Products of Independent Quaternion Ginibre Matrices and their Correlation Functions

We discuss the product of independent induced quaternion ($\beta=4$) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single Ginibre matrix, but with a more complicated weight function. We find the skew-orthogonal polynomials corresponding to the weight function of the product matrix, and use the method of skew-orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size. The radial behavior of the density of states is studied in the limit of large matrices, and the macroscopic density is discussed. The microscopic limit at the origin, at the edge(s) and in the bulk is also discussed for the radial behavior of the density of states.

[1]  A. Soshnikov,et al.  Products of Independent non-Hermitian Random Matrices , 2010, 1012.4497.

[2]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[3]  A. Zee,et al.  Non-gaussian non-hermitian random matrix theory: Phase transition and addition formalism , 1997 .

[4]  A. Crisanti,et al.  Products of random matrices in statistical physics , 1993 .

[5]  Romuald A. Janik,et al.  Multiplying unitary random matrices—universality and spectral properties , 2003 .

[6]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[7]  T. Wettig,et al.  Chiral random matrix theory for two-color QCD at high density , 2009, 0912.4999.

[8]  Eigenvalue correlations in non-Hermitean symplectic random matrices , 2001, cond-mat/0109287.

[9]  T. Wettig,et al.  Random matrix theory of unquenched two-colour QCD with nonzero chemical potential , 2010, 1012.4461.

[10]  J. Baik,et al.  The Oxford Handbook of Random Matrix Theory , 2011 .

[11]  Navinder Singh,et al.  Non-Hermitean Wishart random matrices (I) , 2010, 1006.3096.

[12]  M. Nowak,et al.  Large-Nc confinement and turbulence. , 2008, Physical Review Letters.

[13]  Jerry L. Fields,et al.  The Asymptotic Expansion of the Meijer G-Function* , 1972 .

[14]  Eugene Strahov,et al.  Hole Probabilities and Overcrowding Estimates for Products of Complex Gaussian Matrices , 2012, 1211.1576.

[15]  R. Narayanan,et al.  Universality of large N phase transitions in Wilson loop operators in two and three dimensions , 2007, 0711.4551.

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

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

[18]  Z. Burda,et al.  Universal microscopic correlation functions for products of independent Ginibre matrices , 2012, 1208.0187.

[19]  Ofer Zeitouni,et al.  The single ring theorem , 2009, 0909.2214.

[20]  Z. Burda,et al.  Spectrum of the product of independent random Gaussian matrices. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Angelo Vulpiani,et al.  Products of Random Matrices , 1993 .

[22]  J. Osborn.  Universal results from an alternate random-matrix model for QCD with a baryon chemical potential. , 2004, Physical review letters.

[23]  Wojciech T. Bruzda,et al.  Induced Ginibre ensemble of random matrices and quantum operations , 2011, 1107.5019.

[24]  MATRIX MODELS AND QCD WITH CHEMICAL POTENTIAL , 2007, hep-th/0701175.

[25]  Jerzy Jurkiewicz,et al.  Infinite products of large random matrices and matrix-valued diffusion , 2003 .

[26]  de Ng Dick Bruijn.  On some multiple integrals involving determinants , 1955 .

[27]  G. Akemann,et al.  Characteristic polynomials in real Ginibre ensembles , 2008, 0810.1458.

[28]  U. Haagerup,et al.  Brown's Spectral Distribution Measure for R-Diagonal Elements in Finite von Neumann Algebras☆ , 2000 .

[29]  C. Beenakker.  Random-matrix theory of quantum transport , 1996, cond-mat/9612179.

[30]  G. Akemann,et al.  The chiral Gaussian two-matrix ensemble of real asymmetric matrices , 2009, 0911.1276.

[31]  B. Lautrup,et al.  Products of random matrices. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[33]  G. Akemann.  The complex Laguerre symplectic ensemble of non-Hermitian matrices , 2005, hep-th/0507156.

[34]  Alexander Tikhomirov,et al.  On the Asymptotic Spectrum of Products of Independent Random Matrices. , 2010, 1012.2710.

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

[36]  Z. Burda,et al.  Eigenvalues and singular values of products of rectangular gaussian random matrices. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.