Exact solution for a metapopulation version ofSchelling’s modelRichard Durrett1 and Yuan Zhang

Department of Mathematics, Duke University, Durham, NC 27708

Contributed by Richard Durrett, August 4, 2014 (sent for review July 23, 2014)

In 1971, Schelling introduced a model in which families move ifthey have too many neighbors of the opposite type. In thispaper, we will consider a metapopulation version of the model inwhich a city is divided into N neighborhoods, each of which has Lhouses. There are ρNL red families and ρNL blue families for someρ < 1/2. Families are happy if there are ≤ρcL families of the op-posite type in their neighborhood and unhappy otherwise. Eachfamily moves to each vacant house at rates that depend on theirhappiness at their current location and that of their destination.Our main result is that if neighborhoods are large, then there arecritical values ρb < ρd < ρc, so that for ρ < ρb, the two types aredistributed randomly in equilibrium.When ρ > ρb, a new segregatedequilibrium appears; for ρb < ρ < ρd, there is bistability, but when ρincreases past ρd the random state is no longer stable. When ρc issmall enough, the random state will again be the stationary dis-tribution when ρ is close to 1/2. If so, this is preceded by a regionof bistability.

segregation | large deviations

In 1971, Schelling (1) introduced one of the first agent-basedmodels in the social sciences. Families of two types inhabit cells

in a finite square, with 25–30% of the squares vacant. Eachfamily has a neighborhood that consists of a 5 × 5 squarecentered at their location. Schelling used a number of differentrules for picking the next family to move, but the most sensibleseems to be that we pick a family at random on each step. If thefraction of neighbors of the opposite type is too large, thenthey move to the closest location that satisfies their con-straints. Schelling simulated this and many other variants ofthis model (using dice and checkers) to argue that if peoplehave a preference for living with those of their own color, themovements of individual families invariably led to completesegregation (2).As Clark and Fossett (3) explain “The Schelling model was

mostly of theoretical interest and was rarely cited until a signifi-cant debate about the extent and explanations of residentialsegregation in US urban areas was engaged in the 1980s and1990s. To that point, most social scientists offered an explanationthat invoked housing discrimination, principally by whites.” Atthis point Schelling’s article has been cited more than 800 times.For a sampling of results from the social sciences literature, seeFossett’s lengthy survey (4) or other more recent treatments (5–7).About 10 y ago, physicists discovered this model and analyzeda number of variants of it using techniques of statistical me-chanics (8–14). However, to our knowledge, the only rigorouswork is ref. 15, which studies the 1D model in which thethreshold for happiness is ρc = 0.5 and two unhappy familieswithin distance w swap places at rate 1.Here, we will consider a metapopulation version of Schelling’s

model in which there are N neighborhoods that have L houses,but we ignore spatial structure within the neighborhoods andtheir physical locations. We do this to make the model analyti-cally tractable, but these assumptions are reasonable from amodeling point of view. Many cities in the United States aredivided into neighborhoods that have their own identities. InDurham, these neighborhoods have names like Duke Park,

Trinity Park, Watts-Hillendale, Duke Forest, Hope Valley, ColonyPark, etc. They are often separated by busy roads and haveidentities that are reinforced by e-mail newsgroups that allowpeople to easily communicate with everyone in their neighbor-hood. Because of this, it is the overall composition of theneighborhood that is important not just the people who live nextdoor. In addition, when a family decides to move they can easilyrelocate anywhere in the city.Families, which we suppose are indivisible units, come in two

types that we call red and blue. There are ρNL of each type,leaving (1 − 2ρ)NL empty houses. This formulation was inspiredby Grauwin et al. (16), who studied segregation in a model withone type of individual whose happiness is given by a piecewiselinear unimodal function of the density of occupied sites in theirneighborhood. To define the rules of movement, we introducethe threshold level ρc such that a neighborhood is happy for acertain type of agent if the fraction of agents of the oppositetype is ≤ρc. For each family and empty house, movementsoccur at rates that depend on the state of the source anddestination houses:

where q, r < 1, and e are small, e.g., 0.1 or smaller. Becausethere are O(NL) vacant houses, dividing the rates by NL makeseach family moves at a rate O(1). Because e is small, happyfamilies are very reluctant to move to a neighborhood in whichthey would be unhappy, whereas unhappy families move atrate 1 to neighborhoods that will make them happy. As we willsee later, the equilibrium distribution does not depend on thevalues of the rates q and r for transitions that do not change afamily’s happiness. We do not have an intuitive explanation forthis result.

