User menu

Accès à distance ? S'identifier sur le proxy UCLouvain

Correlation kernels for sums and products of random matrices

Primary tabs

download
  • Open access
  • PDF
  • 358.88 K
Claeys, Tom [UCL] Kuijlaars, Arno [KULeuven] Wang, Dong [University of Singapore]

Let X be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of GX and TX, where G is a complex Ginibre matrix and T is a truncated unitary matrix. We also consider the product of X and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum H+M where H is a GUE matrix and M is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of H+M follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès libre
Publication date 2015
Language Anglais
Journal information "Random Matrices: Theory and Applications" - Vol. 4, no. 4, p. 1550017 (2015)
Peer reviewed yes
Publisher World Scientific Publishing Co. Pte. Ltd. (Singapore)
issn 2010-3263
e-issn 2010-3271
Publication status Publié
Affiliation UCL - SST/IRMP - Institut de recherche en mathématique et physique
Links
  1. ADLER MARK, VAN MOERBEKE PIERRE, WANG DONG, RANDOM MATRIX MINOR PROCESSES RELATED TO PERCOLATION THEORY, 10.1142/s2010326313500081
  2. Akemann G., Ipsen J.R., Recent Exact and Asymptotic Results for Products of Independent Random Matrices, 10.5506/aphyspolb.46.1747
  3. Akemann Gernot, Ipsen Jesper R., Kieburg Mario, Products of rectangular random matrices: Singular values and progressive scattering, 10.1103/physreve.88.052118
  4. Akemann Gernot, Kieburg Mario, Wei Lu, Singular value correlation functions for products of Wishart random matrices, 10.1088/1751-8113/46/27/275205
  5. Baik Jinho, Ben Arous Gérard, Péché Sandrine, Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, 10.1214/009117905000000233
  6. Beals Richard, Szmigielski Jacek, Meijer G-Functions: A Gentle Introduction, 10.1090/noti1016
  7. M. BLEHER Pavel, B.J. Kuijlaars Arno, Integral representations for multiple Hermite and multiple Laguerre polynomials, 10.5802/aif.2148
  8. Borodin Alexei, Péché Sandrine, Airy Kernel with Two Sets of Parameters in Directed Percolation and Random Matrix Theory, 10.1007/s10955-008-9553-8
  9. Brézin E., Hikami S., Correlations of nearby levels induced by a random potential, 10.1016/0550-3213(96)00394-x
  10. Deift P. A., Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, 3 (1999)
  11. Deift Percy, Gioev Dimitri, Random Matrix Theory: Invariant Ensembles and Universality, ISBN:9780821847374, 10.1090/cln/018
  12. Dieker A. B., ALEA Lat. Amer. J. Probab. Math. Stat., 6, 369 (2009)
  13. Duits Maurice, Painlevé Kernels in Hermitian Matrix Models, 10.1007/s00365-013-9201-7
  14. Duits Maurice, Geudens Dries, A critical phenomenon in the two-matrix model in the quartic/quadratic case, 10.1215/00127094-2208757
  15. Fokas A. S., Its A. R., Kitaev A. V., The isomonodromy approach to matric models in 2D quantum gravity, 10.1007/bf02096594
  16. Forrester Peter J., Log-Gases and Random Matrices (LMS-34) : , ISBN:9781400835416, 10.1515/9781400835416
  17. Forrester Peter J, Eigenvalue statistics for product complex Wishart matrices, 10.1088/1751-8113/47/34/345202
  18. Forrester Peter J., Liu Dang-Zheng, Raney Distributions and Random Matrix Theory, 10.1007/s10955-014-1150-4
  19. Forrester Peter J, Nagao Taro, Determinantal correlations for classical projection processes, 10.1088/1742-5468/2011/08/p08011
  20. Hirschman I. I., The Convolution Transform (1955)
  21. Johansson Kurt Johansson, Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices, 10.1007/s002200000328
  22. Kuijlaars Arno B. J., Stivigny Dries, Singular values of products of random matrices and polynomial ensembles, 10.1142/s2010326314500117
  23. Kuijlaars Arno B. J., Zhang Lun, Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits, 10.1007/s00220-014-2064-3
  24. Luke Y. L., The Special Functions and Their Approximations, 53 (1969)
  25. Nica Alexandru, Speicher Roland, Lectures on the Combinatorics of Free Probability, ISBN:9780511735127, 10.1017/cbo9780511735127
  26. Zyczkowski Karol, Sommers Hans-Jürgen, Truncations of random unitary matrices, 10.1088/0305-4470/33/10/307
Bibliographic reference Claeys, Tom ; Kuijlaars, Arno ; Wang, Dong. Correlation kernels for sums and products of random matrices. In: Random Matrices: Theory and Applications, Vol. 4, no. 4, p. 1550017 (2015)
Permanent URL http://hdl.handle.net/2078.1/167932