Schelling's model of segregation looks to explain the way in which particles or agents of two types in an initially random configuration may come to arrange themselves spatially into configurations consisting of large homogeneous clusters, i.e. connected regions consisting of only one type. As one of the earliest agent based models studied by economists and perhaps the most famous model of self-organising behaviour, it also has direct links to areas at the interface between computer science and statistical mechanics, such as the Ising model and the study of contagion and cascading phenomena in networks.
While the model has been extensively studied it has largely resisted rigorous analysis, prior results from the literature generally pertaining to variants of the model which are tweaked so as to be amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. In their STOC'12 paper, Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. We have now provided a rigorous analysis of the model's behaviour much more generally and have established some surprising forms of threshold behaviour, notably the existence of situations where an increased level of intolerance for neighbouring agents of opposite type leads almost certainly to decreased segregation.
Joint work with George Barmpalias and Richard Elwes.