Eigenvalue Fluctuations for Lattice Anderson Hamiltonians

In this talk, we will discuss a homogenization problem for the Schrödinger operator with a random potential: the so-called Anderson Hamiltonian. This type of problem has been well-studied in a similar but more singular setting called "the crushed ice problem", that is, the Laplacian in a randomly perforated domain. Kac and Rauch-Taylor established the convergence (homogenization) of eigenvalues in a certain limiting regime. Later Figari-Orlandi-Teta and Ozawa found a Gaussian fluctuation of the eigenvalues around the limits in the three dimensional case. The proof of homogenization is based on the analysis of the Wiener sausage whereas the fluctuation result is proved by a rather heavy perturbation method.
We propose a probabilistic approach to the fluctuation result based on a martingale central limit theorem. It is carried out in a slightly different setting where the Laplacian is perturbed by a random potential, yielding a central limit theorem in general dimensions. Our results partially extend a previous work by Bal, based on the perturbation method, which covers the dimensions less than or equal to three.
Based on a joint work with Marek Biskup and Wolfgang König.