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

Pierre Yves Gaudreau Lamarre, Yuanyuan Pan

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. Poincaré 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.

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

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 . We compute the small time asymptotics of the AH's exponential trace up to order , and of the PAM's mass up to order . Our proof is probabilistic, and relies on the asymptotics of intersection local times of Brownian motions and bridges in . Applications of our main result include the following: (i) If the boundary is sufficiently regular, then 's area and '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. Poincaré Probab. Statist. 58(3): 1385-1425). (ii) If is simply connected and is fractal, then '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.

Paper Structure

This paper contains 45 sections, 20 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. The Anderson Models
  3. Main Result
  4. Outline of Applications
  5. Outline of Proof
  6. Step 1: Construction of $H_\kappa$ and $u_\kappa$
  7. Step 2: Feynman-Kac Formulas
  8. Step 3: Intersection Local Times Asymptotics
  9. Organization
  10. Past Results, Motivation, and Applications
  11. Past Results
  12. Localization/Intermittency in Small Eigenvalues/Large Time
  13. Delocalization in Large Eigenvalues/Small Time
  14. Motivation and Applications
  15. Spectral Geometry
  16. ...and 30 more sections

Key Result

Theorem 1.3

Let $\kappa>0$ be fixed, and suppose that $H_\kappa$ and $u_\kappa$ are constructed as described in Assumption Assumption: AH. As $t\to0$, it holds that

Theorems & Definitions (43)

  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.4: KacPleijel
  • Remark 2.5
  • Corollary 2.6
  • Theorem 2.7: VanDenBerg
  • ...and 33 more