Spanning tree generating functions and Mahler measures

Abstract

We define the notion of a spanning tree generating function (STGF) $\sum a_n{z}^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.

26 pages. Dedicated to F Y Wu on the occasion of his 80th birthday. This version has additional references, additional calculations, and minor corrections

References in corpus (3)

• Some Exact Results for Spanning Trees on Lattices
• Spanning Trees on Lattices and Integration Identities
• Exact Solution of a Monomer-Dimer Problem: A Single Boundary Monomer on a Non-Bipartite Lattice

Cited by in corpus (5)

• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• The double hypergeometric series for the partition function of the 2D anisotropic Ising model
• Correspondence between spanning trees and the Ising model on a square lattice
• An integral arising from the chiral sl(n) Potts model
• On rationality of generating function for the number of spanning trees in circulant graphs