  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. In \cite{BK},
Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of
the unperturbed model, for a specific set of input parameters. Here we provide
a rigorous analysis of the model's behaviour much more generally and establish
some surprising forms of threshold behaviour, notably the existence of
situations where an \emph{increased} level of intolerance for neighbouring
agents of opposite type leads almost certainly to \emph{decreased} segregation.
