Electronic Journal of Probability

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Peter J. Forrester and Dong Wang

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Abstract

Muttalib–Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the form $\prod _{1 \le j < k \le N}(\lambda _k - \lambda _j) (\lambda _k^\theta - \lambda _j^\theta )$. We study the Laguerre and Jacobi versions of this model — so named by the form of the one-body interaction terms — and show that for $\theta \in \mathbb Z^+$ they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general $\theta > 0$, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlation functions contained in a paper by Adler, van Moerbeke and Wang to analyse the global density (which we also analyse by studying characteristic polynomials), and the hard edge scaled correlation functions. For the global density functional equations for the corresponding resolvents are obtained; solving this gives the moments in terms of Fuss–Catalan numbers (Laguerre case — a known result) and particular binomial coefficients (Jacobi case). For $\theta \in \mathbb Z^+$ the Laguerre and Jacobi cases are closely related to the squared singular values for products of $\theta $ standard Gaussian random matrices, and truncations of unitary matrices, respectively. At the hard edge the double contour integral formulas provide a double contour integral form of the scaled correlation kernel obtained by Borodin in terms of Wright’s Bessel function.

Article information

Source
Electron. J. Probab. Volume 22 (2017), paper no. 54, 43 pp.

Dates
Received: 8 October 2016
Accepted: 26 April 2017
First available in Project Euclid: 23 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1498183245

Digital Object Identifier
doi:10.1214/17-EJP62

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 15B52: Random matrices 82D35: Metals 30E10: Approximation in the complex domain

Keywords
Muttalib–Borodin ensembles determinantal processes biorthogonal polynomials

Rights
Creative Commons Attribution 4.0 International License.

Citation

Forrester, Peter J.; Wang, Dong. Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions. Electron. J. Probab. 22 (2017), paper no. 54, 43 pp. doi:10.1214/17-EJP62. https://projecteuclid.org/euclid.ejp/1498183245.


