Completing the picture for the smallest eigenvalue of real Wishart matrices.

Rectangular real N×(N+ν) matrices W with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of WW^{T}. The extreme eigenvalues of WW^{T} are of particular interest. We explicitly compute the distribution and the gap probability of the smallest nonzero eigenvalue in this ensemble, both for arbitrary fixed N and ν, and in the universal large N limit with ν fixed. We uncover an integrable Pfaffian structure valid for all even values of ν≥0. This extends previous results for odd ν at infinite N and recursive results for finite N and for all ν. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.

Wigner surmise for Hermitian and non-Hermitian chiral random matrices.

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral random matrix theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large- N limit, we find an excellent agreement valid for a small number of exact zero eigenvalues. Compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue distributions are a useful tool in lattice gauge theory, and we illustrate this by showing that our results can describe data from two-color quantum chromodynamics simulations with chemical potential in the symplectic class.

Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential.

We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.

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.

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 spectra 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.

Nonuniversality of compact support probability distributions in random matrix theory.

The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with that of the canonical Gaussian ensemble by a nonuniversal part that is given explicitly for all monomial potentials V(M)=M(2p). Moreover, we prove that for the generalized fixed and bounded trace ensemble all k-point resolvents agree in the large-n limit, despite their nonuniversality.