Weighted graphs, spanning tree generating functions and anisotropic lattice systems: illustrative results for the Ising and dimer models

Abstract

Abstract Anisotropic lattice models are mathematically more involved and often phenomenologically richer than the isotropic counterparts. As a consequence, their analyses tend to be considerably more demanding with a smaller number of exact results available. It has been shown that certain important quantities for isotropic statistical physics models on a lattice L can be cast in terms of the spanning tree generating function STGF (a pure combinatorial topological function) of L. A possible way to formulate an anisotropic lattice model is by defining it on a weighted lattice. Very recently (2021 J. Stat. Mech. 073104), it has been speculated that if a STGF could be extended to such lattices, perhaps the previously mentioned association would hold for the anisotropic case as well. Hence, the aim of this contribution is twofold. To properly define and construct a weighted spanning tree generating function wSTGF for general periodic lattices. To show that the free energy for the anisotropic Ising and dimer models can be mapped onto the wSTGF for some particular, but representative, lattice structures. These findings might represent a novel approach to treat such class of problems.