Spanning Trees on Lattices and Integration Identities

Abstract

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