Asymptotic Determinant of Discrete Laplace-Beltrami Operators

TL;DR

This paper investigates the asymptotic behavior of determinants of discrete Laplace-Beltrami operators on graph approximations of Riemannian manifolds, linking combinatorial Laplacians to scalar field theories.

Contribution

It introduces a framework for defining Riemannian metrics on graphs and analyzes the asymptotic determinants of associated Laplacians as graph mesh size tends to zero.

Findings

Derived asymptotic formulas for determinants of discrete Laplacians

Connected combinatorial Laplacians with scalar field theory partition functions

Provided insights into graph-based approximations of geometric operators

Abstract

We study combinatorial Laplacians on rectangular subgraphs of $ \epsilon \mathbb{Z}^2 $ that approximate Laplace-Beltrami operators of Riemannian metrics as $ \epsilon \rightarrow 0 $. These laplacians arise as follows: we define the notion of a Riemmanian metric structure on a graph. We then define combinatorial free field theories and describe how these can be regarded as finite dimensional approximations of scalar field theory. We focus on the Gaussian field theory on rectangular subgraphs of $ \mathbb{Z}^2 $ and study its partition function by computing the asymptotic determinant of the discrete laplacian.