On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Lamarre, Pierre Yves Gaudreau Pan, Yuanyuan |
| author_facet | Lamarre, Pierre Yves Gaudreau Pan, Yuanyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains Lamarre, Pierre Yves Gaudreau Pan, Yuanyuan Probability Mathematical Physics Spectral Theory 60H25, 60L50, 58J50 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. |
| title | On the Spectral Geometry and Small Time Mass of Anderson Models on Planar Domains |
| topic | Probability Mathematical Physics Spectral Theory 60H25, 60L50, 58J50 |
| url | https://arxiv.org/abs/2507.06186 |