Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

[1]  Sommers,et al.  Eigenvalue statistics of random real matrices. , 1991, Physical review letters.

[2]  C. Beenakker,et al.  Random-matrix theory of thermal conduction in superconducting quantum dots , 2010, 1004.2438.

[3]  Nam-Gyu Kang,et al.  The Random Normal Matrix Model: Insertion of a Point Charge , 2018, Potential Analysis.

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

[5]  Sung-Soo Byun,et al.  Universal scaling limits of the symplectic elliptic Ginibre ensemble , 2021 .

[6]  G. Akemann,et al.  Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices , 2010, 1005.2983.

[7]  V. Bargmann.  On a Hilbert space of analytic functions and an associated integral transform part I , 1961 .

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

[9]  Sommers,et al.  Spectrum of large random asymmetric matrices. , 1988, Physical review letters.

[10]  Massive partition functions and complex eigenvalue correlations in matrix models with symplectic symmetry , 2006, math-ph/0606060.

[11]  Y. Fyodorov,et al.  Universality in the random matrix spectra in the regime of weak non-hermiticity , 1998, chao-dyn/9802025.

[12]  H. Amann,et al.  Ordinary Differential Equations: An Introduction to Nonlinear Analysis , 1990 .

[13]  Walter Van Assche.  Orthogonal polynomials in the complex plane and on the real line , 1997 .

[14]  F. H. Szafraniec,et al.  Coherence, Squeezing and Entanglement: An Example of Peaceful Coexistence , 2017, 1801.00352.

[15]  H. Sommers,et al.  Truncations of random orthogonal matrices. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[17]  Elizabeth Meckes,et al.  The Random Matrix Theory of the Classical Compact Groups , 2019 .

[18]  G. Akemann,et al.  Gegenbauer and Other Planar Orthogonal Polynomials on an Ellipse in the Complex Plane , 2019, Constructive Approximation.

[19]  D. Karp.  Holomorphic Spaces Related to Orthogonal Polynomials and Analytic Continuation of Functions , 2001 .

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

[21]  P. Forrester,et al.  Classical Skew Orthogonal Polynomials and Random Matrices , 1999, solv-int/9907001.

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

[23]  J. Dicapua.  Chebyshev Polynomials , 2019, Fibonacci and Lucas Numbers With Applications.

[24]  J. R. Ipsen.  Products of Independent Quaternion Ginibre Matrices and their Correlation Functions , 2013 .

[25]  P. Moerbeke,et al.  The Pfaff lattice and skew-orthogonal polynomials , 1999, solv-int/9903005.

[26]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[27]  LAUGHLIN'S WAVE FUNCTIONS, COULOMB GASES AND EXPANSIONS OF THE DISCRIMINANT , 1994, hep-th/9401163.

[28]  Eugene Strahov,et al.  Products and ratios of characteristic polynomials of random Hermitian matrices , 2003 .

[29]  H. Sommers,et al.  Truncations of random unitary matrices , 1999, chao-dyn/9910032.

[30]  B. Rider.  A limit theorem at the edge of a non-Hermitian random matrix ensemble , 2003 .

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

[32]  Characteristic Polynomials of Complex Random Matrix Models , 2002, hep-th/0212051.

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

[34]  F. M. Cholewinski.  Generalized Fock Spaces and Associated Operators , 1984 .

[35]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[36]  H. Sommers,et al.  Non-Hermitian Random Matrix Ensembles , 2009, 0911.5645.

[37]  Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations , 2013, 1302.2638.

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

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

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

[41]  N. Makarov,et al.  Scaling limits of random normal matrix processes at singular boundary points , 2015, Journal of Functional Analysis.

[42]  P. Alam.  ‘W’ , 2021, Composites Engineering.

[43]  RANDOM RIGHT EIGENVALUES OF GAUSSIAN QUATERNIONIC MATRICES , 2011, 1104.4455.

[44]  van Stef Stef Eijndhoven,et al.  New Orthogonality Relations for the Hermite Polynomials and Related Hilbert Spaces , 1990 .

[45]  K. Efetov,et al.  DISTRIBUTION OF COMPLEX EIGENVALUES FOR SYMPLECTIC ENSEMBLES OF NON-HERMITIAN MATRICES , 1998, cond-mat/9809173.

[46]  P. Alam.  ‘N’ , 2021, Composites Engineering: An A–Z Guide.

[47]  D. Khavinson,et al.  Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem , 2010 .

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

[49]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[50]  Gernot Akemann,et al.  Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems. , 2019, Physical review letters.

[51]  F. Haake.  Quantum signatures of chaos , 1991 .

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