The Random Phase Property and the Lyapunov Spectrum for Disordered Multi-channel Systems

Abstract

A random phase property establishing a link between quasi-one-dimensional random Schrödinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the full hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum.

Random perturbations of hyperbolic dynamics

“…This turns out to be possible by comparing the random dynamics (1) generated by (3) with the random dynamics generated by 1 + λr n U n instead of T n , that is, the case of R = 1 which has no hyperbolicity. The comparison of the cumulative distribution function (16) under these two random dynamics is based on the next lemma.…”

Random Lie group actions on compact manifolds: A perturbative analysis

“…Remark 5) such that ρ 0 = 1 M . This property is called the random phase property in [22] which is related to the maximal entropy Ansatz in the physics literature. Section 5 can be read directly at this point if Theorem 2 is accepted without proof.…”

“…Let us recall how the general framework of the Introduction is applied in the present situation: the Lie group is G = U(L, L) acting on the compact flag manifold M by (23); equation (22) shows that the rotation is R = R k and the random perturbation P 1,σ = P σ , while P n,σ = 0 for n ≥ 2. Objects of interest are now the L positive Lyapunov exponents γ l,λ (E), l = 1, .…”

Random Schrödinger operators on long boxes, noise explosion and the GOE

“…The noise term in the transfer matrix evolution in this scaling regime for block Jacobi matrices has been studied in the paper by Römer and Schulz-Baldes [15] using a language different from SDEs. The first arxiv version of the present paper was followed by the preprint of the paper by Bachmann and De Roeck [3], who, in independent work, also study SDE limits of transfer matrices.…”