Smoothness of Correlations in the Anderson Model at Strong Disorder

TL;DR

This paper proves that in the strong disorder regime of the Anderson model, certain correlation functions are analytic functions of energy, leading to the analyticity of the density of states and current-current correlations outside coincident energies.

Contribution

It establishes the analyticity of higher-order correlation functions and the density of states in the strong disorder regime of the Anderson model under certain conditions.

Findings

Correlation functions are analytic outside coincident energies.

Density of states is analytic outside the diagonal.

Current-current correlation function has an analytic density at strong disorder.

Abstract

We study the higher-order correlation functions of covariant families of observables associated with random Schr\"odinger operators on the lattice in the strong disorder regime. We prove that if the distribution of the random variables has a density analytic in a strip about the real axis, then these correlation functions are analytic functions of the energy outside of the planes corresponding to coincident energies. In particular, this implies the analyticity of the density of states, and of the current-current correlation function outside of the diagonal. Consequently, this proves that the current-current correlation function has an analytic density outside of the diagonal at strong disorder.