Lyapunov exponents of randomly coupled wires: a perturbative calculation
Seminar Room 1, Newton Institute
AbstractWe consider L discrete wires with a random potential and random hopping terms across the wires with magnetic phases. The random terms are centered, independent, identically distributed along the wires and coupled with some small constant. Associated to such a Hamiltonian are 2L by 2L transfer matrices. To calculate the mean Lyapunov exponent, one has to consider the induced Markov process on a Lagrangian Grassmanian manifold which is diffeomorphic to the unitary group U(L). We obtain that the lowest order invariant measure for this Markov process on U(L) is given by the Haar measure for non-zero energies |E|<2 and some smooth density for E=0. Furthermore we get a perturbative expression for the mean Lyapunov exponent. In case L=1 the dynamics on U(1) is just the dynamics of the Pruefer phases and the model would be the one-dimensional Anderson model. The result away from E=0 is known as random phase approximation and the anomaly at the band-center, E=0, was first found by Kappus and Wegner. To get this result we first consider an abstract setting. Let be given a Markov process on some compact homogeneous space induced by the action of a random family of Lie group elements, where the random terms are coupled with some small constant. Under certain conditions we can prove, that the invariant measure for this process is unique to lowest order and given by a smooth density which is the ground state of a Fokker-Planck operator. Finally we will see, that this theorem can be used in the example described above.
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