Analysis of the Anderson operator

TL;DR

The paper constructs and analyzes the 2D Anderson operator $H=Δ+ξ$ on a closed surface, establishing a rigorous self‑adjoint realization with discrete spectrum through renormalization and meromorphic Fredholm theory. It proves sharp two‑sided heat‑kernel bounds, small‑time asymptotics, Schauder estimates, and a Weyl law, and introduces the Anderson Gaussian free field $φ$ whose Wick square encodes spectral information via the partition function $Z(λ)=\mathbb{E}[e^{-λ<:φ^2:,1>}] = {\det}_2(\mathrm{Id}+λ(H+c)^{-1})^{-1/2}$. The work further develops probabilistic structures: a polymer measure and an Anderson diffusion, connected to loop occupation measures through Le Jan’s identity, and derives large‑deviation results for the polymer measure. Overall, it provides a comprehensive analytic and probabilistic framework for a rough singular operator in 2D, with implications for singular SPDEs, random Gibbs measures, and spectral analysis under randomness.

Abstract

We consider the continuous Anderson operator $H=Δ+ξ$ on a two dimensional closed Riemannian manifold $\mathcal{S}$. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on $L^2(\mathcal{S})$ and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of $H$ that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called 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 simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges.