Quantitative Furstenberg Theory for Large Random Matrices

Abstract

We consider the $2W\times 2W$ transfer matrices associated to the block Anderson model with GOE potential blocks. We make the classical Lyapunov exponent theory quantitative in two ways. First, we prove a quantitative limit theorem for the top Lyapunov exponent. Second, we prove every gap between Lyapunov exponents is at least $c/W$. As a corollary, this implies the localization length of this $1$d block Anderson model is at most $C{W}^2$. The proof uses Furstenberg type formulas for the Lyapunov exponents, and Malliavin calculus style arguments in the symplectic group to show the product of sufficiently many transfer matrices has a sufficiently smooth density. The main technical input for the latter is a least singular value estimate for a structured random matrix.

54 Pages, 1 figure