Spanning Trees on Lattices and Integration Identities

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.

15 pages, 3 figures, 1 table

References in corpus (4)

• Spanning Trees on Graphs and Lattices in d Dimensions
• Lower Bounds and Series for the Ground State Entropy of the Potts Antiferromagnet on Archimedean Lattices and their Duals
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• Some Exact Results for Spanning Trees on Lattices

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• Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs
• Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices
• Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices