Abstract
Bleher & Kuijlaars and Daems & Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the Christoffel-Darboux kernel for multiple orthogonal polynomials. We obtain a representation of this Christoffel-Darboux kernel in terms of the usual orthogonal polynomials.
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References
A. Aptekarev, P. Bleher and A. Kuijlaars, Large n limit of Gaussian random matrices with external source II, Comm. Math. Phys. 259 (2005) no.2, 367–389.
J. Baik, G. Ben Arous and S. Péché, Phase transition of the largest eigenvalue for non-null complex sample covariance matricesm Ann. Probab. 33 (2005) no.5, 1643–1697.
P. Bleher and A. Kuijlaars. Large n limit of Gaussian random matrices with external source I, Comm. Math. Phys. 252 (2004) no.1-3, 43–76.
P. Bleher and A. Kuijlaars, Random matrices with external source and multiple orthogonal polynomials, Int. Math. Res. Not. 3 (2004), 109–129.
P. Bleher and A. Kuijlaars, Large n limit of Gaussian random matrices with external source III, Comm. Math. Phys. 270 (2007) no.2, 481–517.
E. Brézin and S. Hikamim, Level spacing of random matrices in an external source, Phys. Rev. E. 3 (1998) no.6, 7176–7185.
E. Daems and A. Kuijlaars, A Christoffel-Darboux formula for multiple orthogonal polynomials, J. Approx. Theory 130 (2004) no.2, 190–202.
E. Daems, A. Kuijlaars and W. Veys, Asymptotics of non-intersecting Brownian motions and a 4 × 4 Riemann-Hilbert problem, http://xxx.lanl.gov/abs/math/0701923+.
P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math. 52 (1999) no.12, 1491–1552.
P. Deift, T. Kriecherbauer, K. McLaughlin, S. Venakides and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999) no.11, 1335–1425.
M. Duits and A. Kuijlaars, Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis, http://xxx.lanl.gov/abs/0807.4814+.
A. Fokas, A. Its and V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313–344.
Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87–120.
C. Itzykson and J. B. Zuber, The planar approximation II, J. Math. Phys. 21 (1980), 411–421.
M. Mehta. Random Matrices, Academic press, San Diago, second edition, 1991.
S. Péché. The largest eigenvalue of small rank perturbations of hermitian random matrices, Probab. Theory Related Fields 134 (2006) no.1, 127–173.
W. Van Assche, J. Geronimo and A. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, in: J. Bustoz, M. Ismail and S. Suslov (edss), Special Functions 2000: Current Perspective and Future Directions volume 30 of NATO Science Series II: Mathematics, Physics and Chemistry, Kluwer Acadgemic Publishers, 2001, 23–59.
P. Zinn-Justin, Random Hermitian matrices in an external field, Nuclear Phys. B 497 (1998) no.3, 725–732.
P. Zinn-Justin, Universality of correlation functions of Hermitian random matrices in an external field, Comm. Math. Phys. 194 (1998) no.3, 631–650.
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The work of the author was supported in part by NSF grants DMS0457335, DMS075709.
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Baik, J. On the Christoffel-Darboux Kernel for Random Hermitian Matrices with External Source. Comput. Methods Funct. Theory 9, 455–471 (2009). https://doi.org/10.1007/BF03321740
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DOI: https://doi.org/10.1007/BF03321740