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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.

Author Bio

I am currently a senior at Saint Michael㤼㸲s College. I am from Pittsfield, Massachusetts. I love to ski, play softball, and play the drums. I also love mathematics. This is why when I heard about the opportunity to work at Saint Michael’s for the summer conducting mathematical research I quickly told my professor of my interest. She gave me a job and I soon started to investigate the Potts model full time. My two main goals were to develop an understanding of the mathematics involved in the Potts model and to explore as many applications of the model as I could find. My professor eventually wanted me to write a paper on the model that could be understood and appreciated by undergraduate mathematics majors.

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