Communications in Mathematical Physics

, Volume 344, Issue 1, pp 333–368 | Cite as

Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

  • Peter J. Forrester
  • Dang-Zheng LiuEmail author


The singular values squared of the random matrix product \({Y = {G_{r} G_{r-1}} \ldots G_{1} (G_{0} + A)}\), where each \({G_{j}}\) is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of \({0 < b < 1}\) is independent of b, and is in fact the same as that known for the case b =  0 due to Kuijlaars and Zhang. The critical regime of b =  1 allows for a double scaling limit by choosing \({{b = (1 - \tau/\sqrt{N})^{-1}}}\), and for this the critical kernel and outlier phenomenon are established. In the simplest case r =  0, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of \({b > 1}\) with two distinct scaling rates. Similar results also hold true for the random matrix product \({T_{r} T_{r-1} \ldots T_{1} (G_{0} + A)}\), with each \({T_{j}}\) being a truncated unitary matrix.


Random Matrice Random Matrix Random Matrix Theory Eigenvalue Density Global Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler M., Delépine J., van Moerbeke P.: Dyson’s nonintersecting Brownian motions with a new outliers. Commun. Pure Appl. Math. 62, 334–395 (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Akemann G., Ipsen J.R.: Recent exact and asymptotic results for products of independent random matrices. Acta Phys. Pol. B. 46(9), 1747–1784 (2015)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Akemann, G., Ipsen J., Kieburg, M.: Products of rectangular random matrices: singular values and progressive scattering. Phys. Rev. E 88, 052118 (2013) (p 13)Google Scholar
  4. 4.
    Akemann, G., Kieburg, M., Wei, L.: Singular value correlation functions for products of Wishart matrices. J. Phys. A 46, 275205 (2013) (p 22)Google Scholar
  5. 5.
    Alexeev, N., Götze, F., Tikhomirov, A.: On the asymptotic distribution of singular values of products of large rectangular random matrices. arXiv:1012.2586v2
  6. 6.
    Anderson G.W., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Andrews G.E., Askey R., Roy R.: Special Functions. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  8. 8.
    Beals R., Szmigielski J.: Meijer G-functions: a gentle introduction. Not. Am. Math. Soc. 60, 866–872 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)CrossRefzbMATHGoogle Scholar
  10. 10.
    Beenakker C.W.J.: Universality of Brézin and Zee’s spectral correlator. Nucl. Phys. B 422, 515–520 (1994)ADSCrossRefGoogle Scholar
  11. 11.
    Beenakker C.W.J.: Random-matrix theory of quantum transport. Rev. Mod. Phys. 69(3), 731–808 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    Bertola M., Bothner T.: Universality conjecture and results for a model of several coupled positive-definite matrices. Commun. Math. Phys. 337(3), 1077–1141 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Blaizot J.-P., Nowak M.A., Warchoł P.: Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions. Phys. Rev. E 89, 042130 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    Borodin A.: Biorthogonal ensembles. Nuclear Phys. B 536, 704–732 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Borodin A., Kuan J.: Random surface growth with a wall and Plancherel measures for \({O(\infty)}\). Commun. Pure Appl. Math. 63, 831–894 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bougerol P., Lacroix J.: Products of random matrices with applications to Schrödinger operators. Progress in Probability and Statistics, vol. 8. Birkhäuser, Boston (1985)zbMATHGoogle Scholar
  18. 18.
    Cheliotis, D.: Triangular random matrices and biothogonal ensembles. arXiv:1404.4730
  19. 19.
    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)Google Scholar
  20. 20.
    Crisanti, A., Paladin, G., Vulpiani, A.: Products of random matrices in statistical physics. In: Springer Series in Solid-State Sciences, vol. 104. Springer, Berlin (1993) (with a foreword by Giorgio Parisi)Google Scholar
  21. 21.
    Deift P.: Orthogonal polynomials and random matrices: a Riemann–Hilbert approach. Courant Lecture Notes in Mathematics, vol. 3. American Mathematical Society, Providence (1999)Google Scholar
  22. 22.
    Delvaux, S., Vető, B.: The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths. Random Matrices Theory Appl. 04(2), 1550008 (57 pages, preprint). arXiv:1412.0831 (2015)
  23. 23.
    Desrosiers, P., Forrester, P.J.: Asymptotic correlations for Gaussian and Wishart matrices with external source. Int. Math. Res. Not. 27395, 1–43 (2006)Google Scholar
  24. 24.
    Desrosiers P., Forrester P.J.: A note on biorthogonal ensembles. J. Approx. Theory 152, 167–187 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dorokhov, O.N.: Transmission coefficient and the localization length of an electron in N bound disordered chains. Pis’ma Zh. Eksp. Teor. Fiz. 36, 259–262 (1982) (JETP Lett 36, 318–321)Google Scholar
  26. 26.
    Falkovich G., Gawdzki K., Vergassola M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Forrester P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  28. 28.