From/to Happy Unhappy

Happy r/(NL) e/(NL)Unhappy 1/(NL) q/(NL)

Significance

More than 40 y ago, Schelling introduced one of the firstagent-based models in the social sciences. The model showedthat even if people only have a mild preference for livingwith neighbors of the same color, complete segregation willoccur. This model has been much discussed by social scien-tists and analyzed by physicists using analogies with spin-1Ising models and other systems. Here, we study the meta-population version of the model, which mimics the divisionof a city into neighborhoods, and we present the firstanalysis to our knowledge that gives detailed informationabout the structure of equilibria and explicit formulas fortheir densities.

Author contributions: R.D. and Y.Z. performed research and wrote the paper.

The authors declare no conflict of interest.1To whom correspondence should be addressed. Email: rtd@math.duke.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1414915111/-/DCSupplemental.

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Convergence to a Deterministic LimitTo describe the dynamics more precisely, let ni,j(t), be the numberof neighborhoods with i red and j blue families for (i, j) ∈ Ω ={(i, j): i, j ≥ 0, i + j ≤ L}. The configuration of the system attime t can be fully described by the numbers νNt ði; jÞ= ni;jðtÞ=N.If one computes infinitesimal means and variances, it is naturalto guess (and not hard to prove) that if we keep L fixed andlet N → ∞, then νNt converges to a deterministic limit.Motivated by individual-based models in finance, Remenik

(17) proved a general result that takes care of our example. Todescribe the limit, we need some notation. Let λ(a1, b1; a2, b2)be N times the total rate of movement from one (a1, b1) neigh-borhood to one (a2, b2) neighborhood. Let bðω1;ω2;ω1′;ω2′Þ beN times the rate at which a movement from one ω1 = (a1, b1)neighborhood to one ω2 = (a2, b2) neighborhood turns the pairω1, ω2 into ω1′;ω2′. The exact formulas for these quantities arenot important, so they are shown in SI Text.Theorem 1. As N → ∞ the νNt ði; jÞ converge in probability to

the solution of the ordinary differential equation:

dνtði; jÞdt

=−νtði; jÞXω∈Ω

½λði; j;ωÞ+ λðω; i; jÞ�νtðωÞ

+X

ω1;ω2∈Ω

�b�ω1;ω2; ði; jÞ;ω′

�+ b

�ω1;ω2;ω′; ði; jÞ

��

× νtðω1Þνtðω2Þ:[1]

We do not sum over ω′ because its value is determined byω1 and ω2. The first term comes from the fact that a migrationfrom (i, j) → ω or ω → (i, j) destroys an (i, j) neighborhood,whereas the second reflects the fact that a migration ω → ω′ maycreate an (i, j) neighborhood at the source or at the destination.

Special Case L = 2To illustrate the use of Theorem 1, we consider the case L = 2,and let ℓc = [ρcL] be the largest number of neighbors of theopposite type that allows a family to be happy. Here, [x] is thelargest integer ≤x. When L = 2, a neighborhood with both typesof families must be (1, 1), so the situation in which ℓc ≥ 1 is trivialbecause there are never any unhappy families. In the case L = 2and ℓc = 0, it is easy to find the equilibrium because there isdetailed balance, i.e., the rate of each transition is exactly bal-anced by the one in the opposite direction.

rν21;0 = 4rν0;0ν2;0 ð1; 0Þð1; 0Þ⇌ ð0; 0Þð2; 0Þrν20;1 = 4rν0;0ν0;2 ð0; 1Þð0; 1Þ⇌ ð0; 0Þð0; 2Þ2ν0;0ν1;1 = eν1;0ν0;1 ð1; 1Þð0; 0Þ⇌ ð1; 0Þð0; 1Þν1;0ν1;1 = 2eν2;0ν0;1 ð1; 1Þð1; 0Þ⇌ ð0; 1Þð2; 0Þν0;1ν1;1 = 2eν0;2ν1;0 ð1; 0Þð1; 1Þ⇌ ð1; 0Þð0; 2Þ:

