Spectral Properties of the Jacobi Ensembles via the Coulomb Gas approach

Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.

[1]  Iain M Johnstone,et al.  APPROXIMATE NULL DISTRIBUTION OF THE LARGEST ROOT IN MULTIVARIATE ANALYSIS. , 2010, The annals of applied statistics.

[2]  K. Johansson.  Shape Fluctuations and Random Matrices , 1999, math/9903134.

[3]  SYSTEMATIC ANALYTICAL APPROACH TO CORRELATION FUNCTIONS OF RESONANCES IN QUANTUM CHAOTIC SCATTERING , 1999, cond-mat/9903043.

[4]  E. Wigner,et al.  On the statistical distribution of the widths and spacings of nuclear resonance levels , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  S. Majumdar,et al.  How many eigenvalues of a Gaussian random matrix are positive? , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Brian D. Sutton,et al.  The stochastic operator approach to random matrix theory , 2005 .

[7]  R. Lippert,et al.  A matrix model for the -Jacobi ensemble , 2003 .

[8]  Ioana Dumitriu,et al.  Distributions of the Extreme Eigenvaluesof Beta-Jacobi Random Matrices , 2008, SIAM J. Matrix Anal. Appl..

[9]  M. Ledoux.  DIFFERENTIAL OPERATORS AND SPECTRAL DISTRIBUTIONS OF INVARIANT ENSEMBLES FROM THE CLASSICAL ORTHOGONAL POLYNOMIALS: THE DISCRETE CASE , 2019 .

[10]  F. Dyson.  Statistical Theory of the Energy Levels of Complex Systems. I , 1962 .

[11]  Holger Dette,et al.  Some Asymptotic Properties of the Spectrum of the Jacobi Ensemble , 2009, SIAM J. Math. Anal..

[12]  P. Forrester.  Log-Gases and Random Matrices (LMS-34) , 2010 .

[13]  Spectrum of the QCD Dirac operator and chiral random matrix theory. , 1994, Physical review letters.

[14]  Michel Ledoux,et al.  Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case , 2004 .

[15]  S. Majumdar,et al.  Large deviations of the maximum eigenvalue in Wishart random matrices , 2007, cond-mat/0701371.

[16]  Isaac Pérez Castillo,et al.  Large deviations of the smallest eigenvalue of the Wishart-Laguerre ensemble. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  P. Forrester.  Log-Gases and Random Matrices , 2010 .

[18]  P. Forrester.  Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles , 2012, 1201.3055.

[19]  Massimo Vergassola,et al.  Large deviations of the maximum eigenvalue for wishart and Gaussian random matrices. , 2008, Physical review letters.

[20]  Ioana Dumitriu.  Smallest eigenvalue distributions for two classes of $\beta$-Jacobi ensembles , 2010 .

[21]  From Wishart to Jacobi ensembles: statistical properties and applications , 2008 .

[22]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[23]  S. M. Manning,et al.  Asymptotic level spacing of the Laguerre ensemble: a Coulomb fluid approach , 1994 .

[24]  Massimo Vergassola,et al.  Statistical Distribution of Quantum Entanglement for a Random Bipartite State , 2010, 1006.4091.

[25]  S. Majumdar,et al.  Extreme value statistics of eigenvalues of Gaussian random matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Alan Edelman,et al.  The efficient evaluation of the hypergeometric function of a matrix argument , 2006, Math. Comput..

[27]  Benoit Collins.  Product of random projections, Jacobi ensembles and universality problems arising from free probability , 2005 .

[28]  S. Majumdar,et al.  Large deviations of the maximal eigenvalue of random matrices , 2010, 1009.1945.

[29]  Tiefeng Jiang,et al.  Limit theorems for beta-Jacobi ensembles , 2009, 0911.2262.

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

[31]  I. Johnstone.  MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY-WIDOM LIMITS AND RATES OF CONVERGENCE. , 2008, Annals of statistics.

[32]  Giacomo Livan,et al.  Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities , 2011, 1103.2638.

[33]  C'eline Nadal,et al.  Right tail expansion of Tracy-Widom beta laws , 2011, 1111.2761.