Exact solution for a metapopulation version of Schelling's model

In 1971, Schelling introduced a model in which families move if they have too many neighbors of the opposite type. In this paper, we will consider a metapopulation version of the model in which a city is divided into N neighborhoods, each of which has L houses. There are ρNL red families and ρNL blue families for some ρ < 1/2. Families are happy if there are ≤ ρ(c)L families of the opposite type in their neighborhood and unhappy otherwise. Each family moves to each vacant house at rates that depend on their happiness at their current location and that of their destination. Our main result is that if neighborhoods are large, then there are critical values ρ(b) < ρ(d) < ρ(c), so that for ρ < ρ(b), the two types are distributed randomly in equilibrium. When ρ > ρ(b), a new segregated equilibrium appears; for ρ(b) < ρ < ρ(d), there is bistability, but when ρ increases past ρ(d) the random state is no longer stable. When ρ(c) is small enough, the random state will again be the stationary distribution when ρ is close to 1/2. If so, this is preceded by a region of bistability.