Principal Eigenvalue and Landscape Function of the Anderson Model on a Large Box

Abstract

We state a precise formulation of a conjecture concerning the product of the principal eigenvalue and the sup-norm of the landscape function of the discrete Anderson model restricted to a large box. We first provide the asymptotic of the principal eigenvalue as the size of the box grows, and then use it to give a partial proof of the conjecture. For the one dimensional case, we give a complete proof by means of Green function bounds.