Spanning Trees on Lattices and Integration Identities

TL;DR

This paper derives exact integral formulas for the asymptotic growth constant of spanning trees on various lattices, revealing multiple integration identities and extending results to expanded lattices with inserted vertices.

Contribution

It provides new exact integral expressions for the growth constants of spanning trees on different lattices and explores their relationships through integration identities.

Findings

Exact integral expressions for $z_ ext{Lambda}$ on several lattices.

Many integration identities derived from different unit cell choices.

Extension of results to homeomorphic expansions with inserted vertices.

Abstract

For a lattice $\Lambda$ with $n$ vertices and dimension $d$ equal or higher than two, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present exact integral expressions for the asymptotic growth constant $z_\Lambda$ for spanning trees on several lattices. By taking different unit cells in the calculation, many integration identities can be obtained. We also give $z_{\Lambda (p)}$ on the homeomorphic expansion of $k$-regular lattices with $p$ vertices inserted on each edge.