Quantitative Furstenberg Theory for Large Random Matrices
Reuben Drogin
Abstract
We consider the 2W× 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 1d block Anderson model is at most CW2. 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.
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