Analysis of the Anderson operator

We consider the continuous Anderson operator H=−Δ+ξ on a closed Riemannian compact surface S. We provide a short self-contained functional analytic construction of the operator as an unbounded operator on L2(S) based on its resolvent as a meromorphic family of operators. Our main result is a precise description of the Anderson heat semigroup (e−tH)t>0 with two-sided Gaussian bounds and sharp Gaussian small time asymptotics for its kernel with a number of consequences on the spectrum of H. Using these results, we introduce and study the associated Gaussian field that we call the Anderson Gaussian free field and prove that the law of its random partition function characterizes the law of the spectrum of H. We also give a construction of two measures on path space associated to the Anderson operator, the polymer measure and the ground state diffusion, as path in the random environment given by ξ. We relate the Wick square of the Anderson Gaussian free field to the renormalized occupation measure of a Poisson process of loops of diffusion paths and we further prove some large deviation results for the Anderson diffusion and its bridges.