Schelling’s model of segregation looks to explain the way in which particles or agents of two types 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. Recently Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. In the following sequence of papers my co-authors George Barmpalias, Richard Elwes and I provide a rigorous analysis of the model’s behaviour much more generally and establish some surprising forms of threshold behaviour, for the two and three dimensional as well as the one-dimensional model.
Digital morphogenesis via Schelling segregation, with Barmpalias and Elwes, FOCS 2014, 55th Annual IEEE Symposium on Foundations of Computer Science, Oct. 18-21, Philadelphia, pdf.
Tipping points in Schelling segregation, with Barmpalias and Elwes, Journal of Statistical Physics 158, 806-852, 2016, pdf.
From randomness to order: Schelling segregation in two or three dimensions, with Barmpalias and Elwes, Journal of Statistical Physics 164 (6), 1460-1487, 2016, pdf.
Minority population in the one-dimensional Schelling model of segregation, with Barmpalias and Elwes, Journal of Statistical Physics 173(5), 1408–1458, 2018, pdf.