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

Abstract

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.

Spectral zeta function on discrete tori and Epstein-Riemann conjecture

“…In the special case of a discrete torus, [CJK10] as well as the second named author in [Ver17] identified the constant term in that expansion in terms of the zetaregularized determinant of the Laplace-Beltrami operator. This relation to the zeta-regularized determinant was shown to be true in a much more general setting in [IzKh20]. Let us also mention [Sri15], which studied the asymptotic determinant for variations of the Riemannian metric, and a recent result in [TrSa19], where this problem was discussed on a symmetric discretization of surfaces glued together by a finite number of equilateral triangles.…”

“…Finally, an interesting question is if our main result, Theorem 1.10, can be generalized to other geometries in the spirit of [IzKh20] beyond tori.…”