On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains

TL;DR

This paper analyzes the small time asymptotics of the Anderson Hamiltonian and parabolic Anderson model on planar domains, revealing how geometric features can be inferred from spectral data.

Contribution

It provides probabilistic methods to recover geometric and spectral properties of planar domains from the eigenvalues of Anderson models.

Findings

Eigenvalues determine the area and boundary length of regular domains.

Minkowski dimension of fractal boundaries can be inferred from small time asymptotics.

Variance of white noise can be recovered from spectral data.

Abstract

We consider the Anderson Hamiltonian (AH) and the parabolic Anderson model (PAM) with white noise and Dirichlet boundary condition on a bounded planar domain $D\subset\mathbb R^2$. We compute the small time asymptotics of the AH's exponential trace up to order $O(\log t)$, and of the PAM's mass up to order $O(t\log t)$. Our proof is probabilistic, and relies on the asymptotics of intersection local times of Brownian motions and bridges in $\mathbb R^2$. Applications of our main result include the following: (i) If the boundary $\partial D$ is sufficiently regular, then $D$'s area and $\partial D$'s length can both be recovered almost surely from a single observation of the AH's eigenvalues. This extends Mouzard's Weyl law in the special case of bounded domains (Ann. Inst. H. Poincar\'e Probab. Statist. 58(3): 1385-1425). (ii) If $D$ is simply connected and $\partial D$ is fractal, then $\partial D$'s Minkowski dimension (if it exists) can be recovered almost surely from the PAM's small time asymptotics. (iii) The variance of the white noise can be recovered almost surely from a single observation of the AH's eigenvalues.