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References

  • [1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.
  • [2] Mark Adler, Pierre van Moerbeke, and Dong Wang, Random matrix minor processes related to percolation theory, Random Matrices Theory Appl. 2 (2013), no. 4, 1350008, 72.
  • [3] Gernot Akemann, Jesper R. Ipsen, and Mario Kieburg, Products of rectangular random matrices: Singular values and progressive scattering, Phys. Rev. E 88 (2013), no. 5, 052118, 13.
  • [4] N. Alexeev, F. Götze, and A. Tikhomirov, Asymptotic distribution of singular values of powers of random matrices, Lith. Math. J. 50 (2010), no. 2, 121–132.
  • [5] Kazuhiko Aomoto and Kazumoto Iguchi, On quasi-hypergeometric functions, Methods Appl. Anal. 6 (1999), no. 1, 55–66, Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I.
  • [6] Kazuhiko Aomoto and Kazumoto Iguchi, Wu’s equations and quasi-hypergeometric functions, Comm. Math. Phys. 223 (2001), no. 3, 475–507.
  • [7] T. Banica, S. T. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), no. 1, 3–37.
  • [8] C. W. J. Beenakker and B. Rejaei, Nonlogarithmic repulsion of transmission eigenvalues in a disordered wire, Phys. Rev. Lett. 71 (1993), 3689–3692.
  • [9] Hari Bercovici and Vittorino Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), no. 3, 1023–1060, With an appendix by Philippe Biane.
  • [10] M. Bertola, M. Gekhtman, and J. Szmigielski, The Cauchy two-matrix model, Comm. Math. Phys. 287 (2009), no. 3, 983–1014.
  • [11] M. Bertola, M. Gekhtman, and J. Szmigielski, Cauchy-Laguerre two-matrix model and the Meijer-G random point field, Comm. Math. Phys. 326 (2014), no. 1, 111–144.
  • [12] Marco Bertola and Thomas Bothner, Universality conjecture and results for a model of several coupled positive-definite matrices, Comm. Math. Phys. 337 (2015), no. 3, 1077–1141.
  • [13] Alexei Borodin, Biorthogonal ensembles, Nuclear Phys. B 536 (1999), no. 3, 704–732.
  • [14] Gaëtan Borot, Alice Guionnet, and Karol K. Kozlowski, Large-$N$ asymptotic expansion for mean field models with Coulomb gas interaction, Int. Math. Res. Not. IMRN (2015), no. 20, 10451–10524.
  • [15] L. Carlitz, Problem 72-17, “Biorthogonal conditions for a class of polynomials”, SIAM Rev. 15 (1973), 670–672.
  • [16] W. A. Chat, Biorthogonal conditions for a class of polynomials, SIAM Rev. 14 (1972), no. 3, 494–495.
  • [17] Dimitris Cheliotis, Triangular random matrices and biorthogonal ensembles, 2014, arXiv:1404.4730.
  • [18] Tom Claeys and Stefano Romano, Biorthogonal ensembles with two-particle interactions, Nonlinearity 27 (2014), no. 10, 2419–2444.
  • [19] Tom Claeys and Dong Wang, Random matrices with equispaced external source, Comm. Math. Phys. 328 (2014), no. 3, 1023–1077.
  • [20] P. A. Deift, Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, vol. 3, New York University Courant Institute of Mathematical Sciences, New York, 1999.
  • [21] Ioana Dumitriu and Alan Edelman, Matrix models for beta ensembles, J. Math. Phys. 43 (2002), no. 11, 5830–5847.
  • [22] Ken Dykema and Uffe Haagerup, DT-operators and decomposability of Voiculescu’s circular operator, Amer. J. Math. 126 (2004), no. 1, 121–189.
  • [23] P. J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010.
  • [24] Peter J. Forrester and Dang-Zheng Liu, Raney distributions and random matrix theory, J. Stat. Phys. 158 (2015), no. 5, 1051–1082.
  • [25] Peter J. Forrester, Dang-Zheng Liu, and Paul Zinn-Justin, Equilibrium problems for Raney densities, Nonlinearity 28 (2015), no. 7, 2265–2277.
  • [26] Peter J. Forrester and Eric M. Rains, Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Related Fields 131 (2005), no. 1, 1–61.
  • [27] Wolfgang Gawronski, Thorsten Neuschel, and Dries Stivigny, Jacobi polynomial moments and products of random matrices, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5251–5263.
  • [28] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, second ed., Addison-Wesley Publishing Company, Reading, MA, 1994, A foundation for computer science.
  • [29] Adrien Hardy, Average characteristic polynomials of determinantal point processes, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), no. 1, 283–303.
  • [30] Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, Continuous univariate distributions. Vol. 2, second ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1995, A Wiley-Interscience Publication.
  • [31] Mario Kieburg, Arno B. J. Kuijlaars, and Dries Stivigny, Singular value statistics of matrix products with truncated unitary matrices, Int. Math. Res. Not. IMRN (2016), no. 11, 3392–3424.
  • [32] Heiner Kohler, Exact diagonalization of 1D interacting spinless fermions, J. Math. Phys. 52 (2011), no. 3, 032107, 24.
  • [33] Joseph D. E. Konhauser, Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 21 (1967), 303–314.
  • [34] Arno B. J. Kuijlaars and Dries Stivigny, Singular values of products of random matrices and polynomial ensembles, Random Matrices Theory Appl. 3 (2014), no. 3, 1450011, 22.
  • [35] Arno B. J. Kuijlaars and Lun Zhang, Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, Comm. Math. Phys. 332 (2014), no. 2, 759–781.
  • [36] Vadim B. Kuznetsov, Vladimir V. Mangazeev, and Evgeny K. Sklyanin, $Q$-operator and factorised separation chain for Jack polynomials, Indag. Math. (N.S.) 14 (2003), no. 3-4, 451–482.
  • [37] Dang-Zheng Liu, Dong Wang, and Lun Zhang, Bulk and soft-edge universality for singular values of products of Ginibre random matrices, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 4, 1734–1762.
  • [38] T. Lueck, H.-J. Sommers, and M. R. Zirnbauer, Energy correlations for a random matrix model of disordered bosons, J. Math. Phys. 47 (2006), no. 10, 103304, 24.
  • [39] Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969.
  • [40] H. C. Madhekar and N. K. Thakare, Biorthogonal polynomials suggested by the Jacobi polynomials, Pacific J. Math. 100 (1982), no. 2, 417–424.
  • [41] Wojciech Młotkowski, Fuss-Catalan numbers in noncommutative probability, Doc. Math. 15 (2010), 939–955.
  • [42] Wojciech Młotkowski and Karol A. Penson, Probability distributions with binomial moments, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014), no. 2, 1450014, 32.
  • [43] K. A. Muttalib, Random matrix models with additional interactions, J. Phys. A 28 (1995), no. 5, L159–L164.
  • [44] Thorsten Neuschel, Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution, Random Matrices Theory Appl. 3 (2014), no. 1, 1450003, 18.
  • [45] Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006.
  • [46] A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), no. 1, 69–78.
  • [47] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010, With 1 CD-ROM (Windows, Macintosh and UNIX).
  • [48] Jack W. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab. 13 (1985), no. 4, 1364–1368.
  • [49] H. M. Srivastava, Some biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 98 (1982), no. 1, 235–250.
  • [50] Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
  • [51] Hale F. Trotter, Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő, Adv. in Math. 54 (1984), no. 1, 67–82.
  • [52] Lun Zhang, Local Universality in Biorthogonal Laguerre Ensembles, J. Stat. Phys. 161 (2015), no. 3, 688–711.
  • [53] Karol Życzkowski and Hans-Jürgen Sommers, Truncations of random unitary matrices, J. Phys. A 33 (2000), no. 10, 2045–2057.