Eigenvalue separation in some random matrix models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble an...

[1]  M. A. Girshick.  On the Sampling Theory of Roots of Determinantal Equations , 1939 .

[2]  Polynuclear growth model with external source and random matrix model with deterministic source. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  P. Forrester,et al.  Hermite and Laguerre β-ensembles: Asymptotic corrections to the eigenvalue density , 2005, math-ph/0509021.

[4]  Peter J. Forrester,et al.  A note on biorthogonal ensembles , 2008, J. Approx. Theory.

[5]  Alexei Borodin.  Biorthogonal ensembles , 1998 .

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

[7]  First Colonization of a Hard-Edge in Random Matrix Theory , 2008, 0804.1111.

[8]  J. Cowan,et al.  Some mathematical questions in biology , 1974 .

[9]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

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

[11]  János Komlós,et al.  The eigenvalues of random symmetric matrices , 1981, Comb..

[12]  Mylene Maida,et al.  Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles , 2006, math/0609738.

[13]  G. Akemann,et al.  Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem , 2007, math-ph/0703019.

[14]  Joel Anderson.  A SECULAR EQUATION FOR THE EIGENVALUES OF A DIAGONAL MATRIX PERTURBATION , 1996 .

[15]  J. W. Silverstein,et al.  Eigenvalues of large sample covariance matrices of spiked population models , 2004, math/0408165.

[16]  H. Sommers,et al.  General eigenvalue correlations for the real Ginibre ensemble , 2008, 0806.2756.

[17]  P. Forrester,et al.  Eigenvalue statistics of the real Ginibre ensemble. , 2007, Physical review letters.

[18]  ROBERT M. MAY,et al.  Will a Large Complex System be Stable? , 1972, Nature.

[19]  P. Krishnaiah,et al.  Multivariate Analysis III. , 1975 .

[20]  V. Jansen,et al.  Complexity and stability revisited , 2003 .

[21]  Marc Timme,et al.  Speed of synchronization in complex networks of neural oscillators: analytic results based on Random Matrix Theory. , 2005, Chaos.

[22]  Jianfeng Feng,et al.  Synchronization in networks with random interactions: theory and applications. , 2006, Chaos.

[23]  M. Adler,et al.  Dyson's nonintersecting Brownian motions with a few outliers , 2007, 0707.0442.

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

[25]  Arno B.J. Kuijlaars,et al.  Large n Limit of Gaussian Random Matrices with External Source, Part I , 2004 .

[26]  D. W. Lang.  ISOLATED EIGENVALUE OF A RANDOM MATRIX , 1964 .

[27]  Z. Bai,et al.  Central limit theorems for eigenvalues in a spiked population model , 2008, 0806.2503.

[28]  Spiked Models in Wishart Ensemble , 2008, 0804.0889.

[29]  C. Sinclair,et al.  The Ginibre Ensemble of Real Random Matrices and its Scaling Limits , 2008, 0805.2986.

[30]  S. Péché,et al.  Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices , 2004, math/0403022.

[31]  K. Wachter.  The Strong Limits of Random Matrix Spectra for Sample Matrices of Independent Elements , 1978 .

[32]  F. Dyson.  A Brownian‐Motion Model for the Eigenvalues of a Random Matrix , 1962 .

[33]  Spyros Makridakis,et al.  On the synthesis of general systems , 1971 .

[34]  D. Thouless,et al.  Spherical Model of a Spin-Glass , 1976 .

[35]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[36]  Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges , 1998, cond-mat/9811142.

[37]  L. Abbott,et al.  Eigenvalue spectra of random matrices for neural networks. , 2006, Physical review letters.

[38]  S. Péché.  The largest eigenvalue of small rank perturbations of Hermitian random matrices , 2004, math/0411487.

[39]  D. Thouless,et al.  The eigenvalue spectrum of a large symmetric random matrix with a finite mean , 1978 .

[40]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[41]  K. Pillai,et al.  Asymptotic Expansions for the Distributions of Characteristic Roots When the Parameter Matrix Has Several Multiple Roots , 1973 .

[42]  R. Stanley,et al.  Enumerative Combinatorics: Index , 1999 .