B. Mehlig

EIGENVECTOR STATISTICS IN NON-HERMITIAN RANDOM MATRIX ENSEMBLES

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibres complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random variables. Calculating ensemble averages based on the quantity

Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles.

, where

Universal spectral properties of spatially periodic quantum systems with chaotic classical dynamics

are left and right eigenvectors of J, we show for large N that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications.

Critical dynamics of the hybrid Monte Carlo algorithm

In this paper the Hybrid Monte Carlo algorithm for condensed-matter systems is described as a generalized Langevin algorithm and compared to the Langevin scheme. Furthermore the parallel implementation of this class of algorithms is described.

Classical phase-space analysis of vibronically coupled systems

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre’s ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using N−1 as an expansion parameter, where N is the rank of the random matrix: this is applied to Girko’s ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time evolution governed by a non-Hermitian random matrix, transients are controlled by eigenvector correlations.

Universal eigenvector statistics in a quantum scattering ensemble.

We determine the statistical properties of wave functions in disordered quantum systems by exact diagonalization of one-, two- and quasi-one dimensional tight-binding Hamiltonians. In the quasi-one dimensional case we find that the tails of the distribution of wave-function amplitudes are described by the non-linear �-model. In two dimensions, the tails of the distribution function are consistent with a recent prediction based on a direct optimal fluctuation method. 72.15.Rn,71.23.An,05.40.-a

Spectral correlations in systems undergoing a transition from periodicity to disorder

We consider a quasi one-dimensional chain of N chaotic scattering elements with periodic boundary conditions. The classical dynamics of this system is dominated by diffusion. The quantum theory, on the other hand, depends crucially on whether the chain is disordered or invariant under lattice translations. In the disordered case, the spectrum is dominated by Anderson localization whereas in the periodic case, the spectrum is arranged in bands. We investigate the special features in the spectral statistics for a periodic chain. For finite N, we define spectral form factors involving correlations both for identical and non-identical Bloch numbers. The short-time regime is treated within the semiclassical approximation, where the spectral form factor can be expressed in terms of a coarse-grained classical propagator which obeys a diffusion equation with periodic boundary conditions. In the long-time regime, the form factor decays algebraically towards an asymptotic constant. In the limit N → ∞, we derive a universal scaling function for the form factor. The theory is supported by numerical results for quasi one-dimensional periodic chains of coupled Sinai billiards.

Universality in the critical two-dimensionalφ4-model

Abstract In this letter, the critical dynamics of the hybrid Monte Carlo algorithm is discussed for the φ4-model in d = 2 dimensions. It is pointed out that care has to be taken in choosing a consistent unit of time. With a correct unit of time the dynamical critical exponent is found to be z ≈ 2. Moreover, the computational effort per site required to generate a new, statistically independent configuration is found to diverge according to Lχ near critically, with χ>2, in contrast to χ = 2 for a single-site Monte Carlo algorithm.