007/s10955-014-1150-4
  • [11] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457-469. MR 0121828
  • [12] Yan V. Fyodorov and H.-J. Sommers, Random matrices close to Hermitian or unitary: overview of methods and results, J. Phys. A 36 (2003), no. 12, 3303-3347. Random matrix theory. MR 1986421, https://doi.org/10.1088/0305-4470/36/12/326
  • [13] F. Götze and A. Tikhomirov, On the asymptotic spectrum of products of independent random matrices, preprint arXiv:1012.2710.
  • [14] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498
  • [15] Mourad E. H. Ismail and Dennis Stanton, Classical orthogonal polynomials as moments, Canad. J. Math. 49 (1997), no. 3, 520-542. MR 1451259, https://doi.org/10.4153/CJM-1997-024-9
  • [16] Mourad E. H. Ismail and Dennis Stanton, More orthogonal polynomials as moments, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), Progr. Math., vol. 161, Birkhäuser Boston, Boston, MA, 1998, pp. 377-396. MR 1627382
  • [17] Mourad E. H. Ismail and Dennis Stanton, $ q$-integral and moment representations for $ q$-orthogonal polynomials, Canad. J. Math. 54 (2002), no. 4, 709-735. MR 1913916, https://doi.org/10.4153/CJM-2002-027-2
  • [18] M. Kieburg, A. Kuijlaars, and D. Stivigny, Singular Value Statistics of Matrix Products with Truncated Unitary Matrices, International Mathematics Research Notices, to appear.
  • [19] Arno B. J. Kuijlaars and Dries Stivigny, Singular values of products of random matrices and polynomial ensembles, Random Matrices Theory Appl. 3 (2014), no. 3, 1450011, 22. MR 3256862, https://doi.org/10.1142/S2010326314500117
  • [20] Wojciech Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math. 15 (2010), 939-955. MR 2745687
  • [21] W. Mlotkowski, M.A. Nowak, K.A. Penson and K. Życskowski, Spectral density of generalized Wishart matrices and free multiplicative convolution, Phys. Rev. E 92, 012121.
  • [22] Thorsten Neuschel, Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution, Random Matrices Theory Appl. 3 (2014), no. 1, 1450003, 18. MR 3190209, https://doi.org/10.1142/S2010326314500038
  • [23] T. Neuschel, Spectral Densities of Singular Values of Products of Gaussian and Truncated Unitary Random Matrices, preprint arXiv:1511.03491v3, 2015.
  • [24] T. Neuschel and D. Stivigny, Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices, Journal of Multivariate Analysis, to appear.
  • [25] K.A. Penson and K. Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev. E 83 (2011), 061118, 9 pp.
  • [26] R. Speicher, Free probability and random matrices, preprint arXiv: 1404.3393.
  • [27] Roland Speicher, Free probability theory, The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, pp. 452-470. MR 2932642
  • [28] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
  • [29] A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications, Commun. Inf. Theory 1 (2004), no. 1, 1-182.
  • [30] Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201-220. MR 1094052, https://doi.org/10.1007/BF01245072
  • [31] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253

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Additional Information

Wolfgang Gawronski
Affiliation: Department of Mathematics, University of Trier, 54286 Trier, Germany
Email: gawron@uni-trier.de

Thorsten Neuschel
Affiliation: Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, B-1348 Louvain-La-Neuve, Belgium
Email: thorsten.neuschel@uclouvain.be

Dries Stivigny
Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium
Email: dries.stivigny@wis.kuleuven.be

DOI: https://doi.org/10.1090/proc/13153
Keywords: Moment problem, Jacobi polynomials, Raney distributions, random matrices, distribution of eigenvalues, free probability theory, free multiplicative convolution
Received by editor(s): August 26, 2014
Received by editor(s) in revised form: February 15, 2016
Published electronically: June 10, 2016
Communicated by: Mourad Ismail
Article copyright: © Copyright 2016 American Mathematical Society

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