A Review of the Potts Model

Abstract

This paper examines a mathematical modeling tool for complex systems with nearest neighbor interactions known as the Potts model. We begin by explaining the structure of the model and defining its Hamiltonian, probability function, and partition function. We then focus on the partition function, giving examples and showing the equivalence of two different formulations. We then introduce the Tutte polynomial, a well known graph invariant. We give details of the equivalence of the Tutte polynomial and the Potts model partition function. Since the Tutte polynomial, and hence the Potts model partition function, is computationally intractable, we explore Monte Carlo simulations of the Potts model. Finally, we discuss three applications illustrating how these simulations model real world situations. The Potts Model Page 2 5/4/2007