A Polyakov Formula for Sectors

Aldana, Clara L.; Rowlett, Julie

doi:10.1007/s12220-017-9888-y

Cited by 10 publications

(35 citation statements)

References 46 publications

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“…Proof. We use a patchwork parametrix construction, as discussed in section 3.2 of [1]. This is a general technique that works to construct heat kernels whenever we have exact geometric matches for each part of a domain.…”

Section: Theorem 4 (Locality Principle For Neumann Boundary Condition)mentioning

confidence: 99%

“…This locality principle is incredibly useful, because if one has exact geometric matches for which one can explicitly compute the heat kernel, then one can use these to compute the short time asymptotic expansion of the heat trace. Moreover, in addition to being able to compute the heat trace expansion, one can also use this locality principle to compute the zeta regularized determinant of the Laplacian as in [1].…”

Section: Introductionmentioning

confidence: 99%

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How to Hear the Corners of a Drum

2019

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We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.

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Section: Theorem 4 (Locality Principle For Neumann Boundary Condition)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

How to Hear the Corners of a Drum

2019

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“…In particular, the boundary ∂Ψ get's tiled by the boundaries of the tiles and the angles between smooth components of the boundary are of the form kπ 2 , k ∈ N * \ {2}. Such surfaces are also called pillowcase covers, and in case if there is no boundary, they can be characterized as certain ramified coverings of CP 1 .…”

Section: Introductionmentioning

confidence: 99%

“…Remark that in [5] authors consider smooth manifolds, and thus their results cannot be directly applied for manifolds with conical singularities and corners. However, considerable advances has been done to extend the anomaly formula for manifolds in this case, see Aurell-Salomonson [2], Kokotov-Korotkin [29], Aldana-Rowlett [1], Kalvin [23].…”

Section: Introductionmentioning

confidence: 99%

Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

Finski¹

2020

Preprint

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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.

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“…where n ∈ N is arbitrary and δ α, π n denotes the Kronecker delta. It therefore follows that the list of examples given following Theorem 4 in [1] should be revised accordingly:…”

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confidence: 99%

Correction to: A Polyakov Formula for Sectors

Aldana

Rowlett

2019

J Geom Anal

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Let S α denote a finite circular sector of opening angle α ∈ (0, π) and radius one, and let e −t α denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term + 14π . This calculation will appear in [2]. The corrected statements of these theorems are given below.

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A Polyakov Formula for Sectors

Cited by 10 publications

References 46 publications

How to Hear the Corners of a Drum

How to Hear the Corners of a Drum

Spanning trees, cycle-rooted spanning forests on discretizations of flat surfaces and analytic torsion

Correction to: A Polyakov Formula for Sectors

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