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

Abstract

Consider a surface Omega 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 } be a discretization of this surface, in which each triangle or square is discretized by a bi-periodic lattice of mesh size delta , 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 A|Ωδ|+B|∂Ωδ|+Clogδ+D+o(1).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} A|\\Omega ^{\\delta }|+B|\\partial \\Omega ^{\\delta }|+C\\log \\delta +D+o(1). \\end{aligned}$$\\end{document}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.