Asymptotic analysis of determinant of discrete Laplacian

Hou, Yingwei; Kandel, Santosh

doi:10.1007/s11005-019-01208-5

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…Recently, Finski [14,15] obtained, by a different method, a slightly weaker version of Theorem 1.1 in the case of the square lattice quadrangulations of Riemann surfaces with Neumann boundary conditions and cone angles restricted to integer multiples of π. For other related recent work, see [17,26,27,29,10] Theorem 1.1 above is both sharper and more general than the previous results, and we propose a new, relatively short and elementary proof. The idea is similar to that used by Chinta-Jorgenson-Karlsson [5,6] and Friedli [16] who studied the square lattice Laplacians on a torus: we use an integral representation for log det ∆ Ω δ ,ϕ in terms of theta function and then break the integral into parts that we analyze separately.…”

Section: Introductionsupporting

confidence: 49%

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Izyurov¹,

Khristoforov²

2020

Preprint

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Consider a surface Ω with a boundary obtained by gluing together a finite number of equilateral triangles, or squares, along their boundaries, equipped with a flat unitary vector bundle. Let Ω δ be the discretization of this surface by a bi-periodic lattice with enough symmetries, scaled to have mesh size δ. We show that the logarithm of the product of non-zero eigenvalues of the discrete Laplacian acting on the sections of the bundle is asymptotic toHere A and B are lattice-dependent constants; C is an explicit constant depending on the bundle, the angles at conical singularities and at corners of the boundary, and D is a sum of lattice-dependent contributions from singularities and a universal term that can be interpreted as a zeta-regularization of the continuum Laplacian on Ω. We allow for Dirichlet or Neumann boundary conditions, or mixtures thereof. Our proof is based on an integral formula for the determinant in terms of theta function, and the functional Central limit theorem.

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Section: Introductionsupporting

confidence: 49%

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Izyurov¹,

Khristoforov²

2020

Preprint

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“…On the physics side, its importance stems from its role as a partition function of conformal field theories, and in particular, from its conformal transformation properties given by the Polyakov–Alvarez formula [ 2 , 39 , 40 ], see also [ 1 ] and the references therein for the most recent developments. For other related recent work, see [ 12 , 22 , 43 , 44 , 46 ].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the Determinant of Discrete Laplacians on Triangulated and Quadrangulated Surfaces

Izyurov

Khristoforov

2022

Commun. Math. Phys.

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Consider a surface $$\Omega $$ Ω with a boundary obtained by gluing together a finite number of equilateral triangles, or squares, along their boundaries, equipped with a vector bundle with a flat unitary connection. Let $$\Omega ^{\delta }$$ Ω δ be a discretization of this surface, in which each triangle or square is discretized by a bi-periodic lattice of mesh size $$\delta $$ δ , possessing enough symmetries so that these discretizations can be glued together seamlessly. We show that the logarithm of the product of non-zero eigenvalues of the discrete Laplacian acting on the sections of the bundle is asymptotic to $$\begin{aligned} A|\Omega ^{\delta }|+B|\partial \Omega ^{\delta }|+C\log \delta +D+o(1). \end{aligned}$$ A | Ω δ | + B | ∂ Ω δ | + C log δ + D + o ( 1 ) . Here A and B are constants that depend only on the lattice, C is an explicit constant depending on the bundle, the angles at conical singularities and at corners of the boundary, and D is a sum of lattice-dependent contributions from singularities and a universal term that can be interpreted as a zeta-regularization of the determinant of the continuum Laplacian acting on the sections of the bundle. We allow for Dirichlet or Neumann boundary conditions, or mixtures thereof. Our proof is based on an integral formula for the determinant in terms of theta function, and the functional Central limit theorem.

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Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

Greenblatt

2023

Journal of Mathematical Physics

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For [Formula: see text], a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on [Formula: see text] with Dirichlet boundary conditions has an asymptotic expression for large L involving the zeta-regularized determinant of the associated continuum Laplacian. When Π is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.

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Asymptotic analysis of determinant of discrete Laplacian

Cited by 3 publications

References 12 publications

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces

Asymptotics of the Determinant of Discrete Laplacians on Triangulated and Quadrangulated Surfaces

Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

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