Hermitean matrix ensembles and orthogonal polynomials

Studies in Applied Mathematics

Volumn 100, Issue 1, 1998, Pages 33-52

  Chen, Yang ^b   Ismail, Mourad E H ^a  

^a UNIVERSITY OF SOUTH FLORIDA   (United States)

^b NONE

Author keywords

[No Author keywords available]

Indexed keywords

JACOBIAN MATRICES; ORTHOGONAL FUNCTIONS; POLYNOMIALS;

FLUID THEORY; GENERAL FORMULAS; JACOBI MATRIX; LARGEST EIGENVALUES; MATRIX ENSEMBLES; NATURAL REPRESENTATION; ORTHOGONAL POLYNOMIAL; POTENTIAL THEORY; TWO-LEVEL CORRELATION FUNCTION; WEIGHT FUNCTIONS;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 0040710945     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9590.00068     Document Type: Article

References (29)

1. N. I. AKHIEZER, The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, Edinburgh, 1965 [English translation].
2. G. D. BIRKHOFF and W. J. TRJITZINSKY, Analytic theory of singular difference equations, Acta Math. 60:1-89 (1933).
3. O. BOHIGAS and M. J. GIANNONI, Chaotic motion and random matrices, in Mathematical and Computational Methods in Nuclear Physics (J. S. Dehesa et al., Eds.), Lecture Notes in Physics, Vol. 209, pp. 1-99, Springer Verlag, Berlin, 1984.
4. Y. CHEN and K. J. ERIKSEN, Gap formation probability of the α-ensemble, Int. J. Mod. Phys. B 9:1205-1225 (1995).
5. Y. CHEN, K. J. ERIKSEN, and C. A. TRACY, Largest eigenvalue distribution in the double scaling limit of matrix models: A Coulomb fluid approach, J. Phys. A 28:L207-211 (1995).
6. Y. CHEN and M. E. H. ISMAIL, Some indeterminate moment problems and Freud-like weights, Constructive Approximation, 1998, to appear.
7. Y. CHEN and M. E. H. ISMAIL, Asymptotics of the largest zeros of some orthogonal polynomials, Preprint, 1996.
8. T. S. CHIHARA, An Introduction to Orthogonal Polynomials, Gordan & Breach, New York, 1978.
9. F. J. DYSON, Statistical theory of energy levels of complex systems, I, II, III, J. Math. Phys. 3:140-156, 157-165, 166-175 (1962).
10. G. FREUD, On the greatest zero of an orthogonal polynomial, I, II, Acta Sci. Math. (Szeged) 34:91-97 (1971); 36:49-54 (1974).
11. G. FREUD, On the greatest zero of an orthogonal polynomial, I, II, Acta Sci. Math. (Szeged) 34:91-97 (1971); 36:49-54 (1974).
12. M. E. H. ISMAIL and X. LI, Bounds on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 115:131-140 (1992).
13. K. IWASAKI, H. KIMURA, S. SHIMOMURA, and M. YOSHIDA, From Gauss to Painlevé: A Modern Theory of Special Functions, Braunschweig, Viewweg, 1991.
14. M. JIMBO, T. MIWA, Y. MÔRI, and M. SATO, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1:80-158 (1980).
15. α), α ≤ 1, J. Approx. Theory 80:219-252 (1995).
16. D. S. LUBINSKY, A survey of general orthogonal polynomials for weights on finite and infinite interval, Acta Appl. Math. 10:237-296 (1987).
17. M. L. MEHTA, Random Matrices, 2nd ed., Academic Press, New York, 1991.
18. M. L. MEHTA, A non-linear differential equation and a Fredholm determinant, J. Phys. 1 (France) 2:1721-1729 (1992).
19. M. MHASKAR and E. B. SAFF, Extremal problems for polynomials with exponential weight, Trans. Amer. Soc. 285:203-234 (1984).
20. G. MOORE, Matrix models of 2D gravity and isomonodromic deformation, Proc. Theor. Phys. Suppl., No. 102:255-285 (1990).
21. C. E. PORTER, Statistical Theory of Spectra: Fluctuations, Academic Press, New York, 1991.
22. E. A. RAKHMANOV, On the asymptotic properties of polynomials orthogonal on the real axis, Math. USSR-Sb. 46:155-193 (1984).
23. J. D. SHOHAT and J. D. TAMARKIN, The Problem of Moments, American Mathematical Society, Providence, RI, 1950.
24. G. SZEGÔ, Orthogonal Polynomials, 4th ed., American Mathematical Society, Providence, RI, 1975.
25. C. TRACY and H. WIDOM, Level spacing distribution and the Airy kernel, Commun. Math. Phys. 159:151-174 (1994); Level spacing distribution and the Bessel kernel 161:289-309 (1994); Fredholm determinants, differential equations and matrix models 163:33-72 (1994).
26. C. TRACY and H. WIDOM, Level spacing distribution and the Airy kernel, Commun. Math. Phys. 159:151-174 (1994); Level spacing distribution and the Bessel kernel 161:289-309 (1994); Fredholm determinants, differential equations and matrix models 163:33-72 (1994).
27. C. TRACY and H. WIDOM, Level spacing distribution and the Airy kernel, Commun. Math. Phys. 159:151-174 (1994); Level spacing distribution and the Bessel kernel 161:289-309 (1994); Fredholm determinants, differential equations and matrix models 163:33-72 (1994).
28. J. L. ULLMAN, On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. 24:119-148 (1972).
29. J. WIMP, Computing with Recurrence Relations, Pitman, London, 1984.