Finite difference method on flat surfaces with a flat unitary vector bundle
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866912979244548096 |
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| author | Finski, Siarhei |
| author_facet | Finski, Siarhei |
| contents | We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions.
As an interesting byproduct of our study, we obtain Harnack-type estimates on "almost harmonic" discrete functions, defined on the graphs, which approximate a given surface.
The results of this paper will be later used to relate the asymptotic expansion of the number of spanning trees, spanning forests and weighted cycle-rooted spanning forests on the discretizations to the corresponding zeta-regularized determinants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2001_04862 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Finite difference method on flat surfaces with a flat unitary vector bundle Finski, Siarhei Differential Geometry Functional Analysis Spectral Theory 58A99, 47N30, 31A05 We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, we obtain Harnack-type estimates on "almost harmonic" discrete functions, defined on the graphs, which approximate a given surface. The results of this paper will be later used to relate the asymptotic expansion of the number of spanning trees, spanning forests and weighted cycle-rooted spanning forests on the discretizations to the corresponding zeta-regularized determinants. |
| title | Finite difference method on flat surfaces with a flat unitary vector bundle |
| topic | Differential Geometry Functional Analysis Spectral Theory 58A99, 47N30, 31A05 |
| url | https://arxiv.org/abs/2001.04862 |