Central limit theorem for the integrated density of states of the Anderson model on lattice

Dhriti Ranjan Dolai

Published 2023-09-14Version 1

We consider the existence of integrated density of states (IDS) of the Anderson model on the Hilbert space $\ell^2(\mathbb{Z}^d)$ as analogues to the law of large numbers (LLN). In this work, we prove the analogues central limit theorem (CLT) for integrated density of states when the test functions are polynomials. Then, we extend the result for a subclass of infinitely smooth functions on the spectrum. We also show that for absolutely continuous single site distribution (SSD), which has log-concave density, the CLT holds for a subclass of $C^1(\mathbb{R})$.