[Paper Review] Analysis of the Anderson operator

Ismaël Bailleul, N. V. Dang+1 more|arXiv (Cornell University)|Jan 12, 2022

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper presents a self-contained functional analytic construction of the Anderson operator $ H = \Delta + \xi $ on a 2D compact Riemannian manifold, using a fixed-point approach to the resolvent and paracontrolled calculus at order one. It establishes almost sure spectral gap estimates, sharp Gaussian small-time asymptotics for the heat kernel, and introduces the Anderson Gaussian free field, whose partition function characterizes the spectrum of $ H $.

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.

Motivation & Objective

• To provide a self-contained, functional-analytic construction of the Anderson operator $ H = \Delta + \xi $ on a 2D closed Riemannian manifold $ \mathcal{S} $, avoiding reliance on full regularity structures.
• To establish almost sure spectral gap estimates under mild geometric assumptions on $ \mathcal{S} $, ensuring the operator has a discrete spectrum diverging to $ +\infty $.
• To derive sharp small-time asymptotics for the heat kernel of $ H $, which enables norm estimates for eigenfunctions and quasimodes.
• To introduce and characterize the Anderson Gaussian free field, showing its partition function fully determines the law of the spectrum of $ H $.
• To construct a polymer measure and a diffusion process (Anderson diffusion) on path space, and relate them to renormalized loop measures via Wick products.

Proposed method

• Construct the Anderson operator via a fixed-point equation for the resolvent, leveraging analytic Fredholm theory with a parameter.
• Use a paracontrolled ansatz at order one, relying only on the fundamental continuity estimate from Gubinelli, Imkeller & Perkowski (2012), avoiding higher-order paracontrolled calculus.
• Define the heat kernel $ p_t(x,y) $ of $ H $ as a solution to the parabolic Anderson equation with singular initial data, using the resolvent construction.
• Introduce the Anderson Gaussian free field as a random distribution whose Wick-ordered square relates to the renormalized occupation measure of a Poisson loop process.
• Construct the polymer measure and Anderson diffusion via path-space measures, with the latter arising as a time-changed diffusion process.
• Apply meromorphic Fredholm theory and Littlewood-Paley decompositions to control operator norms and convergence in pseudodifferential operator spaces.

Experimental results

Research questions

• RQ1 Can the Anderson operator $ H = \Delta + \xi $ be constructed as an unbounded self-adjoint operator on $ L^2(\mathcal{S}) $ using only functional analytic tools and low-order paracontrolled calculus?
• RQ2 What are the almost sure spectral gap estimates for $ H $, and how do they depend on the geometry of the 2D manifold $ \mathcal{S} $?
• RQ3 What is the small-time asymptotic behavior of the heat kernel $ p_t(x,y) $, and how does it yield norm estimates for eigenfunctions and quasimodes?
• RQ4 How does the law of the partition function of the Anderson Gaussian free field characterize the spectrum of $ H $?
• RQ5 What is the relationship between the Wick square of the Anderson Gaussian free field and the renormalized occupation measure of a Poisson process of diffusion paths?

Key findings

• The Anderson operator $ H $ is constructed as an unbounded self-adjoint operator on $ L^2(\mathcal{S}) $ via a fixed-point argument on the resolvent, requiring only first-order paracontrolled calculus.
• The spectrum of $ H $ is almost surely discrete and diverges to $ +\infty $, with the random eigenvalues forming continuous functions of the white noise $ \xi $.
• A sharp Gaussian small-time asymptotic is established for the heat kernel $ p_t(x,y) $, which implies $ p_t(x,y) \sim \frac{1}{4\pi t} e^{-|x-y|^2/(4t)} $ as $ t \to 0^+ $, up to a random correction involving the coupling function $ h $.
• The law of the partition function of the Anderson Gaussian free field uniquely characterizes the law of the spectrum of $ H $, providing a non-perturbative spectral invariant.
• The Wick square of the Anderson Gaussian free field equals the renormalized occupation measure of a Poisson process of diffusion paths, linking the field to stochastic processes.
• Large deviation principles are proven for the Anderson diffusion and its bridges, establishing the stochastic stability of the process under time and space scaling.