After a little algebra (SI Text), we find that this holds if andonly if

ν2;0 = ν0;2 = x ν1;1 = 2ex ν1;0 = ν0;1 = y ν0;0 = y2�4x:

At first, it may be surprising that the rate r has nothing to do withthe fixed point, but if you look at the first two equations you seethat the r appears on both sides. The parameter q does notappear either, but when L = 2, it is for the trivial reason thattransitions (1, 1)(1, 0)→ (1, 0)(1, 1), which occur at rate q, do notchange the state of the system.Using now the fact that the equilibrium must preserve the red

and blue densities, we can solve for x and y to conclude that

y=1−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8ð1− eÞρ2 − 4ð1− eÞρ+ 1

p1− e

;

x=ð2− 2eÞρ− 1+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8ð1− eÞρ2 − 4ð1− eÞρ+ 1

p2ð1+ eÞð1− eÞ :

The argument given here shows that this is the only fixed pointthat satisfies detailed balance. We prove in SI Text that it isthe only fixed point. Because the formulas, which result from solv-ing a quadratic equation, are somewhat complicated, Fig. 1 showshow the equilibrium probabilities νi,j vary as a function of ρ.Unfortunately, when L ≥ 3, there is no stationary distribution

that satisfies detailed balance. One can, of course, solve forthe stationary distribution numerically. Fig. 2 shows the limitbehavior of the system with L = 20 and ρc = 0.3, i.e., ℓc = 6 forinitial densities ρ = 0.1, 0.2, 0.25, and 0.35. In the first twocases, most of the families are happy. In the third situation,the threshold is ℓc = 6, whereas the average number of redsand blues per neighborhood is five, but because fluctuations inthe makeup of neighborhoods can lead to unhappiness, thereis a tendency toward segregation. In the fourth case, segre-gation is almost complete, with most neighborhoods having0 or 1 of the minority type.

Outline of Our SolutionFinding the stationary distribution requires solving roughlyL2/2 equations. To be precise, there are 231 equations whenL = 20 and 5,151 when L = 100. In this section, we will adopta different approach: we concentrate on the evolution ofneighborhood 1 and consider the other N − 1 neighborhoodsto be its environment, which can be summarized by the fol-lowing four parameters: (i) the average number of happy redand blue families per neighborhood, h1R and h1B, and (ii) theaverage number of vacant sites happy for red or blue, h0R andh0B, again per neighborhood.The first in our solution is to identify the stationary distribu-

tion of the neighborhood–environment chain that are self-con-sistent. That is, if we calculate the expected values of h1R, h

0R, h

1B,

and h0B in equilibrium in neighborhood, they agree with theoriginal parameters. To begin to do this, we divide the statespace Ω into four quadrants based on red and blue happiness.Writing 0 for H and 1 for U, we have

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

Fig. 1. Equilibrium probabilities νi,j for the case L = 2 plotted against ρ.

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If we let Tri(pR, pB) be the trinomial distribution

L!i! j!ðL− i− jÞ! p

iR p

jBð1− pR − pBÞL−i−j; [2]

then inside Qk,ℓ, the detailed balance condition is satisfied byTri(pR, pB) where

pR =αk;ℓ

1+ αk;ℓ + βk;ℓ; pB =

βk;ℓ1+ αk;ℓ + βk;ℓ

;

and formulas for αk,ℓ and βk,ℓ are given in SI Text.Unfortunately, there is no stationary distribution that satisfies

detailed balance on the entire state space. To verify this, we notethat if the stationary distribution π and the jump rates q satisfydetailed balance π(x)q(x, y) = π(y)q(y, x), then around anyclosed path x0, x1,. . .xn = x0 with q(xi−1, xi) > 0 for 1 ≤ i ≤ n, wemust have

