Exact solution for a metapopulation
version of Schelling's model
Rick Durrett and Yuan Zhang
Abstract.
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 ρ NL red families and
ρ NL blue families. Families are happy if there are ≤ ρc 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.
Let Tri(pR,pB) be a trinomial distribution with probability pR and pB of red and blue,
and probability 1-pR-pB of empty. Suppose first that ρc >0.25963. In this case,
if neighborhoods are large then there are critical values ρb, and ρd so that
- For ρ < ρb the two types are distributed randomly in equilibrium, i.e, neighborhoods are Tri(ρ,ρ).
- When ρ > ρb a new segregated equilibrium (1/2)Tri(ρ1,ρ2)
+(1/2)Tri(ρ2,ρ1) appears.
- For ρd < ρ < ρd there is a bistability, but beyond that point the segregated state
is the unique stationary distribution. When ρc ≤ 0.25963,
Tri(ρ,ρ) may be the stationary distribution when ρ
is close to 1/2, and if so there is a region of bistability.
New version July 2014
Back to Durrett's home page