Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

Abstract

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\beta =2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically simplifying the expressions for numerical applications. It also serves as a good starting point for studying the Wilson-Dirac operator in the $\epsilon$-regime of lattice quantum chromodynamics.

50 pages, PACS: 02.10.Yn, 02.50.-r, 05.50.+q, 71.10.Fd, 12.38.-t

References in corpus (18)

• Phase of the Fermion Determinant at Nonzero Chemical Potential
• The QCD Sign Problem for Small Chemical Potential
• Superbosonization of invariant random matrix ensembles
• Matrix Models and QCD with Chemical Potential
• Arbitrary Rotation Invariant Random Matrix Ensembles and Supersymmetry
• A Random Matrix Study of the QCD Sign Problem
• Discretization errors in the spectrum of the Hermitian Wilson-Dirac operator
• Random matrix analysis of the QCD sign problem for general topology
• Characteristic polynomials in real Ginibre ensembles
• Chiral Condensate at Nonzero Chemical Potential in the Microscopic Limit of QCD
• Arbitrary rotation invariant random matrix ensembles and supersymmetry: orthogonal and unitary-symplectic case
• The Realization of the Sharpe-Singleton Scenario
• The Microscopic Twisted Mass Dirac Spectrum
• Phase Diagram of the Dirac Spectrum at Nonzero Chemical Potential
• Determining low-energy constants in partially quenched Wilson chiral perturbation theory
• Staggered chiral random matrix theory
• Wilson Fermions, Random Matrix Theory and the Aoki Phase
• Topology, Random Matrix Theory and the spectrum of the Wilson Dirac operator

Cited by in corpus (14)

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• Phase Diagram of Dynamical Twisted Mass Wilson Fermions at Finite Isospin Chemical Potential
• Spectral Properties of the Wilson Dirac Operator and random matrix theory
• New Ways to Determine Low-Energy Constants with Wilson Fermions
• Dirac Spectrum of the Wilson Dirac Operator for QCD with Two Colors
• Random Matrix Theory and Quantum Chromodynamics
• GUE-chGUE Transition preserving Chirality at finite Matrix Size
• Universality Conjecture for all Airy, Sine and Bessel Kernels in the Complex Plane
• Chiral Dynamics With Wilson Fermions
• Consecutive level spacings in the chiral Gaussian unitary ensemble: From the hard and soft edge to the bulk
• Universal microscopic spectrum of the unquenched QCD Dirac operator at finite temperature
• Complex symmetric, self-dual, and Ginibre random matrices: Analytical results for three classes of bulk and edge statistics
• The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator