论文标题
离散Laplacians决定因素在三角剖分和四边形表面上的渐近差异
Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces
论文作者
论文摘要
考虑一个表面$ω$，其边界通过将有限数量的等边三角形或正方形粘合在一起，并沿其边界配备了平坦的统一矢量束。令$ω^δ$为具有足够对称性的双周期晶格对该表面的离散化，并缩放以具有网格尺寸$δ$。我们表明，作用于束的截面的离散laplacian非零特征值的乘积的对数是渐近的[a |ω^e |+b |+b | \partialΩ \]这里$ a $和$ b $是依赖格子的常数； $ c $是一个明确的常数，具体取决于捆绑包，圆锥形奇异性和边界角的角度，而$ d $是奇异性的依赖格子的贡献的总和，可以将其解释为Zeta-laplacian zeta-continuum laplacian in $ω$。我们允许Dirichlet或Neumann边界条件或其混合物。我们的证明是基于根据theta函数和功能性中心极限定理的确定因素的积分公式。
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 to \[ A|Ω^δ|+B|\partialΩ^δ|+C\logδ+D+o(1). \] Here $A$ 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.
