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Abstract:

We analyse correlations of eigenvectors in Ginibre's and Girko's ensembles of Gaussian, non-Hermitian random N x N matrices J. We study the ensemble average of [L-alpha/L-beta] [R-beta/R-alpha], where [L-alpha\ and \R-beta] are the left and right eigenvectors of J. The case of Ginibre's ensemble, in which the real and imaginary parts of each element of J are independent random variables, is sufficiently symmetric to allow for an exact solution. In the more general case of Girko's ensemble, we rely on approximations which become exact in the limit of N --> infinity.

Abstract:

We study the low-temperature behaviour of the classical Heisenberg antiferromagnet with nearest neighbour interactions on the pyrochlore lattice. Because of geometrical frustration, the ground state of this model has an extensive number of degrees of freedom. We show, by analysing the effects of small fluctuations around the ground-state manifold, and from the results of Monte Carlo and molecular dynamics simulations, that the system is disordered at all temperatures, T, and has a finite relaxation time, which varies as 1/T for small T.

Abstract:

We study spectral properties of the Fokker-Planck operator that describes particles diffusing in a quenched random velocity field. This random operator is non-Hermitian and has eigenvalues occupying a finite area in the complex plane. We calculate the eigenvalue density and averaged one-particle Green's function, for weak disorder and dimension d>2. We relate our results to the time-evolution of particle density, and compare them with numerical simulations.

Abstract:

The statistics of energy levels for a disordered conductor are considered in the critical energy window near the mobility edge. It is shown that, if critical wave functions are multifractal, the one-dimensional gas of levels on the energy axis is ``compressible'', in the sense that the variance of the level number in an interval is $< (δN)^{2} > = χ$ for $ >> 1$. The compressibility, $χ=η/2d$, is given ``exactly'' in terms of the multifractal exponent $η=d-D_2$ at the mobility edge in a $d$-dimensional system.