Matrix models for beta ensembles

This paper constructs tridiagonal random matrix models for general (β>0) β-Hermite (Gaussian) and β-Laguerre (Wishart) ensembles. These generalize the well-known Gaussian and Wishart models for β=1,2,4. Furthermore, in the cases of the β-Laguerre ensembles, we eliminate the exponent quantization present in the previously known models. We further discuss applications for the new matrix models, and present some open problems.

[1]  Freeman J. Dyson,et al.  The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics , 1962 .

[2]  A. James.  Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[3]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[4]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[5]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[6]  Gene H. Golub,et al.  Matrix computations , 1983 .

[7]  H. Trotter.  Eigenvalue distributions of large Hermitian matrices; Wigner's semi-circle law and a theorem of Kac, Murdock, and Szegö , 1984 .

[8]  P. Gallagher.  Pair correlation of zeros of the zeta function. , 1985 .

[9]  J. W. Silverstein.  The Smallest Eigenvalue of a Large Dimensional Wishart Matrix , 1985 .

[10]  A. Odlyzko.  On the distribution of spacings between zeros of the zeta function , 1987 .

[11]  K. Aomoto,et al.  Jacobi polynomials associated with Selberg integrals , 1987 .

[12]  A. Edelman.  Eigenvalues and condition numbers of random matrices , 1988 .

[13]  R. Stanley.  Some combinatorial properties of Jack symmetric functions , 1989 .

[14]  Y. Chikuse,et al.  Properties of Hermite and Laguerre polynomials in matrix argument and their applications , 1992 .

[15]  Jyoichi Kaneko,et al.  Selberg integrals and hypergeometric functions associated with Jack polynomials , 1993 .

[16]  C. Tracy,et al.  Mathematical Physics © Springer-Verlag 1996 On Orthogonal and Symplectic Matrix Ensembles , 1995 .

[17]  Grigori Olshanski,et al.  Shifted Jack polynomials, binomial formula, and applications , 1996 .

[18]  Michel Carpentier,et al.  Polynômes de Jacobi généralisés et intégrales de Selberg , 1995, Electron. J. Comb..

[19]  Riemannian symmetric superspaces and their origin in random‐matrix theory , 1996, math-ph/9808012.

[20]  C. Tracy,et al.  The Distribution of the Largest Eigenvalue in the Gaussian Ensembles: β = 1, 2, 4 , 1997, solv-int/9707001.

[21]  A. Edelman.  The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law , 1997 .

[22]  Kevin W. J. Kadell.  The Selberg–Jack Symmetric Functions , 1997 .

[23]  P. Forrester,et al.  The Calogero-Sutherland Model and Generalized Classical Polynomials , 1996, solv-int/9608004.

[24]  Madan Lal Mehta,et al.  Probability density of the determinant of a random Hermitian matrix , 1998 .

[25]  Craig A. Tracy,et al.  Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices , 1998 .

[26]  Statistical physics on the eve of the 21st century : in honour of J B McGuire on the occasion of his 65th birthday , 1999 .

[27]  Craig A. Tracy,et al.  Universality of the distribution functions of random matrix theory , 1999 .

[29]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[30]  I. Johnstone.  On the distribution of the largest principal component , 2000 .

[31]  Delannay,et al.  Distribution of the determinant of a random real-symmetric matrix from the gaussian orthogonal ensemble , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[33]  Random-Matrix Ensembles in p-Wave Vortices , 2001, cond-mat/0103089.