Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits.(English)Zbl 1303.15046
G. Akemann, {J. R. Ipsen} andM. Kieburg [“Products of rectangular random matrices: singular values and progressive scattering”, Phys. Rev. E 88, Article ID 052118, 13 p. (2013)] 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. The authors show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. They give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows 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 [M. Bertola et al., Commun. Math. Phys. 326, No. 1, 111–144 (2014;Zbl 1303.82018)] in the Cauchy–Laguerre two–matrix model, which indicates that these kernels represent a new universality class in random matrix theory.
Reviewer: A. Arvanitoyeorgos (Patras)
Citations:
Zbl 1303.82018Software:
DLMFCite
References:
| [1] | Akemann, G., Baik, J., Di Francesco, P. (eds.): The Oxford Handbook of Random Matrix Theory. Oxford University Press, Oxford (2011) ·Zbl 1225.15004 |
| [2] | Akemann G., Burda Z.: Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A Math. Theor. 45, 465201 (2012) ·Zbl 1261.15041 ·doi:10.1088/1751-8113/45/46/465201 |
| [3] | Akemann, G., Burda, Z., Kieburg, M., Nagao, T.: Universal microscopic correlation functions for products of truncated unitary matrices, preprint. arXiv:1310.6395 ·Zbl 1296.15016 |
| [4] | Akemann G., Ipsen J.R., Kieburg M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013) ·doi:10.1103/PhysRevE.88.052118 |
| [5] | Akemann G., Kieburg M., Wei L.: Singular value correlation functions for products of Wishart random matrices. J. Phys. A Math. Theor. 46, 275205 (2013) ·Zbl 1271.15022 ·doi:10.1088/1751-8113/46/27/275205 |
| [6] | Akemann G., Strahov E.: Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. J. Stat. Phys. 151, 987-1003 (2013) ·Zbl 1314.15026 ·doi:10.1007/s10955-013-0750-8 |
| [7] | Anderson G.W., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009) ·Zbl 1184.15023 ·doi:10.1017/CBO9780511801334 |
| [8] | Banica T., Belinschi S., Capitaine M., Collins B.: Free Bessel laws. Can. J. Math. 63, 3-37 (2011) ·Zbl 1218.46040 ·doi:10.4153/CJM-2010-060-6 |
| [9] | Beals R., Szmigielski J.: Meijer G-functions: a gentle introduction. Not. Am. Math. Soc. 60, 866-872 (2013) ·Zbl 1322.33001 ·doi:10.1090/noti1016 |
| [10] | Bertola M., Gekhtman M., Szmigielski J.: The Cauchy two-matrix model. Comm. Math. Phys. 287, 983-1014 (2009) ·Zbl 1197.82037 ·doi:10.1007/s00220-009-0739-y |
| [11] | Bertola M., Gekhtman M., Szmigielski J.: Cauchy biorthogonal polynomials. J. Approx. Theory 162, 832-867 (2010) ·Zbl 1202.33012 ·doi:10.1016/j.jat.2009.09.008 |
| [12] | 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) ·Zbl 1303.82018 ·doi:10.1007/s00220-013-1833-8 |
| [13] | Borodin A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704-732 (1999) ·Zbl 0948.82018 ·doi:10.1016/S0550-3213(98)00642-7 |
| [14] | 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) ·Zbl 0572.60001 |
| [15] | Burda Z., Janik R.A., Waclaw B.: Spectrum of the product of independent random Gaussian matrices. Phys. Rev. E 81, 041132 (2010) ·doi:10.1103/PhysRevE.81.041132 |
| [16] | 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)) ·Zbl 1371.60014 |
| [17] | 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) ·Zbl 1165.33007 ·doi:10.1090/S0002-9947-08-04535-2 |
| [18] | Crisanti A., Paladin G., Vulpiani A.: Products of Random Matrices in Statistical Physics. Springer Series in Solid-State Sciences, vol. 104. Springer, Heidelberg (1993) ·Zbl 0784.58003 |
| [19] | Daems E., Kuijlaars A.B.J.: A Christoffel-Darboux formula for multiple orthogonal polynomials. J. Approx. Theory 130, 188-200 (2004) ·Zbl 1063.42014 |
| [20] | Deift, P.: Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, vol. 3. American Mathematical Society, Providence (1999) ·Zbl 0997.47033 |
| [21] | Flajolet P., Gourdon X., Dumas P.: Mellin transforms and asymptotics: harmonic sums. Theor. Comput. Sci. 144, 3-58 (1995) ·Zbl 0869.68057 ·doi:10.1016/0304-3975(95)00002-E |
| [22] | Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010) ·Zbl 1217.82003 |
| [23] | Furstenberg H., Kesten H.: Products of random matrices. Ann. Math. Stat. 31, 457-469 (1960) ·Zbl 0137.35501 ·doi:10.1214/aoms/1177705909 |
| [24] | Götze, F., Tikhomirov, A.: On the asymptotic spectrum of products of independent random matrices, preprint. arXiv:1012.2710 ·Zbl 1517.60011 |
| [25] | Ipsen J.R.: Products of independent quaternion Ginibre matrices and their correlation functions. J. Phys. A Math. Theor. 46, 265201 (2013) ·Zbl 1271.15023 ·doi:10.1088/1751-8113/46/26/265201 |
| [26] | 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) ·Zbl 1082.42016 ·doi:10.1017/CBO9781107325982 |
| [27] | 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) ·Zbl 0719.35091 ·doi:10.1142/S0217979290000504 |
| [28] | 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) ·Zbl 1218.60005 |
| [29] | 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) ·Zbl 1230.42034 |
| [30] | Luke Y.L.: The Special Functions and their Approximations. Academic Press, New York (1969) ·Zbl 0193.01701 |
| [31] | 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]) ·Zbl 1198.00002 |
| [32] | O’Rourke S., Soshnikov A.: Products of independent non-Hermitian random matrices. Electron. J. Probab. 81, 2219-2245 (2011) ·Zbl 1244.60011 |
| [33] | Penson K.A., K.: Product of Ginibre matrices: Fuss-Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011) ·doi:10.1103/PhysRevE.83.061118 |
| [34] | Tao, T.: Topics in Random Matrix Theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012) ·Zbl 1256.15020 |
| [35] | Tracy C., Widom H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161, 289-309 (1994) ·Zbl 0808.35145 ·doi:10.1007/BF02099779 |
| [36] | 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) ·Zbl 1143.94303 |
| [37] | 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) ·Zbl 0997.42012 |
| [38] | Van Assche W., Yakubovich S.B.: Multiple orthogonal polynomials associated with Macdonald functions. Integral Transforms Spec. Funct. 9, 229-244 (2000) ·Zbl 0959.42016 ·doi:10.1080/10652460008819257 |
| [39] | 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) ·Zbl 1305.15080 |
| [40] | Zhang L., Román P.: The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions. J. Approx. Theory 163, 143-162 (2011) ·Zbl 1259.33020 ·doi:10.1016/j.jat.2010.08.003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
