Quantitative Lyapunov Exponents for Large Matrices

It is a classical fact that the singular values of random matrix products tend to grow or decay exponentially at asymptotically deterministic rates, referred to as the Lyapunov exponents. The landmark works of Furstenberg and Goldsheid-Margulis give qualitative control over these Lyapunov exponents (positivity of the top Lyapunov exponent and simplicity of the Lyapunov spectrum), however in some settings a quantitative estimate would be helpful. 

 

In this talk we discuss some ideas in this direction focusing particularly on large transfer matrices, as well as other aspects of products of high dimensional random matrices.