Asymptotics for the determinant of the combinatorial Laplacian on hypercubic lattices

TL;DR

This paper derives asymptotic formulas for the determinant of the combinatorial Laplacian on high-dimensional lattices, linking it to spanning trees and forests, and provides explicit asymptotics for various related graph structures.

Contribution

It introduces new asymptotic results for the Laplacian determinant on hypercubic lattices and relates spanning tree counts to other lattice structures and boundary conditions.

Findings

Asymptotic formulas for the Laplacian determinant on d-dimensional lattices.

Explicit asymptotics for spanning forests in these graphs.

Connections between spanning trees in square lattices and other combinatorial structures.

Abstract

In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of $d$-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the number of spanning trees by the well-known matrix tree theorem. Asymptotics for $2$ and $3$ component rooted spanning forests in these graphs are also derived. Moreover, we express the number of spanning trees in a $2$-dimensional square lattice in terms of the one in a $2$-dimensional discrete torus and also in the quartered Aztec diamond. As a consequence, we find an asymptotic expansion of the number of spanning trees in a subgraph of $\mathbb{Z}^2$ with a triangular boundary.