Jacobi polynomial moments and products of random matrices

Gawronski, Wolfgang; Neuschel, Thorsten; Stivigny, Dries

doi:10.1090/proc/13153

Jacobi polynomial moments and products of random matrices
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by Wolfgang Gawronski, Thorsten Neuschel and Dries Stivigny PDF

Proc. Amer. Math. Soc. 144 (2016), 5251-5263 Request permission

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.

References

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, DOI 10.1088/1751-8113/47/25/255202
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, DOI 10.1088/1751-8113/46/27/275205
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
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, DOI 10.1007/978-1-4684-9172-2
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, DOI 10.1103/PhysRevE.81.041132
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, DOI 10.5506/APhysPolB.42.939
Romain Couillet and Mérouane Debbah, Random matrix methods for wireless communications, Cambridge University Press, Cambridge, 2011. MR 2884783, DOI 10.1017/CBO9780511994746
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.
Peter J. Forrester, Eigenvalue statistics for product complex Wishart matrices, J. Phys. A 47 (2014), no. 34, 345202, 22. MR 3251989, DOI 10.1088/1751-8113/47/34/345202
Peter J. Forrester and Dang-Zheng Liu, Raney distributions and random matrix theory, J. Stat. Phys. 158 (2015), no. 5, 1051–1082. MR 3313617, DOI 10.1007/s10955-014-1150-4
H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469. MR 121828, DOI 10.1214/aoms/1177705909
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, DOI 10.1088/0305-4470/36/12/326
F. Götze and A. Tikhomirov, On the asymptotic spectrum of products of independent random matrices, preprint arXiv:1012.2710.
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
Mourad E. H. Ismail and Dennis Stanton, Classical orthogonal polynomials as moments, Canad. J. Math. 49 (1997), no. 3, 520–542. MR 1451259, DOI 10.4153/CJM-1997-024-9
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
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, DOI 10.4153/CJM-2002-027-2
M. Kieburg, A. Kuijlaars, and D. Stivigny, Singular Value Statistics of Matrix Products with Truncated Unitary Matrices, International Mathematics Research Notices, to appear.
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, DOI 10.1142/S2010326314500117
Wojciech Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math. 15 (2010), 939–955. MR 2745687, DOI 10.1016/j.cnsns.2009.05.004
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.
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, DOI 10.1142/S2010326314500038
T. Neuschel, Spectral Densities of Singular Values of Products of Gaussian and Truncated Unitary Random Matrices, preprint arXiv:1511.03491v3, 2015.
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.
K.A. Penson and K. Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev. E 83 (2011), 061118, 9 pp.
R. Speicher, Free probability and random matrices, preprint arXiv: 1404.3393.
Roland Speicher, Free probability theory, The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, pp. 452–470. MR 2932642
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications, Commun. Inf. Theory 1 (2004), no. 1, 1–182.
Dan Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104 (1991), no. 1, 201–220. MR 1094052, DOI 10.1007/BF01245072
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, DOI 10.1090/crmm/001