Random Lie group actions on compact manifolds: A perturbative analysis

Abstract

A random Lie group action on a compact manifold generates a discrete time Markov process. The main object of this paper is the evaluation of associated Birkhoff sums in a regime of weak, but sufficiently effective coupling of the randomness. This effectiveness is expressed in terms of random Lie algebra elements and replaces the transience or Furstenberg's irreducibility hypothesis in related problems. The Birkhoff sum of any given smooth function then turns out to be equal to its integral w.r.t. a unique smooth measure on the manifold up to errors of the order of the coupling constant. Applications to the theory of products of random matrices and a model of a disordered quantum wire are presented.

Published in at http://dx.doi.org/10.1214/10-AOP544 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Multidimensional continued fractions, dynamical renormalization and KAM theory
• Rotation numbers for Jacobi matrices with matrix entries
• Scaling diagram for the localization length at a band edge
• The random phase property and the Lyapunov spectrum for disordered multi-channel systems

Cited by in corpus (8)

• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• From the Anderson model on a strip to the DMPK equation and random matrix theory
• Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems of non-interacting Fermions
• Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case
• Relations between Transfer and Scattering Matrices in the presence of Hyperbolic Channels
• A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
• Regular expansion for the characteristic exponent of a product of $2\times 2$ random matrices
• Disordered quantum wires: microscopic origins of the DMPK theory and Ohm's law