Spanning tree generating functions and Mahler measures

Abstract

We define the notion of a spanning tree generating function (STGF) 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. 1

Correspondence between spanning trees and the Ising model on a square lattice

“…In what follows, we adopt some of Typeset by REVT E X the notation of ref. [9]. Let Λ L denote the random walk structure function for the lattice L, i.e., the Fourier transform of the discrete step probability distribution, and let z L denote the coordination number of the lattice.…”

“…(57) in ref. [9]). The question we address here is whether the above relation holds only at the critical point or whether it generalizes in some way.…”

“…The question we address here is whether the above relation holds only at the critical point or whether it generalizes in some way. Specifically, we ask here whether or not (9) can be extended to non-critical temperatures. It is well known that the partition function of the Ising model is closely related to the generating function of "loops" or multipolygons on the square lattice, i.e.…”

“…What is the correct or "natural" way to define this spanning tree generating function that generalizes the spanning tree constant? Fortunately, the answer is known [9,13].…”

“…Consider for example the simple cubic lattice. The random walk structure function and the spanning tree generating function are known [9,13]. Specifically, the structure function for the simple cubic lattice is given by…”