Abstract
Skew orthogonal polynomials arise in the calculation of the n-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials. In the case that the eigenvalue probability density function involves a classical weight function, explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials, and the structure is used to give a closed-form expression for the sum. This theory treates all classical cases on an equal footing, giving formulas applicable at once to the Hermite, Laguerre, and Jacobi cases.
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REFERENCES
M. Adler and P. van Moerbeke, Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials, Duke Math. Journal 80:863–911 (1995).
M. Adler and P. van Moerbeke, The Pfaff lattice, matrix integrals and a map from Toda to Pfaff. Preprint, 1999.
M. Adler, E. Horozov, and P. van Moerbeke, The Pfaff lattice and skew-orthogonal polynomials, Int. Math. Res. Notices 11:569–588 (1999).
E. Brézin and H. Neuberger, Multicritical points of unoriented random surfaces, Nucl. Phys. B 350:513–553 (1991).
F. J. Dyson, Correlations between the eigenvalues of a random matrix, Commun. Math. Phys. 19:235–250 (1970).
P. J. Forrester, T. Nagao, and G. Honner, Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nucl. Phys. B 553:601–643 (1999).
K. Frahm and J. L. Pichard, Brownian motion ensembles and parametric correlations of the transmission eigenvalues: applications to coupled quantum billiards and to disordered systems, J. Phys. I (France) 5:877–906 (1995).
G. Mahoux and M. L. Mehta, A method of integration over matrix variables IV, J. Physique I (France) 1:1093–1108 (1991).
M. L. Mehta, Random Matrices (Academic Press, New York, 2nd ed., 1991).
R. J. Muirhead, Aspects of Multivariable Statistical Theory (Wiley, New York, 1982).
T. Nagao and P. J. Forrester, Asymptotic correlations at the spectrum edge of random matrices, Nucl. Phys. B 435:401–420 (1995).
T. Nagao and P. J. Forrester, Transitive ensembles of random matrices related to orthogonal polynomials, Nucl. Phys. B 530:742–762 (1998).
T. Nagao and M. Wadati, Correlation functions of random matrix ensembles related to classical orthogonal polynomials, J. Phys. Soc. Japan 60:3298–3322 (1991); 61:78–88 (1992); 61:1910–1918 (1992).
C. A. Tracy and H. Widom, Correlation functions, cluster functions and spacing distributions in random matrices, J. Stat. Phys. 92:809–835 (1998).
H. Widom, On the relation between orthogonal, symplectic and unitary random matrices, J. Stat. Phys. 94:601–643 (1999).
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Adler, M., Forrester, P.J., Nagao, T. et al. Classical Skew Orthogonal Polynomials and Random Matrices. Journal of Statistical Physics 99, 141–170 (2000). https://doi.org/10.1023/A:1018644606835
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DOI: https://doi.org/10.1023/A:1018644606835