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

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.