Angular Gelfand--Tzetlin Coordinates for the Supergroup UOSp(k_1/2k_2)

We construct Gelfand--Tzetlin coordinates for the unitary orthosymplectic supergroup UOSp(k_1/2k_2). This extends a previous construction for the unitary supergroup U(k_1/k_2). We focus on the angular Gelfand--Tzetlin coordinates, i.e. our coordinates stay in the space of the supergroup. We also present a generalized Gelfand pattern for the supergroup UOSp(k_1/2k_2) and discuss various implications for representation theory

Between Poisson and GUE statistics: Role of the Breit-Wigner width

We consider the spectral statistics of the superposition of a random diagonal matrix and a GUE matrix. By means of two alternative superanalytic approaches, the coset method and the graded eigenvalue method, we derive the two-level correlation function $X_2(r)$ and the number variance $\Sigma^2(r)$. The graded eigenvalue approach leads to an expression for $X_2(r)$ which is valid for all values of the parameter $\lambda$ governing the strength of the GUE admixture on the unfolded scale. A new twofold integration representation is found which can be easily evaluated numerically. For $\lambda \gg 1$ the Breit-Wigner width $\Gamma_1$ measured in units of the mean level spacing $D$ is much larger than unity. In this limit, closed analytical expression for $X_2(r)$ and $\Sigma^2(r)$ can be derived by (i) evaluating the double integral perturbatively or (ii) an ab initio perturbative calculation employing the coset method. The instructive comparison between both approaches reveals that random fluctuations of $\Gamma_1$ manifest themselves in modifications of the spectral statistics. The energy scale which determines the deviation of the statistical properties from GUE behavior is given by $\sqrt{\Gamma_1}$. This is rigorously shown and discussed in great detail. The Breit-Wigner $\Gamma_1$ width itself governs the approach to the Poisson limit for $r\to\infty$. Our analytical findings are confirmed by numerical simulations of an ensemble of $500\times 500$ matrices, which demonstrate the universal validity of our results after proper unfolding.Comment: 25 pages, revtex, 5 figures, Postscript file also available at http://germania.ups-tlse.fr/frah

Spectral correlations of the massive QCD Dirac operator at finite temperature

We use the graded eigenvalue method, a variant of the supersymmetry technique, to compute the universal spectral correlations of the QCD Dirac operator in the presence of massive dynamical quarks. The calculation is done for the chiral Gaussian unitary ensemble of random matrix theory with an arbitrary Hermitian matrix added to the Dirac matrix. This case is of interest for schematic models of QCD at finite temperature.Comment: 19 pages, no figures, LaTeX (elsart.cls) minor changes, one reference adde