Abstract
Consider the product GX of two rectangular complex random matrices coupled by a constant matrix \(\Omega \), where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin’s sense, and further that for X being Gaussian, the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of \(\Omega ^{} \Omega ^{*}\) are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel, and the finite coupled product kernel associated with GX. In the special case when X is also a Gaussian matrix and \(\Omega \) is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.
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Akemann, G., Damgaard, P.H., Osborn, J.C., Splittorff, K.: A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential. Nucl. Phys. B 766, 34–76 (2007)
Akemann, G., Ipsen, J.R.: Recent exact and asymptotic results for products of independent random matrices. Acta Physica Polonica B 46(9), 1747–1784 (2015)
Akemann, G., Ipsen, J., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013). [13pp]
Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart matrices. J. Phys. A 46, 275205 (2013). [22pp]
Akemann, G., Strahov, E.: Dropping the independence: singular values for products of two coupled random matrices. Commun. Math. Phys. 345, 101–140 (2016)
Akemann, G., Strahov, E.: Hard edge limit of the product of two strongly coupled random matrices. Nonlinearity 29, 3743–3776 (2016)
Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)
Anderson, T.W.: An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York (2003)
Atkin, M.R., Claeys, T., Mezzadri, F.: Random matrix ensembles with singularities and a hierarchy of Painlevé III equations. Int. Math. Res. Not. 2016(8), 2320–2375 (2016)
Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Prob. 33(5), 1643–1697 (2005)
Bao, Z., Hu, J., Pan, G., Zhou, W.: Canonical correlation coefficients of high-dimensional normal vectors: finite rank case, arXiv: 1407.7194
Bertola, M., Gekhtman, M., Szmigielski, J.: Cauchy–Laguerre two-matrix model and the Meijer-G random point field. Commun. Math. Phys. 326, 111–144 (2014)
Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1998)
Borodin, A., Péché, S.: Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys. 132, 275–290 (2008)
Chen, Y., Its, A.: Painlevé III and a singular linear statistics in Hermitian random matrix ensembles I. J. Approx. Theory 162(2), 270–297 (2010)
Claeys, T., Doeraene, A.: Gaussian perturbations of hard edge random matrix ensembles. Nonlinearity 29(11), 3385–3416 (2016)
Claeys, T., Kuijlaars, A.B.J., Wang, D.: Correlation kernels for sums and products of random matrices. Random Matrices Theory Appl. 4(4), 1550017 (2015). (31pp)
Constantine, A.G.: Some non-central distribution problems in multivariate analysis. Ann. Math. Stat. 34(4), 1270–1285 (1963)
Deift, P., Gioev, G.: Random Matrix Theory: Invariant Ensembles and Universality, Courant Lecture Notes in Mathematics vol. 18, Amr. Math. Soc., Providence R.I., (2009)
Delvaux, S., Geudens, D., Zhang, L.: Universality and critical behaviour in the chiral two-matrix model. Nonlinearity 26, 2231–2298 (2013)
Desrosiers, P., Forrester, P.J.: Asymptotic correlations for Gaussian and Wishart matrices with external source, Int. Math. Res. Notices (2006), ID 27395, 1–43
Desrosiers, P., Forrester, P.J.: A note on biorthogonal ensembles. J. Approx. Theory 152, 167–187 (2008)
Eynard, B., Mehta, M.L.: Matrices coupled in a chain: I. Eigenvalue correlations. J. Phys. A Math. Gen. 31, 4449–56 (1998)
Fischmann, J., Bruzda, W., Khoruzhenko, B.A., Sommers, H.-J., Życzkowski, K.: Induced Ginibre ensemble of random matrices and quantum operations. J. Phys. A Math. Theor. 45, 075203 (2012). (31pp)
Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)
Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)
Forrester, P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A 47, 345202 (2014)
Forrester, P.J., Liu, D.-Z.: Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition. Commun. Math. Phys. 344(1), 333–368 (2016)
