Gaussian fluctuations of products of random matrices distributed close to the identity

Abstract

Products of random 2 × 2 matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighbourhood of order $\lambda >0$ of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in $\lambda$ and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schrödinger operators.

Keywords::

• products of random matrices
• Lyapunov exoponent
• variances
• perturbation theory

Acknowledgements

We thank the referee for careful reading and several comments that improved the presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work profited from financial support by the DFG (SCHU 1358/3-1).