240 research outputs found
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 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
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