Gross, K.I., Richards, D.S.P.: Total positivity, spherical series, and hypergeometric functions of matrix argument. J. Approx. Theory 59, 224–246 (1989)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, Cambridge (2007)
Guhr, T., Wettig, T.: An Itzykson-Zuber like integral and diffusion for complex ordinary and supermatrices. J. Math. Phys. 37, 6395–6413 (1996)
Harish-Chandra, : Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)
Itoi, C.: Universal wide correlators in non-Gaussian orthogonal, unitary and symplectic random matrix ensembles. Nucl. Phys. B 493, 651–659 (1997)
Its, A.R., Isergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990)
Itzykson, C., Zuber, J.-B.: The planar approximation II. J. Math. Phys. 21, 411–421 (1980)
Jackson, A.D., Şener, M.K., Verbaarschot, J.J.M.: Finite volume partition functions and Itzyson-Zuber integrals. Phys. Lett. B 387, 355–360 (1997)
James, A.T.: Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 35(2), 475–501 (1964)
Johnstone, I.M.: Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Stat. 36(6), 2638–2716 (2008)
Johnstone, I.M., Onatski, A.: Testing in high-dimensional spiked models, arXiv:1509.07269
Kieburg, K., Kuijlaars, A.B.J., Stivigny, D.: Singular value statistics of matrix products with truncated unitary matrices. Int. Math. Res. Not. 2016(11), 3392–3424 (2016)
Kuijlaars, A.B.J.: Transformations of Polynomial Ensembles, Contemporary Mathematics, vol. 661. Amer. Math. Soc, Providence, RI (2016)
Kuijlaars, A.B.J., Stivigny, D.: Singular values of products of random matrices and polynomial ensembles. Random Matrices Theory Appl. 3(3), 1450011 (2014). (22pp)
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)
Liu, D.-Z.: Limits for circular Jacobi beta-ensembles. J. Approx. Theory 215, 40–67 (2017)
Liu, D.-Z., Wang, D., Zhang, L.: Bulk and soft-edge universality for singular values of products of Ginibre random matrices. Ann. Inst. Henri Poincarés Prob. Stat. 52(4), 1734–1762 (2016)
Luke, Y.L.: The Special Functions and Their Approximations, vol. 1. Academic Press, New York (1969)
Mathai, A.M.: Jacobians of Matrix Transformation and Functions of Matrix Argument. World Scientific Pub Co., Inc., Singapore (1997)
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
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])
Osborn, J.C.: Universal results from an alternate random matrix model for QCD with a baryon chemical potential. Phys. Rev. Lett. 93, 222001 (2004). 4pp
Strahov, E.: Differential equations for singular values of products of Ginibre random matrices. J. Phys. A Math. Theor. 47, 325203 (2014). (27pp)
Tracy, C., Widom, H.: Level-spacing distributions and the Bessel kernel. Commun. Math. Phys. 161, 289–309 (1994)
Vinayak, Benet, L.: Spectral domain of large nonsymmetric correlated Wishart matrices. Phys. Rev. E 90, 042109 (2014)
Wachter, K.W.: The limiting empirical measure of multiple discriminant ratios. Ann. Stat. 8(5), 937–957 (1980)
Witte, N.S., Forrester, P.J.: Singular values of products of Ginibre random matrices. Stud. Appl. Math. 138(2), 133–244 (2017)
Wong, R.: Asymptotic Approximations of Integrals, vol. 34, SIAM, 2001
Xu, S.-X., Dai, D., Zhao, Y.-Q.: Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble. Commun. Math. Phys. 332(3), 1257–1296 (2014)
Acknowledgements
We are grateful to Gernot Akemann, Peter J. Forrester, Jiang Hu, Mario Kieburg, Dong Wang, and Lun Zhang for helpful discussions. We also thank the anonymous referees for their careful reading and constructive suggestions. The work was partially supported by the National Natural Science Foundation of China #11771417, the Youth Innovation Promotion Association CAS #2017491, Anhui Provincial Natural Science Foundation #1708085QA03 and the Fundamental Research Funds for the Central Universities (Grants WK0010450002 and WK3470000008).
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Communicated by Arno Kuijlaars.
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Liu, DZ. Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition. Constr Approx 47, 487–528 (2018). https://doi.org/10.1007/s00365-017-9389-z
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DOI: https://doi.org/10.1007/s00365-017-9389-z