| Abstract: | We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix \({\mathcal{T}_0}\). Focusing on the eigenvalues of \({\mathcal{T}_0}\) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required \({\mathcal{T}_0}\) 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 Schrödinger operators we can improve some results 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. |