    Forrester P.J.: The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensemble with a source. J. Phys. A 46, 345204 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Forrester P.J.: Eigenvalue statistics for product complex Wishart matrices. J. Phys. A 47, 345202 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Forrester, P.J., Kieburg, M.: Relating the Bures measure to the Cauchy two-matrix model. Commun. Math. Phys. 1–37 (2015). doi: 10.1007/s00220-015-2435-4. arXiv:1410.6883v3
  31. 31.
    Forrester P.J., Liu D.-Z.: Raney distributions and random matrix theory. J. Stat. Phys. 158, 1051–1082 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Forrester, P.J., Wang, D.: Muttalib–Borodin ensembles in random matrix theory—realisations and correlation functions. arXiv:1502.07147v2
  33. 33.
    Furstenberg H., Kesten H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ginibre J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Götze, F., Naumov, A., Tikhomirov, A.: Distribution of linear statistics of singular values of the product of random matrices. arXiv:1412.3314
  36. 36.
    Itoi C.: Universal wide correlators in non-Gaussian orthogonal, unitary and symplectic random matrix ensembles. Nucl. Phys. B 493, 651–659 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    König W., O’Connell N.: Eigenvalues of the Laguerre process as non-colliding squared Bessel process. Electron. Commun. Probab. 6, 107–114 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Katori M., Tanemura H.: Noncolliding squared Bessel process. J. Stat. Phys. 142, 592–615 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kieburg, M.: Supersymmetry for products of random matrices. arXiv:1502.00550
  40. 40.
    Kieburg, M., Kuijlaars, A.B.J., Stivigny, D.: Singular value statistics of matrix products with truncated unitary matrices. Int. Math. Res. Not. (to appear). doi: arXiv:1501.03910
  41. 41.
    Kuijlaars, A.B.J.: Transformations of polynomial ensembles. Contemp. Math. (to appear). arXiv:1501.05506
  42. 42.
    Kuijlaars A.B.J.: Multiple orthogonal polynomial ensembles. Recent trends in orthogonal polynomials and approximation theory. Contemp. Math. 507, 155–176 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kuijlaars A.B.J., Martínez-Finkelshtein A., Wielonsky F.: Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights. Commun. Math. Phys. 286, 217–275 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Kuijlaars A.B.J., Martínez-Finkelshtein A., Wielonsky F.: Non-intersecting squared Bessel paths: critical time and double scaling limit. Commun. Math. Phys. 308, 227–279 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kuijlaars, A.B.J., Stivigny, D.: Singular values of products of random matrices and polynomial ensembles. Random Matrices Theory Appl. 3(3), 1450011 (2014) (22 pages)Google Scholar
  46. 46.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Liu, D.-Z., Wei, L., Zhang, L.: In preparationGoogle Scholar
  48. 48.
    Liu, D.-Z., Wang, D., Zhang, L.: Bulk and soft-edge universality for singular values of products of Ginibre random matrices. Ann. l’IHP Probab. Stat. (to appear). arXiv:1412.6777v2
  49. 49.
    Luke Y.L.: The Special Functions and Their Approximations, vol. 1. Academic Press, New York (1969)zbMATHGoogle Scholar
  50. 50.
    May R.M.: Will a large complex system be stable?. Nature 238, 413–414 (1972)ADSCrossRefGoogle Scholar
  51. 51.
    Mello P.A., Pereyra P., Kumar N.: Macroscopic approach to multichannel disordered conductors. Ann. Phys. (N.Y.) 181, 290–317 (1988)ADSCrossRefGoogle Scholar
  52. 52.
    Muttalib K.A.: Random matrix models with additional interactions. J. Phys. A 28, L159–164 (1995)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Neuschel, T.: Plancherel–Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss–Catalan distribution. Random Matrices Theory Appl. 03(1), 1450003 (2014) (p 18)Google Scholar
  54. 54.
    Pastur L., Shcherbina M.: Eigenvalue distribution of large random matrices. American Mathematical Society, Providence (2011)CrossRefzbMATHGoogle Scholar
  55. 55.
    Péché S.: The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Relat. Fields 134, 127–173 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Penson, K.A., Życzkowski, K.: Product of Ginibre matrices: Fuss–Catalan and Raney distributions. Phys. Rev. E 83, 061118 (2011) (p 9)Google Scholar
  57. 57.
    Strahov, E.: Differential equations for singular values of products of Ginibre random matrices. J. Phys. A Math. Theor. 47, 325203 (2014) (p 27)Google Scholar
  58. 58.
    Tao T.: Topics in random matrix theory. Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  59. 59.
    Tulino, A.M., Verdú, S.: Random matrix theory and wireless communications. In: Foundations and Trends in Communcations and Information Theory, vol. 1, pp. 1–182. Now Publisher, Hanover (2004)Google Scholar
  60. 60.
    Wong R.: Asymptotic Approximations of Integrals, vol. 34. SIAM, Philadelphia (2001)CrossRefGoogle Scholar
  61. 61.
    Zhang, L.: Local universality in biorthogonal Laguerre ensembles. J. Stat. Phys. 161(3), 688–711 (2015)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  2. 2.ARC Centre of Excellence for Mathematical and Statistical FrontiersThe University of MelbourneMelbourneAustralia
  3. 3.Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

Personalised recommendations