Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion
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| Format: | Preprint |
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2020
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| _version_ | 1866914414842609664 |
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| author | Finski, Siarhei |
| author_facet | Finski, Siarhei |
| contents | We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the sum of cycle-rooted spanning forests weighted by the monodromy of the connection of the unitary vector bundle, to the corresponding zeta-regularized determinants.
As one application, by combining our result with a recent work of Kassel-Kenyon, modulo some universal topological constants, we give an explicit formula for the limit of the probability that a cycle-rooted spanning forest with non-contractible loops, sampled uniformly on discretizations approaching a given surface, induces the given lamination by its cycles. We also calculate an explicit value for the limit of certain topological observables on the associated loop measures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_05162 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion Finski, Siarhei Probability Mathematical Physics Combinatorics Differential Geometry Functional Analysis 60B05, 51P05, 58A99, 82B20 We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the sum of cycle-rooted spanning forests weighted by the monodromy of the connection of the unitary vector bundle, to the corresponding zeta-regularized determinants. As one application, by combining our result with a recent work of Kassel-Kenyon, modulo some universal topological constants, we give an explicit formula for the limit of the probability that a cycle-rooted spanning forest with non-contractible loops, sampled uniformly on discretizations approaching a given surface, induces the given lamination by its cycles. We also calculate an explicit value for the limit of certain topological observables on the associated loop measures. |
| title | Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion |
| topic | Probability Mathematical Physics Combinatorics Differential Geometry Functional Analysis 60B05, 51P05, 58A99, 82B20 |
| url | https://arxiv.org/abs/2001.05162 |