Some Exact Results for Spanning Trees on Lattices

TL;DR

This paper derives exact asymptotic growth constants for spanning trees on various high-dimensional lattices, including body-centered cubic, face-centered cubic, and a specific 4-8-8 lattice, advancing understanding of lattice combinatorics.

Contribution

It provides the first exact closed-form results for the asymptotic growth constants of spanning trees on these complex lattices.

Findings

Exact growth constant for bcc(d) lattices derived

Integral expression for fcc lattice growth constant provided

Closed-form expression for 4-8-8 lattice growth constant obtained

Abstract

For $n$-vertex, $d$-dimensional lattices $\Lambda$ with $d \ge 2$, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present an exact closed-form result for the asymptotic growth constant $z_{bcc(d)}$ for spanning trees on the $d$-dimensional body-centered cubic lattice. We also give an exact integral expression for $z_{fcc}$ on the face-centered cubic lattice and an exact closed-form expression for $z_{488}$ on the $4 \cdot 8 \cdot 8$ lattice.