The motion of a particle in a random potential in two or more
dimensions is chaotic, and the trajectories in deterministically
chaotic systems are effectively random. It is therefore no surprise
that there are links between the quantum properties of disordered
systems and those of simple chaotic systems. The question is, how
deep do the connec tions go? And to what extent do the mathematical
techniques designed to understand one problem lead to new insights
into the other? The canonical problem in the theory of disordered
mesoscopic systems is that of a particle moving in a random array
of scatterers. The aim is to calculate the statistical properties
of, for example, the quantum energy levels, wavefunctions, and
conductance fluctuations by averaging over different arrays; that
is, by averaging over an ensemble of different realizations of the
random potential. In some regimes, corresponding to energy scales
that are large compared to the mean level spacing, this can be done
using diagrammatic perturbation theory. In others, where the
discreteness of the quantum spectrum becomes important, such an
approach fails. A more powerful method, devel oped by Efetov,
involves representing correlation functions in terms of a
supersymmetric nonlinear sigma-model. This applies over a wider
range of energy scales, covering both the perturbative and
non-perturbative regimes. It was proved using this method that
energy level correlations in disordered systems coincide with those
of random matrix theory when the dimensionless conductance tends to
infinity.
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