A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes

TL;DR

This paper establishes a central limit theorem for products of small, i.i.d. random matrices near a fixed matrix, deriving a stochastic differential equation limit and applying it to improve understanding of eigenvalue statistics in the Anderson model.

Contribution

It extends previous results by allowing eigenvalues of different moduli in the fixed matrix and derives new SDE limits for eigenvalues and flag manifold actions.

Findings

Derived a new SDE limit for eigenvalues of perturbed matrices.

Improved GOE statistics results for Anderson models on long boxes.

Solved an open problem related to eigenvalue distributions in the Anderson model.

Abstract

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix $T_0$. Focusing on the eigenvalues of $T_0$ of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required $T_0$ to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schr\"odinger operators we can improve some result by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular we solve a problem posed in their work.