The pseudoinverse of the Laplacian matrix: Asymptotic behavior of its trace

TL;DR

This paper investigates the asymptotic behavior of the trace of the pseudoinverse of the Laplacian matrix for a square lattice as the size grows, improving error estimates with higher-degree Taylor approximations.

Contribution

It develops a method using fourth-degree Taylor polynomials to achieve more precise asymptotic estimates of the Laplacian pseudoinverse trace for large lattices.

Findings

Derived an asymptotic formula with improved error bounds

Demonstrated the effectiveness of higher-degree Taylor approximations

Enhanced understanding of Laplacian pseudoinverse behavior in large grids

Abstract

In this paper we are concerned with the asymptotic behavior of \[ \operatorname{tr}(\mathcal{L}^+_{\rm sq}) = \frac{1}{4} \sum_{j,k=0 \atop (j,k) \neq (0,0)}^{n-1} \frac{1}{1-\frac{1}{2} \big( \cos \frac{2\pi j}{n} + \cos \frac{2\pi k}{n} \big)}, \] the trace of the pseudoinverse of the Laplacian matrix related with the square lattice, as $n \to \infty$. The method we developed for such sums in former papers depends on the use of Taylor approximations for the summands. It was shown that the error term depends on whether the Taylor polynomial used is of degree two or higher. Here we carry this out for the square lattice with a fourth degree Taylor polynomial and thereby obtain a result with an improved error term which is perhaps the most precise one can hope for.