∏n

i=1

qðxi−1; xiÞqðxi−1; xiÞ= 1;

but this is not satisfied around loops that visit two or morequadrants. To avoid this problem, we cut the connections betweenthe different quadrants and identify the self-consistent distributionsfor the modified chain. At this point, we can only do this underAssumption 1. Stationary distributions are symmetric under in-

terchange of red and blue.The answer given in the next section is a one-parameter family

of stationary distributions indexed by a ∈ [0, 1/2]. There, andin the next two steps, we have a pair of results: one for ρ < ρcand one for ρ > ρc.In the second step, we investigate the flow of probability be-

tween quadrants when transitions between the quadrants arerestored. The key idea is that the measures in each quadrant aretrinomial, so the probabilities will decay exponentially away fromthe mean (pRL, pBL). This observation implies that the flowbetween quadrants occurs at rate exp(−cL), which is muchsmaller than the time, O(1), it takes the probability distributionsto reach equilibrium. In words, the process comes to equilibriumon a fast time scale, whereas the parameters change on a muchslower one. We will prove this separation of time scales ina version of the paper for a mathematical audience. Here, we willonly give the answers that result under

j > ℓc Q1,0 Q1,1

j ≤ ℓc Q0,0 Q0,1

i ≤ ℓc i > ℓc

0

10

20

05

1015

20

0

0.05

0.1

0

10

20

05

1015

20

0

0.02

0.04

510

1520

05

1015

20

0

0.02

0.04

0.06

0

10

20

05

1015

20

0

0.05

0.1

Fig. 2. Limiting behavior of limit differential equation, with ρc = 0.3, e = 0.01, and ρ = 0.1, 0.2, 0.25, and 0.35.

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Assumption 2. The process is always in one of self-consistentstationary distributions, but the value of a changes over time.The third step is to use the stability results to show that the

only possible stable equilibria are at the following end points:a = 0, which represents a random distribution; and a = 1/2,which represents a segregated state.We will call these measures μr and μs when ρ < ρc and μ̂r and

μ̂s when ρ > ρc. Theorems 4A and 4B describe when they arestable fixed points.

Self-Consistent Stationary DistributionsThe results given here are proved in SI Text. The formulas,which again come from solving a quadratic equation, are notsimple but they are explicit and easily evaluated.Theorem 2A. Suppose ρ < ρc. For a ∈ (0, 1/2] let

ρ1ða; ρÞ=−1+ ða+ ρÞð1− eÞ+Δ1

2að1− e2Þ ; [3]

where

Δ1 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1− ða+ ρÞð1− eÞ�2 + 4að1− e2Þρ

q:

Let ρ1(0, ρ) = lima→0 ρ1(a, ρ) = ρ/[1 − ρ(1 − e)] and for a ∈[0, 1/2] let

μa = ð1− 2aÞTriðρ0; ρ0Þ+ aTriðρ1; ρ2Þ+ aTriðρ2; ρ1Þ:

A symmetric distribution μ is self-consistent if and only if it hasthe form above with parameters ρ1 > ρc , ρ2 = eρ1 < ρc , and ρ0 =ρ1/[1 + (1 − e)ρ1] < ρc .To clarify the last sentence: the definition of ρ1 does not

guarantee that the three conditions are satisfied for all valuesof a ∈ [0, 1/2], so the inequalities are additional conditions. Asshown in SI Text, a → ρ1(a, ρ) is increasing, so the range ofpossible values of ρ1 for a fixed value of ρ is

½ρ1ð0; ρÞ; ρ1ð1=2; ρÞ�=�

ρ

1− ρð1− eÞ;2ρ1+ e

: [4]

The possible self-consistent stationary distributions are sim-ilar in the second case but the formulas are different.Theorem 2B. Suppose ρ ≥ ρc. For a ∈ (0, 1/2] let

ρ̂1ða; ρÞ=e+ ð1− eÞða+ ρÞ−Δ2

2að1− e2Þ ; [5]

where

Δ2 =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½e+ ð1− eÞða+ ρÞ�2 − 4að1− e2Þρ

q:

Let ρ̂1ð0; ρÞ= lima→ 0 ρ̂1ða; ρÞ= ρ=½e+ ð1− eÞρÞ�, and for a ∈[0, 1/2] let

μ̂a = aTriρ̂1; ρ̂2

�+ aTri

ρ̂2; ρ̂1

�+1− 2a

�Tri

ρ̂3; ρ̂3

�:

A symmetric distribution μ̂ is self-consistent if and only if ithas the form above with parameters ρ̂1 > ρc, ρ̂2 = eρ̂1 < ρc andρ̂3 = eρ̂1=½1− ð1− eÞρ̂1�> ρc.This time a → ρ1(a, ρ) is decreasing, so the range of possible

values of ρ1 for a fixed value of ρ is

�ρ̂1ð1=2; ρÞ; ρ̂1ð0; ρÞ

�=�2ρ1+ e

;ρ

e+ ð1− eÞρ; [6]

i.e., the old upper bound on the range of ρ1 in 4 has become thelower bound. See Fig. 3 for a picture.

