Some Counting Theorems in the Theory of the Ising Model and the Excluded Volume Problem

The problem of the exact enumeration of self-avoiding random walks on a lattice is studied and a theorem derived that enables the number of such walks to be calculated recursively from the number of a restricted class of closed graphs more easily enumerated than the walks themselves. The method of Oguchi for deriving a high-temperature expansion for the zero-field susceptibility of the Ising model is developed and a corresponding theorem enabling the successive coefficients to be calculated recursively from a restricted class of closed graphs deduced. The theorem relates the susceptibility to the configurational energy and enables the behavior of the antiferromagnetic susceptibility at the transition point to be inferred.