Spanning trees for three-dimensional lattices.
by Professor Tony Guttmann
Abstract: The number of spanning trees on a regular graph (lattice) grows exponentially with the size of the lattice. The spanning tree constant is a characteristic of the lattice, and varies from lattice to lattice. This constant is known for two-dimensional lattices, but not generally for three-dimensional lattices (apart from the b.c.c. which is a trivial exception). We have recently derived the spanning tree constants for all the usual three-diemnsional lattices. The method used is to make use of some results in number theory pertaining to logarithmic Mahler measures of Laurent polynomials.
In as yet unpublished work we have recently evaluated a similar integral that arises in the study by Baxter and Bazhanov of the sl(n) Potts model on the simple cubic lattice.
This is joint work with Mathew Rogers of Univ. de Montreal.