Abstract
Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M = 2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy–Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akemann, G., Baik, J., Di Francesco, P. (eds.): The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011)
Akemann G., Burda Z.: Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A Math. Theor. 45, 465201 (2012)
Akemann, G., Burda, Z., Kieburg, M., Nagao, T.: Universal microscopic correlation functions for products of truncated unitary matrices, preprint. arXiv:1310.6395
Akemann G., Ipsen J.R., Kieburg M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013)
Akemann G., Kieburg M., Wei L.: Singular value correlation functions for products of Wishart random matrices. J. Phys. A Math. Theor. 46, 275205 (2013)
Akemann G., Strahov E.: Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. J. Stat. Phys. 151, 987–1003 (2013)
Anderson G.W., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)
Banica T., Belinschi S., Capitaine M., Collins B.: Free Bessel laws. Can. J. Math. 63, 3–37 (2011)
Beals R., Szmigielski J.: Meijer G-functions: a gentle introduction. Not. Am. Math. Soc. 60, 866–872 (2013)
Bertola M., Gekhtman M., Szmigielski J.: The Cauchy two-matrix model. Comm. Math. Phys. 287, 983–1014 (2009)
Bertola M., Gekhtman M., Szmigielski J.: Cauchy biorthogonal polynomials. J. Approx. Theory 162, 832–867 (2010)
Bertola M., Gekhtman M., Szmigielski J.: Cauchy–Laguerre two-matrix model and the Meijer-G random point field. Commun. Math. Phys. 326(1), 111–144 (2014)
Borodin A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1999)
Bougerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. In: Huber, P., Rosenblatt, M. (eds) Progress in Probability and Statistics, vol. 8. Birkhäuser, Boston (1985)
Burda Z., Janik R.A., Waclaw B.: Spectrum of the product of independent random Gaussian matrices. Phys. Rev. E 81, 041132 (2010)
Burda, Z., Jarosz, A., Livan, G., Nowak, M.A., Swiech, A.: Eigenvalues and singular values of products of rectangular Gaussian random matrices. Phys. Rev. E 82, 061114 (2010) (the extended version Acta Phys. Polon. B 42, 939–985 (2011))
Coussement E., Coussement J., Van Assche W.: Asymptotic zero distribution for a class of multiple orthogonal polynomials. Trans. Am. Math. Soc. 360, 5571–5588 (2008)
Crisanti A., Paladin G., Vulpiani A.: Products of Random Matrices in Statistical Physics. Springer Series in Solid-State Sciences, vol. 104. Springer, Heidelberg (1993)
Daems E., Kuijlaars A.B.J.: A Christoffel–Darboux formula for multiple orthogonal polynomials. J. Approx. Theory 130, 188–200 (2004)
Deift, P.: Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert approach. Courant Lecture Notes in Mathematics, vol. 3. American Mathematical Society, Providence (1999)
Flajolet P., Gourdon X., Dumas P.: Mellin transforms and asymptotics: harmonic sums. Theor. Comput. Sci. 144, 3–58 (1995)
Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Furstenberg H., Kesten H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)
Götze, F., Tikhomirov, A.: On the asymptotic spectrum of products of independent random matrices, preprint. arXiv:1012.2710
Ipsen J.R.: Products of independent quaternion Ginibre matrices and their correlation functions. J. Phys. A Math. Theor. 46, 265201 (2013)
Ismail M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 University Press. Cambridge University Press, London (2005)
Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990)
Kuijlaars, A.B.J.: Multiple orthogonal polynomial ensembles. In: Arvesú, J., Marcellán, F., Martínez-Finkelshtein, A. (eds.) Recent Trends in Orthogonal Polynomials and Approximation Theory. Contemporary Mathematics, vol. 507, pp. 155–176 (2010)
Kuijlaars, A.B.J.: Multiple orthogonal polynomials in random matrix theory. In: Bhatia, R. (ed.) Proceedings of the International Congress of Mathematicians, vol. III, Hyderabad, India, pp. 1417–1432 (2010)
Luke Y.L.: The Special Functions and their Approximations. Academic Press, New York (1969)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge. (2010) (Print companion to [DLMF])
O’Rourke S., Soshnikov A.: Products of independent non-Hermitian random matrices. Electron. J. Probab. 81, 2219–2245 (2011)
Penson K.A., K.: Product of Ginibre matrices: Fuss-Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011)
Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)
Tracy C., Widom H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161, 289–309 (1994)
Tulino, A.M., Verdú, S.: Random Matrix Theory and Wireless Communications. Foundations and Trends in Communications and Information Theory, vol. 1, pp. 1–182. Now Publisher, Hanover (2004)
Van Assche, W., Geronimo, J.S., Kuijlaars, A.B.J.: Riemann–Hilbert problems for multiple orthogonal polynomials. In: Bustoz J. et al. (eds.) Special Functions 2000: Current Perspectives and Future Directions. Kluwer, Dordrecht, pp. 23–59 (2001)
Van Assche W., Yakubovich S.B.: Multiple orthogonal polynomials associated with Macdonald functions. Integral Transforms Spec. Funct. 9, 229–244 (2000)
Zhang, L.: A note on the limiting mean distribution of singular values for products of two Wishart random matrices. J. Math. Phys. 54, 083303, 8 pp. (2013)
Zhang L., Román P.: The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions. J. Approx. Theory 163, 143–162 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Deift
Rights and permissions
About this article
Cite this article
Kuijlaars, A.B.J., Zhang, L. Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits. Commun. Math. Phys. 332, 759–781 (2014). https://doi.org/10.1007/s00220-014-2064-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2064-3