Ratios of characteristic polynomials in complex matrix models

We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their complex conjugate

Individual Eigenvalue Distributions for the Wilson Dirac Operator

We derive the distributions of individual eigenvalues for the Hermitian Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac Operator DW. The framework we provide is valid in the epsilon regime of chiral perturbation theory for any number of flavours Nf and for non-zero low energy constants W6, W7, W8. It is given as a perturbative expansion in terms of the k-point spectral density correlation functions and integrals thereof, which in some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at fixed chirality nu this expansion truncates after at most nu terms for small lattice spacing "a". Explicit examples for the distribution of the first and second eigenvalue are given in the microscopic domain as a truncated expansion of the Fredholm Pfaffian for quenched D5, where all k-point densities are explicitly known from random matrix theory. For the real eigenvalues of quenched DW at small "a" we illustrate our method by the finite expansion of the corresponding Fredholm determinant of size nu.Comment: 20 pages, 5 figures; v2: typos corrected, refs added and discussion of W6 and W7 extende

The k-th Smallest Dirac Operator Eigenvalue and the Pion Decay Constant

We derive an analytical expression for the distribution of the k-th smallest Dirac eigenvalue in QCD with imaginary isospin chemical potential in the Dirac operator. Because of its dependence on the pion decay constant F through the chemical potential in the epsilon-regime of chiral perturbation theory this can be used for lattice determinations of that low-energy constant. On the technical side we use a chiral Random-Two Matrix Theory, where we express the k-th eigenvalue distribution through the joint probability of the ordered k smallest eigenvalues. The latter can be computed exactly for finite and infinite N, for which we derive generalisations of Dyson's integration Theorem and Sonine's identity.Comment: 27 pages, 5 figures; v2: typos corrected, published versio

Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition

We introduce a random two-matrix model interpolating between a chiral Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE) and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit) this theory is in one to one correspondence to the partition function of Wilson chiral perturbation theory in the epsilon regime, such as the related two matrix-model previously introduced in refs. [20,21]. For a generic number of flavours and rectangular block matrices in the chGUE part we derive an eigenvalue representation for the partition function displaying a Pfaffian structure. In the quenched case with nu=0,1 we derive all spectral correlations functions in our model for finite-n, given in terms of skew-orthogonal polynomials. The latter are expressed as Gaussian integrals over standard Laguerre polynomials. In the weakly non-chiral microscopic limit this yields all corresponding quenched eigenvalue correlation functions of the Hermitian Wilson operator.Comment: 27 pages, 4 figures; v2 typos corrected, published versio

Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials

"Single Ring Theorem" and the Disk-Annulus Phase Transition

Recently, an analytic method was developed to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-hermitean literature. One obtains an explicit algebraic equation for the integrated density of eigenvalues from which the Green's function and averaged density of eigenvalues could be calculated in a simple manner. Thus, that formalism may be thought of as the non-hermitean analog of the method due to Br\'ezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. A somewhat surprising result is the so called "Single Ring" theorem, namely, that the domain of the eigenvalue distribution in the complex plane is either a disk or an annulus. In this paper we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distiobution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-hermitean matrix in terms of moments of the eigenvalue distribution of the associated hermitean matrix. We then present several numerical verifications of the previously obtained analytic results for the quartic ensemble and its phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we demonstrate numerically the "Single Ring" theorem for the sextic potential, namely, the potential of lowest degree for which the "Single Ring" theorem has non-trivial consequences.Comment: latex, 5 eps figures, 41 page

Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent

Character Expansions for the Orthogonal and Symplectic Groups

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand group functions over the characters of the U(N) group. All three expansions have been checked for all N by using them to calculate the known expansions of the generating function of the homogeneous symmetric functions. An expansion of the exponential of the traces of group elements, appearing in the finite-volume gauge field partition functions, is worked out for the orthogonal and symplectic groups.Comment: 20 pages, in REVTE

Kappa-deformed random-matrix theory based on Kaniadakis statistics

We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index {\kappa} (Boltzmann-Gibbs entropy is recovered in the limit {\kappa}\rightarrow0), we propose the non-Gaussian deformations ({\kappa} \neq 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the {\kappa}-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invarient as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index {\kappa} as a measure for deviation from the state of chaos. We also introduce a {\kappa}-deformed Porter-Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution.Comment: 18 pages, 8 figure