RANDOM LIE GROUP ACTIONS ON COMPACT MANIFOLDS: A PERTURBATIVE ANALYSIS

• 成果类型: Article
• 署名作者: Sadel, Christian; Schulz-Baldes, Hermann
• 署名单位: University of Erlangen Nuremberg
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP544
• 发表日期: 2010
• 页码: 2224-2257
• 关键词: lyapunov exponents; invariant-measures; anderson model; anomalies

摘要

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.