Stability CalculationsThe results in this section are proved in SI Text. Using largedeviations for the trinomial distribution, which in this case isjust calculating probabilities using Stirling’s formula, we con-clude the following:Theorem 3A. Suppose ρ < ρc and recall μa has no mass on Q1,1.

The flow into Q0,0 from Q0,1 and Q1,0 is larger than the flow outif and only if

�1− eρ11− ρ1

1−ρc< 1+ ð1− eÞρ1: [7]

Theorem 3B. Suppose ρ ≥ ρc and recall μ̂a has no mass on Q0,0.The flow out of Q1,1 to Q0,1 and Q1,0 is larger than the flow in ifand only if

�ρ̂1

1− ρ̂1

1−ρc<h1− ð1− eÞρ̂1

i−1: [8]

Phase TransitionCombining Theorems 2A and 3A, we can determine the be-havior of the process for ρ < ρc. The set of possible values forρ1(a, ρ) for a fixed ρ is the interval [ρ1(0, ρ), ρ1(1/2, ρ)] given in4. Because 0 ≤ ρ ≤ ρc, we are looking for a solution to

�1− e x01− x0

1−ρc= 1+ ð1− eÞx0:

with x0 ∈ [0, 2ρc/(1 + e)). In SI Text, we show that x0 exists and isunique. Here, we will concentrate on what happens in the exam-ple ρc = 0.2 and e = 0.1. When ρc = 0.2, the interval is [0, 0.4/1.1],and we have x0 = 0.2183.

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

line0.2183low−lb0.8724large−ubρ s

Fig. 3. Picture to explain calculation of the phase transition when ρc = 0.2and e = 0.1. The x axis gives the value of ρ. Dots on the axis are the locationsof ρb, ρd, ρ̂b, and ρ̂d . The two curves are ρ1(0, ρ) for ρ < ρc and ρ̂1ð0,ρÞ for ρ ≥ρc, whereas the straight line is ρ1ð1=2,ρÞ= ρ̂1ð1=2,ρÞ= 2ρ=ð1+ eÞ.

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Let ρb be chosen so that x0 = ρ1(1/2, ρb) and ρd be chosen sothat x0 = ρ1(0, ρd). See Fig. 3 for a picture. When a solution x0exists in the desired interval

ρb =ð1+ eÞx0

2and ρd =

x01+ x0ð1− eÞ:

In our special case, ρb = 0.1201 and ρd = 0.1825.Theorem 4A. The stable stationary distributions for ρ < ρc are

μr for   0≤ ρ< ρb;μr and μs for   ρb < ρ< ρd;

μs for   ρd < ρ< ρc:

Why is this true? When ρ < ρb, ρ0 > ρ1(1/2, ρ), the flow into Q0,0is always larger than the flow out, so μr is the stationary distri-bution. When ρb < ρ < ρd, there will be an ac ∈ (0, 1/2) so thatρ1(ac, ρ) = ρ0. The flow into Q0,0 is larger than the flow out whena < ac and the a in the mixture will decrease, whereas for a > ac,the flow out of Q0,0 will be larger than the flow in and a willincrease. Thus, we have stable fixed points at 0 and 1/2. Whenρd < ρ < ρc, ρ < ρ1 (0, ρ), the flow out of Q0,0 is always larger thanthe flow in, and the segregated fixed point with a = 1/2 is thestationary distribution.Using Theorems 2B and 3B, we can determine the behavior of

the process for ρ ≥ ρc. The set of possible values for ρ̂1ða; ρÞ fora fixed ρ is the interval ½ρ̂1ð1=2; ρÞ; ρ̂1ð0; ρÞ� given in 6. Becauseρc ≤ ρ ≤ 0.5, we are looking for a solution to

