This thesis is organized as two chapters whose contents are closely related yet quite distinct. The first chapter presents a paper "Role of 'Vision' in Neighborhood Racial Segregation: A Variant of the Schelling Segregation Model," authored by myself and Dr. Jaggi, which has been accepted for publication by the journal Urban Studies and is currently in press (as of April 2003). This chapter introduces the well-known Schelling model of neighborhood segregation, outlines the sociopolitical motivation for our work, and presents the key results that we believe are of interest to social scientists. Chapter two, which ought to be of greater interest to the physics community, presents the results of our investigations into the parallels between the Schelling model and critical phenomena.
Our primary extension of the Schelling model was to include social agents who can authentically 'see' their neighbors up to a distance R, called 'vision'. By exploring the consequences of systematically varying R, we have developed an understanding of how vision interacts with racial preferences and minority concentrations and leads to novel, complex segregation behavior. We have discovered three regimes: an unstable regime, where societies invariably segregate; a stable regime, where integrated societies remain stable; and an intermediate regime where a complex behavior is observed.
Since the primary audience of Urban Studies consists of sociologists and economists, we have not elaborated in the first chapter upon the phase transition which was strongly suggested by the "complex behavior" in the intermediate regime. The purpose of chapter two then, is to elucidate these additional physically interesting aspects of our model. Melting is a textbook example of first order (discontinuous) phase transitions. These are marked by two central features: a sharp temperature at which the transition occurs, and the coexistence of the two phases at that melting point. One can study the first-order phase transition that ice undergoes when melting into water by observing the ice while continuously raising its temperature. However, if you were only able to view the system at certain discrete temperatures, you would only see a either a piece of ice or a puddle of water during each observation. Thus in order to study the potential phase transition occurring in our model, we must be able to control the governing parameters continuously. However, in our original 'discrete' model, R measures how far an agent sees from its own home as an integer number of houses. Since we can only assign discrete values to R, it is meaningless to speak of a phase transition occurring as a function of this variable.
To overcome the limitations of our first model, we introduce a continuous model in chapter two where the range of vision (denoted R2 for notational clarity) can be varied continuously. This model uses a utility function that assigns greater weight to neighbors nearer an evaluating agent. The function used to model this decrease in utility contribution with distance is an exponentially decaying curve. We control the steepness of this curve (and thereby control the agents' vision) using R2. Since R2 can be set to equal any positive real number, we can indeed study the possible phase transition in our simulations' behavior as the function of a continuous variable.
Additionally, the continuous model demonstrates the robustness of the sociologically relevant conclusions drawn in chapter one. Our continuous model, a generalization of a model developed by Wasserman and Yohe (2001), is in fact more realistic than our first model. In particular, we were pleased to discover the same three behavioral regimes and all associated trends in both our discrete model and our continuous model. This confirms that our original results were robust and not merely algorithmic artifacts related to the specific treatment of vision used in our discrete model.
Laurie '03, Alexander J., "Phase Transitions Occurring in Models of Neighborhood Racial Segregation" (2003). Honors Projects. Paper 5.