The complex laguerre symplectic ensemble of Non-Hermitian matrices

We solve the complex extension of the chiral Gaussian Symplectic Ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correlation functions of complex eigenvalues are given in terms of the corresponding skew orthogonal polynomials in the complex plane for finite-N, where N is the matrix size or number of eigenvalues, respectively. We also allow for an arbitrary number of complex conjugate pairs of characteristic polynomials in the weight function, corresponding to massive quark flavours in applications to field theory. Explicit expressions are given in the large-N limit at both weak and strong non-Hermiticity for the weight of the Gaussian two-matrix model. This model can be mapped to the complex Dirac operator spectrum with non-vanishing chemical potential. It belongs to the symmetry class of either the adjoint representation or two colours in the fundamental representation using staggered lattice fermions

Microscopic and bulk spectra of Dirac operators from finite-volume partition functions

The microscopic spectrum of the QCD Dirac operator is shown to obey random matrix model statistics in the bulk region of the spectrum close to the origin using finite-volume partition functions

Correlations for non-Hermitian Dirac operators: chemical potential in three-dimensional QCD

In the presence of a non-vanishing chemical potential the eigenvalues of the Dirac opera- tor become complex. We use a Random Matrix Model (RMM) approach to calculate ana- lytically all correlation functions at weak and strong non-Hermiticity for three-dimensional QCD with broken flavor symmetry and four-dimensional QCD in the bulk

Microscopic correlation functions for the QCD Dirac operator with chemical potential

A chiral random matrix model with complex eigenvalues is solved as an effective model for QCD with nonvanishing chemical potential. The new correlation functions derived from it are conjectured to predict the local fluctuations of complex Dirac operator eigenvalues at zero virtuality. The parameter measuring the non-Hermiticity of the random matrix is related to the chemical potential. In the phase with broken chiral symmetry all spectral correlations are calculated for finite matrix size N and in the large-N limit at weak and strong non-Hermiticity. The derivation uses the orthogonality of the Laguerre polynomials in the complex plane

Loop equations for multi-cut matrix models

The loop equation for the complex one-matrix model with a multi-cut structure is derived and solved in the planar limit. An iterative scheme for higher genus contributions to the free energy and the multi-loop correlators is presented for the two-cut model, where explicit results are given up to and including genus two. The double-scaling limit is analyzed and the relation to the one-cut solution of the hermitian and complex one-matrix model is discussed

The solution of a chiral random matrix model with complex eigenvalues

We describe in detail the solution of the extension of the chiral Gaussian unitary ensemble (chGUE) into the complex plane. The correlation functions of the model are first calculated for a finite number of N complex eigenvalues, where we exploit the existence of orthogonal Laguerre polynomials in the complex plane. When taking the large-N limit we derive new correlation functions in the case of weak and strong non-Hermiticity, thus describing the transition from the chGUE to a generalized Ginibre ensemble. We briefly discuss applications to the Dirac operator eigenvalue spectrum in quantum chromodynamics with non-vanishing chemical potential

Multicritical matrix models and the chiral phase transition

Universality of multicritical unitary matrix models is shown and a new scaling behavior is found in the microscopic region of the spectrum, which may be relevant for the low energy spectrum of the Dirac operator at the chiral phase transition

Equivalence of matrix models for complex QCD dirac spectra

Two different matrix models for QCD with a non-vanishing quark chemical potential are shown to be equivalent by mapping the corresponding partition functions. The equivalence holds in the phase with broken chiral symmetry. It is exact in the limit of weak non-Hermiticity, where the chemical potential squared is rescaled with the volume. At strong non-Hermiticity it holds only for small chemical potential. The first model proposed by Stephanov is directly related to QCD and allows to analyze the QCD phase diagram. In the second model suggested by the author all microscopic spectral correlation functions of complex Dirac operators can be calculated in the broken phase. We briefly compare those predictions to complex Dirac eigenvalues from quenched QCD lattice simulations

Chiral random two-matrix theory and QCD with imaginary chemical potential

We summarise recent results for the chiral Random Two-Matrix Theory constructed to describe QCD in the epsilon-regime with imaginary chemical potential. The virtue of this theory is that unquenched Lattice simulations can be used to determine both low energy constants Sigma and F in the leading order chiral Lagrangian, due to their respective coupling to quark mass and chemical potential. We briefly recall the analytic formulas for all density and individual eigenvalue correlations and then illustrate them in detail in the simplest, quenched case with imaginary isospin chemical potential. Some peculiarities are pointed out for this example: i) the factorisation of density and individual eigenvalue correlation functions for large chemical potential and ii) the factorisation of the non-Gaussian weight function of bi-orthogonal polynomials into Gaussian weights with ordinary orthogonal polynomials

Macroscopic and microscopic (non-)universality of compact support random matrix theory

A random matrix model with a σ-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials V(M)=M2p, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs