Transitions In Spectral Statistics

We present long range statistical properties of a recently introduced unitary random matrix ensemble, whose short range correlations were found to describe a transition from Wigner to Poisson type as a function of a single parameter.Comment: 12 pp. late

Spectral properties of a generalized chGUE

We consider a generalized chiral Gaussian Unitary Ensemble (chGUE) based on a weak confining potential. We study the spectral correlations close to the origin in the thermodynamic limit. We show that for eigenvalues separated up to the mean level spacing the spectral correlations coincide with those of chGUE. Beyond this point, the spectrum is described by an oscillating number variance centered around a constant value. We argue that the origin of such a rigid spectrum is due to the breakdown of the translational invariance of the spectral kernel in the bulk of the spectrum. Finally, we compare our results with the ones obtained from a critical chGUE recently reported in the literature. We conclude that our generalized chGUE does not belong to the same class of universality as the above mentioned model.Comment: 12 pages, 3 figure

Spectral Correlations from the Metal to the Mobility Edge

We have studied numerically the spectral correlations in a metallic phase and at the metal-insulator transition. We have calculated directly the two-point correlation function of the density of states $R(s,s')$. In the metallic phase, it is well described by the Random Matrix Theory (RMT). For the first time, we also find numerically the diffusive corrections for the number variance $$ predicted by Al'tshuler and Shklovski\u{\i}. At the transition, at small energy scales, $R(s-s')$ starts linearly, with a slope larger than in a metal. At large separations $|s - s'| \gg 1$, it is found to decrease as a power law $R(s,s') \sim - c / |s -s'|^{2-\gamma}$ with $c \sim 0.041$ and $\gamma \sim 0.83$, in good agreement with recent microscopic predictions. At the transition, we have also calculated the form factor $\tilde K(t)$, Fourier transform of $R(s-s')$. At large $s$, the number variance contains two terms $= B ^\gamma + 2 \pi \tilde K(0) where$\tilde{K}(0)$is the limit of the form factor for$t \to 0$.Comment: 7 RevTex-pages, 10 figures. Submitted to PR

Energy level statistics of a critical random matrix ensemble

We study level statistics of a critical random matrix ensemble of a power-law banded complex Hermitean matrices. We compute numerically the level compressibility via the level number variance and compare it with the analytical formula for the exactly solvable model of Moshe, Neuberger and Shapiro.Comment: 8 pages, 3 figure

Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

We consider orthogonal, unitary, and symplectic ensembles of random matrices with (1/a)(ln x)^2 potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have multifractal eigenstates. If the coefficient $a$ is small, spectral correlations in the bulk are universally governed by a translationally invariant, one-parameter generalization of the sine kernel. We provide analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can be excellently fitted to the numerical data for three symmetry classes of the three-dimensional Anderson Hamiltonians at the metal-insulator transition, previously measured by several groups using exact diagonalization.Comment: 12 pages, 8 figures, REVTeX. Additional figure and text on the level number variance, to appear in Phys.Rev.

Critical statistics for non-Hermitian matrices

We introduce a generalized ensemble of nonhermitian matrices interpolating between the Gaussian Unitary Ensemble, the Ginibre ensemble and the Poisson ensemble. The joint eigenvalue distribution of this model is obtained by means of an extension of the Itzykson-Zuber formula to general complex matrices. Its correlation functions are studied both in the case of weak nonhermiticity and in the case of strong nonhermiticity. In the weak nonhermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics: the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing. To lowest order, its slope does not depend on the degree of nonhermiticity. Close the edge, the spectral correlations are similar to the Hermitian case. In the strong nonhermiticity limit the crossover behavior from the Ginibre ensemble to the Poisson ensemble first appears close to the surface of the spectrum. Our model may be relevant for the description of the spectral correlations of an open disordered system close to an Anderson transition.Comment: 25 pages, 6 figure

Random Matrix Theory of the Energy-Level Statistics of Disordered Systems at the Anderson Transition

We consider a family of random matrix ensembles (RME) invariant under similarity transformations and described by the probability density $P({\bf H})= \exp[-{\rm Tr}V({\bf H})]$. Dyson's mean field theory (MFT) of the corresponding plasma model of eigenvalues is generalized to the case of weak confining potential, $V(\epsilon)\sim {A\over 2}\ln ^2(\epsilon)$. The eigenvalue statistics derived from MFT are shown to deviate substantially from the classical Wigner-Dyson statistics when $A<1$. By performing systematic Monte Carlo simulations on the plasma model, we compute all the relevant statistical properties of the RME with weak confinement. For $A_c\approx 0.4$ the distribution function of the energy-level spacings (LSDF) of this RME coincides in a large energy window with the LSDF of the three dimensional Anderson model at the metal-insulator transition. For the same $A_c$, the variance of the number of levels, $\langle n^2\rangle -\langle n\rangle^2$, in an interval containing $\langle n\rangle$ levels on average, grows linearly with $\langle n\rangle$, and its slope is equal to $0.32 \pm 0.02$, which is consistent with the value found for the Anderson model at the critical point.Comment: 32 pages, REVTEX 3.0, 10 postscript (uuencoded) figures include

Network meta-analysis of (individual patient) time to event data alongside (aggregate) count data

Objectives: Network meta-analysis (NMA) methods extend the standard pair-wise framework to allow simultaneous comparison of multiple interventions in a single statistical model. Despite published work on NMA mainly focussing on the synthesis of aggregate data (AD), methods have been developed that allow the use of individual patient-level data (IPD) specifically when outcomes are dichotomous or continuous. This paper focuses on the synthesis of IPD and AD time to event data, motivated by a real data example looking at the effectiveness of high compression treatments on the healing of venous leg ulcers. Methods: This paper introduces a novel NMA modelling approach that allows IPD (time to event with censoring) and AD (event count for a given follow-up time) to be synthesised jointly by assuming an underlying, common, distribution of time to healing. Alternative model assumptions were tested within the motivating example. Model fit and adequacy measures were used to compare and select models. Results: Due to the availability of IPD in our example we were able to use a Weibull distribution to describe time to healing; otherwise, we would have been limited to specifying a uniparametric distribution. Absolute effectiveness estimates were more sensitive than relative effectiveness estimates to a range of alternative specifications for the model. Conclusions: The synthesis of time to event data considering IPD provides modelling flexibility, and can be particularly important when absolute effectiveness estimates, and not just relative effect estimates, are of interest