Shape and size of clusters in the Ising model

Abstract

The cyclomatic number of a cluster is introduced as a measure of its degree of compactness or ramification. Using Monte Carlo data for a two-dimensional Ising model, estimates are given of the average number of spins and the average number of cycles per cluster as a function of temperature. The results are related to the Whitney polynomial studied recently by Temperley and Lieb (1971). An exact calculation by these authors at the critical temperature enables the pattern of behaviour in the critical region to be conjectured.