Central limit theorem for the random variables associated with the IDS of the Anderson model on lattice

Abstract

We consider the existence of the 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 the collection of random variables associated with the integrated density of states for the class of test functions $C^1_P(\mathbb{R})$, the set of all differentiable (first-order) functions on the real line whose derivative is continuous and has at most polynomial growth. Our work extends the result by Grinshpon-White (J. Spectr. Theory 12 (2022) 591-615), where the CLT is obtained when the test functions are polynomial.