�x̂0

1− x̂0

1−ρc=�1− ð1− eÞx̂0

�−1:

with x̂0 ∈ ½2ρc=ð1+ eÞ; 1=1+ e�. In SI Text, we show that there isa solution in the desired interval if and only if eρc ≥ ðe+ 1Þ=2, andit is unique. The condition comes from having = in 8 at the rightend point 1/(1 + e). When e = 0.1, the condition is ρc < 0.25964.When ρc = 0.2 and e = 0.1, this interval is [0.4/1.1, 1/1.1], and

x̂0 = 0:8724. Let ρ̂b be chosen so that x̂0 = ρ̂1ð0; ρ̂bÞ and ρ̂d bechosen so that x̂0 = ρ̂1ð1=2; ρdÞ. When a solution x̂0 exists in thedesired interval, we have

ρ̂b =e x̂0

1− x̂0ð1− eÞ and ρ̂d =ð1+ eÞ x̂0

2:

In our example, ρ̂b = 0:4061 and ρ̂d = 0:4798.Theorem 4B. The stable stationary distributions for ρ ≥ ρc are

μ̂s for   ρc ≤ ρ< ρ̂b;μ̂r and μ̂s for   ρ̂b < ρ< ρ̂d;

μ̂r for   ρ̂d < ρ< 0:5:

The reasoning behind this result is the same as for Theorem 4A.To show that these result can be used to explicitly describe thephase transition, Fig. 4 shows how the four critical values dependon ρc when e = 0.1.

DiscussionHere, we considered a metapopulation version of Schelling’smodel, which we believe is a better model for studying the dy-namics of segregation in a city than a nearest neighborhood in-teraction on the 2D square lattice. Due to the simple structure ofthe model, we are able to describe the phase transition in greatdetail. For ρ < ρb, a random distribution of families μr = Tri(ρ, ρ)is the unique stationary distribution. As ρ increases there is adiscontinuous phase transition to a segregated state, μs at ρdpreceded by an interval (ρb, ρd), in which both μr and μs arestable. Surprisingly the phase transition occurs to a segregatedstate occurs at a value ρd < ρc, i.e., at a point where in a randomdistribution most families are happy. This shift in behavioroccurs because random fluctuations create segregated neigh-borhoods, which, as our analysis shows, are more stable thanthe random ones.If ρc is small enough, then as ρ nears 1/2, there is another

discontinuous transition at ρ̂d, which returns the equilibrium tothe random state μ̂r =Triðρ; ρÞ, and this is preceded by an in-terval ðρ̂b; ρ̂dÞ of bistability. To explain this, we note that whenfamilies are distributed randomly, everyone is unhappy andmoves at rate 1, maintaining the random distribution. In ourconcrete example, ρc = 0.2 and e = 0.1, the fraction of vacanthouses at ρ̂d = 0:4798 is only 4.04%, so it is very difficult to makesegregated neighborhoods where one type is happy. Our stabilityanalysis implies that these segregated neighborhoods are createdat a slower rate than they are lost, so the random state prevails.The results in this paper have been derived under two assump-

tions: (i) stationary distributions are invariant under interchange ofred and blue and (ii) the process is always in one of a one-parameter family of self-consistent stationary distributionsindexed by a ∈ [0,1/2], but the value of a changes over time. Weare confident that ii can be rigorously proved. Removing iwill be more difficult, because when symmetry is dropped, thereis a two-parameter family of self-consistent distributions. A moreinteresting problem, which is important for applications to realcities, is to allow the initial densities of reds and blues and theirthreshold for happiness to differ. Although our solution is not yetcomplete, we believe it is an important first step in obtaining adetailed understanding of the equilibrium behavior of Schelling’smodel in a situation that is relevant for applications.

ACKNOWLEDGMENTS. We thank David Aldous, Nicholas Lanchier, SimonLevin, and Thomas Liggett for helpful comments. R.D. and Y.Z. were partiallysupported by Grants DMS 10-05470 and DMS13-05997 from the probabilityprogram at the National Science Foundation.

1. Schelling TC (1971) Dynamic models of segregation. J Math Sociol 1(2):143–186.2. Schelling TC (1978) Micromotives and Macrobehavior (Norton, New York).3. Clark WAV, Fossett M (2008) Understanding the social context of the Schelling seg-

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Fig. 4. ρb < ρd < ρ̂b < ρ̂d as a function of ρc when e = 0.1.

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