Parametric level statistics in random matrix theory: Exact solution

An exact solution to the problem of parametric level statistics in non-Gaussian ensembles of N by N Hermitian random matrices with either soft or strong level confinement is formulated within the framework of the orthogonal polynomial technique. Being applied to random matrices with strong level confinement, the solution obtained leads to emergence of a new connection relation that makes a link between the parametric level statistics and the scalar two-point kernel in the thermodynamic limit.Comment: 4 pages (revtex

Microscopic universality with dynamical fermions

It has recently been demonstrated in quenched lattice simulations that the distribution of the low-lying eigenvalues of the QCD Dirac operator is universal and described by random-matrix theory. We present first evidence that this universality continues to hold in the presence of dynamical quarks. Data from a lattice simulation with gauge group SU(2) and dynamical staggered fermions are compared to the predictions of the chiral symplectic ensemble of random-matrix theory with massive dynamical quarks. Good agreement is found in this exploratory study. We also discuss implications of our results.Comment: 5 pages, 3 figures, minor modifications, to appear in Phys. Rev. D (Rapid Commun.

Spectra of massive and massless QCD Dirac operators: A novel link

We show that integrable structure of chiral random matrix models incorporating global symmetries of QCD Dirac operators (labeled by the Dyson index beta=1,2, and 4) leads to emergence of a connection relation between the spectral statistics of massive and massless Dirac operators. This novel link established for beta-fold degenerate massive fermions is used to explicitly derive (and prove the random matrix universality of) statistics of low--lying eigenvalues of QCD Dirac operators in the presence of SU(2) massive fermions in the fundamental representation (beta=1) and SU(N_c >= 2) massive adjoint fermions (beta=4). Comparison with available lattice data for SU(2) dynamical staggered fermions reveals a good agreement

Distribution of the k-th smallest Dirac operator eigenvalue

Based on the exact relationship to Random Matrix Theory, we derive the probability distribution of the k-th smallest Dirac operator eigenvalue in the microscopic finite-volume scaling regime of QCD and related gauge theories.Comment: REVTeX 3.1, 6 pages, 1 figure. Corrected factors in eqs. (16a) and (16c) and very minor typos (v2

Massive random matrix ensembles at beta = 1 & 4 : QCD in three dimensions

The zero momentum sectors in effective theories of three dimensional QCD coupled to pseudoreal (two colors) and real (adjoint) quarks in a classically parity-invariant manner have alternative descriptions in terms of orthogonal and symplectic ensembles of random matrices. Using this correspondence, we compute finite-volume QCD partition functions and correlation functions of Dirac operator eigenvalues in a presence of finite quark masses of the order of the smallest Dirac eigenvalue. These novel correlation functions, expressed in terms of quaternion determinants, are reduced to conventional results for the Gaussian ensembles in the quenched limit.Comment: 26 pages, REVTeX 3.1 + mathlett.sty (attached), 4 figure

Universality in Chiral Random Matrix Theory at β=1\beta =1 and β=4\beta =4

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($\beta =1$) and quaternion real ($\beta = 4$) matrix elements is expressed in terms of the kernel of the corresponding complex Hermitean random matrix ensembles ($\beta=2$). Such identities are exact in case of a Gaussian probability distribution and, under certain smoothness assumptions, they are shown to be valid asymptotically for an arbitrary finite polynomial potential. They are proved by means of a construction proposed by Br\'ezin and Neuberger. Universal behavior at the hard edge of the spectrum for all three chiral ensembles then follows from microscopic universality for $\beta =2$ as shown by Akemann, Damgaard, Magnea and Nishigaki.Comment: 4 pages, modified discussion of edge contributions and corrected typo

Massive chiral random matrix ensembles at beta = 1 & 4 : Finite-volume QCD partition functions

In a deep-infrared (ergodic) regime, QCD coupled to massive pseudoreal and real quarks are described by chiral orthogonal and symplectic ensembles of random matrices. Using this correspondence, general expressions for the QCD partition functions are derived in terms of microscopically rescaled mass variables. In limited cases, correlation functions of Dirac eigenvalues and distributions of the smallest Dirac eigenvalue are given as ratios of these partition functions. When all masses are degenerate, our results reproduce the known expressions for the partition functions of zero-dimensional sigma models.Comment: 6 pages, REVTeX 3.1, no figure; (v2) corrected signatures of c'