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. The model is described precisely here.

Digital morphogenesis via Schelling segregation, to appear in FOCS 2014, pdf.

Tipping points in Schelling segregation, to appear in the Journal of Statistical Physics, pdf.

From randomness to order: Schelling segregation in two or three dimensions, pdf.

The typical Turing degree, Proceedings of the London Mathematical Society, Dec 2013, pdf.

A * computable structure * is given by a computable domain, and then a set of computable relations and functions defined on that domain.
The study of computable structures, going back as far as the work of Frohlich and Shepherdson, Rabin, and Malcev is part of a long-term
programme to understand the algorithmic content of mathematics.

In mathematics generally, the notion of isomorphism is used to determine structures which are * essentially the same. *
Within the context of effective (algorithmic) mathematics, however, one is presented with the possibility that pairs of computable structures
may exist which, while isomorphic, fail to have a * computable * isomorphism between them.
Thus the notion of * computable categoricity * has become of central importance: a computable structure S
is computably categorical if any two computable presentations A and B of S are computably isomorphic.
In this paper, my co-authors Downey, Kach, Lempp, Montalban, Turetsky and I, answer one of the longstanding questions in computable structure theory, showing the class of computably categorical structures has
no simple structural or syntactic
characterisation.

The complexity of computable categoricity, to appear in Advances in Mathematics, pdf.