Weak-Graph Method for Obtaining Formal Series Expansions for Lattice Statistical Problems

Abstract

A unified exposition of the weak‐graph method for obtaining formal series expansions for lattice statistical problems is presented. The prototype of this method is the derivation of the hyperbolictangent high‐temperature expansion for the spin‐½ Ising model. Also, recent expansions of the monomer‐dimer problem and various hydrogen‐bonded problems have been treated by essentially the same method. In this paper the method is further illustrated by obtaining series expansions for the low‐temperature spin‐½ Ising problem, the low‐density hard‐core lattice‐gas problem, the high‐temperature spin‐1 Ising problem, the k‐color problem, and two new model problems, the ramrod model and a special ternary model. The weak‐graph method enables one to obtain especially useful series expansions for a certain class of problems, including the spin‐½ Ising problem and the monomer‐dimer problem, which have essentially a binary nature.