Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 065, 15 pages      arXiv:1604.03070      https://doi.org/10.3842/SIGMA.2016.065
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

Arno B.J. Kuijlaars
Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received April 12, 2016, in final form July 03, 2016; Published online July 05, 2016

Abstract
The Muttalib-Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $\theta$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $\theta$ there is an equivalent vector equilibrium problem.

Key words: biorthogonal ensembles; vector equilibrium problem; random matrix theory; logarithmic potential theory.

pdf (373 kb)   tex (19 kb)

References

  1. Aptekarev A.I., Kuijlaars A.B.J., Hermite-Padé approximations and ensembles of multiple orthogonal polynomials, Russ. Math. Surv. 66 (2011), 1133-1199.
  2. Beckermann B., Kalyagin V., Matos A.C., Wielonsky F., Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses, Constr. Approx. 37 (2013), 101-134, arXiv:1105.3088.
  3. Bloom T., Levenberg N., Totik V., Wielonsky F., Modified logarithmic potential theory and applications, Int. Math. Res. Not., to appear, arXiv:1502.06925.
  4. Borodin A., Biorthogonal ensembles, Nuclear Phys. B 536 (1999), 704-732, math.CA/9804027.
  5. Butez R., Large deviations principle for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distribution, arXiv:1602.07201.
  6. Cheliotis D., Triangular random matrices and biorthogonal ensembles, arXiv:1404.4730.
  7. Claeys T., Romano S., Biorthogonal ensembles with two-particle interactions, Nonlinearity 27 (2014), 2419-2444, arXiv:1312.2892.
  8. Deift P., Kriecherbauer T., McLaughlin K.T.-R., New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95 (1998), 388-475.
  9. Duits M., Kuijlaars A.B.J., An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices, SIAM J. Matrix Anal. Appl. 30 (2008), 173-196, arXiv:0704.0378.
  10. Eichelsbacher P., Sommerauer J., Stolz M., Large deviations for disordered bosons and multiple orthogonal polynomial ensembles, J. Math. Phys. 52 (2011), 073510, 16 pages, arXiv:1102.0792.
  11. Forrester P.J., Liu D.-Z., Raney distributions and random matrix theory, J. Stat. Phys. 158 (2015), 1051-1082, arXiv:1404.5759.
  12. Forrester P.J., Liu D.-Z., Zinn-Justin P., Equilibrium problems for Raney densities, Nonlinearity 28 (2015), 2265-2277, arXiv:1411.4091.
  13. Forrester P.J., Wang D., Muttalib-Borodin ensembles in random matrix theory - realisations and correlation functions, arXiv:1502.07147.
  14. Hardy A., Kuijlaars A.B.J., Weakly admissible vector equilibrium problems, J. Approx. Theory 164 (2012), 854-868, arXiv:1110.6800.
  15. Kuijlaars A.B.J., Multiple orthogonal polynomial ensembles, in Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemp. Math., Vol. 507, Amer. Math. Soc., Providence, RI, 2010, 155-176, arXiv:0902.1058.
  16. Kuijlaars A.B.J., Stivigny D., Singular values of products of random matrices and polynomial ensembles, Random Matrices Theory Appl. 3 (2014), 1450011, 22 pages, arXiv:1404.5802.
  17. Muttalib K.A., Random matrix models with additional interactions, J. Phys. A: Math. Gen. 28 (1995), L159-L164, cond-mat/9405084.
  18. Neuschel T., Van Assche W., Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials, J. Approx. Theory 205 (2016), 114-132, arXiv:1509.04542.
  19. Nikishin E.M., Sorokin V.N., Rational approximations and orthogonality, Translations of Mathematical Monographs, Vol. 92, Amer. Math. Soc., Providence, RI, 1991.
  20. Saff E.B., Totik V., Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, Vol. 316, Springer-Verlag, Berlin, 1997.

Previous article  Next article   Contents of Volume 12 (2016)