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

2025/07/08 by Pierre Yves Gaudreau Lamarre, Yuanyuan Pan, Lamarre, Pierre Yves Gaudreau +1
Economics, Econometrics and Finance · Mathematics · #58J50 #60H25 #60L50 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2507.06186

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊂\mathbb R². 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(tlog t). Our proof is probabilistic, and relies on the asymptotics of intersection local times of Brownian motions and bridges in \mathbb R². Applications of our main result include the following: (i) If the boundary ∂ D is sufficiently regular, then D's area and ∂ 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 ∂ D is fractal, then ∂ 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.

Citations

• Fractal curvatures and short-time asymptotics of heat content
• A simple construction of the Anderson operator via its quadratic form in dimensions two and three
• Anderson Hamiltonians with singular potentials
• Moment Intermittency in the PAM with Asymptotically Singular Noise
• Analysis of the Anderson operator
• Localization crossover for the continuous Anderson Hamiltonian in 1-d
• The delocalized phase of the Anderson Hamiltonian in 1-d
• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Moment estimates for some renormalized parabolic Anderson models
• Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions
• On the trace of Schrödinger heat kernels and regularity of potentials
• Moments of 2D Parabolic Anderson Model
• The parabolic Anderson model on Riemann surfaces
• The continuous Anderson hamiltonian in dimension two
• Paracontrolled distributions and singular PDEs
• Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm
• The parabolic Anderson model

Related

• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Fractal geometry of the PAM in 2D and 3D with white noise potential
• Parabolic Anderson model in bounded domains of recurrent metric measure spaces
• Parabolic Anderson model with rough noise in space and rough initial conditions
• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model