Clustering and Mixing Times for Segregation Models on ℤ2

The Schelling Segregation Model was proposed by Thomas Schelling in 1971 as a means of explaining possible causes of racial segregation in cities. He considered residents of two types, say red and blue, where each person prefers the majority of his or her neighbors to have the same color. He showed through simulations that even mild preferences of this type can lead to segregation if residents move whenever they are not happy with their local environments. Economists have considered many natural variants on the model, with similar findings, but there has been very little rigorous work justifying the claims about segregation. Here we introduce a General Influence Model to capture some of these variants in a utillity function, which we study in the context of a reversible Markov chain in which the more unhappy residents are with the demographics of their local environment the more likely they are to move. We use insights from convergence rates of Markov chains to then characterize the stationary distributions to determine whether cities are likely to be integrated or segregated (or clustered). The General Influence Model considers open cities (where residents can move away) in a saturated or non-saturated setting (so we can allow unoccupied houses), with neighborhoods of any radius, and where moving is based on the product of everyone’s happiness. An individual’s happiness depends on the demographics in his local neighborhood, and the influence function dictates the probability of moving depending on this function. We show that for any influence function, the dynamics will be rapidly mixing and cities will be integrated (i.e., there will not be clustering) if the racial bias is sufficiently low. We also show complementary results for two broad classes of influence functions. The first is for Increasing Bias Functions (IBF), where an individual’s likelihood of moving increases each time someone of the other color moves close or someone of the same color leaves (this does not include Schelling’s threshold models). The second is for Threshold Bias Functions (TBF) when the threshold is at least one half, including the model that Schelling originally proposed. Here a resident is happy as long as the majority of his neighbors share his color, and is unhappy otherwise, regardless of the actual percentage. For both classes (IBF and TBF) we show that when the bias is sufficiently high, the dynamics take exponential time to mix and we will have segregation, which in the case of an open city means that a large ghetto will form. ∗Supported in part by NSF CCF-1219020; [email protected]. †Supported by a DOE Office of Science Graduate Fellowship and NSF CCF-1219020; [email protected]. ‡Supported in part by NSF CCF-1219020; [email protected].