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Physical Considerations on the Schelling model of Social Segregation

Sergio Rica Universidad Adolfo Ibáñez In collaboration with N. Goles-Domic and E. Goles Phys Rev E 83, 056111 (2011)

On the occasion of the 60th birthday of Prof. Eric Goles

Valparaíso, nov 2011

Supported by grant COSTUME, ANR SYSCOM (France)

Plan

Introduction The Schelling Model Qualitative behavior Quantitative behavior Discussion

The Model of Segregation by Shelling

Lattice {i,k}

xk

xk

xk

xk = ±1

State

Vicinity Tolerance threshold

Thomas C. Schelling (1969 - 1972)

Happiness threshold

An individual is unhappy if there are more than individuals of the other type.

θ

• eg. in a vicinity
• f 8 neighbors

and if then :

θ = 5

The rule

At each step, one lists the unhappy individuals of both species, and then randomly one exchanges two individuals of

• pposite specie.

R e m a r k : t h e n u m b e r o f individuals of each specie (N+ & N- ) are conserved.

• i,k

xik = N+ − N− = Cte

θ = 5

Phase diagram

• 4

3 2 5 6 25% 50%

finite time evolution infinite time evolution

12.5% 7 φ = N+ N+ + N−

Comments

A tendency of segregation. A tendency of a diminution of the interfaces But! there is a strong frustration. A length scale ?

256 x 256

128 x 128

32 x 32 64 x 64

512 x 512 Length scale for θ = 5

Case of 44 neighbors

8 neighbors

Case of 68 neighbors

8 neighbors

Summary

8 neighbors 68 neighbors 44 neighbors

Quantitative behavior

E[{x}] = −1 2

N

• k=1

xk

• i∈Vk

xi

θ = 5

decreases strictly during the evolution. Moreover, ∆Ekl ≤ 4 (wkl + 8 − 2θ)

wkl ≤ 1

where For 8 neighbors and, if and higher, then the “energy"

Geometrical interpretation

E[{x}] = −1 2

N

• k=1
• i∈Vk

1 + 1 2

N

• k=1
• i∈Vk

(1 − xkxi) = −1 28N + 1 2

N

• k=1
• i∈Vk

(1 − xkxi) E = −4N + 2 ×

• 3
• edges −
• corners
• = −4N + 2 × (3 × perimeter − Nb. of corners),

Few Consequences

Because the energy is bounded

• 4 N ≤ E ≤ 4 N the dynamics is
• f finite time for and

higher. For the dynamics continues indefinitely The case may posses a complex dynamics The energy ground state.

θ = 5 θ ≤ 3 θ = 4

Phase diagram

• 4

3 2 5 6 25% 50%

finite time evolution infinite time evolution

12.5% 7 φ = N+ N+ + N−

E vs time

E/N t [10 units ]

3

=5 =4 =3 =2 =6

GSE

E vs time

E/N t [10 units ]

3

=2

GSE

=3 =4 =5

Discussion

Var i a nt s o n t h e m o d e l a n d g en eralizatio ns (g rap hs, no n uniform tolerance, various states, protocols...) Q2R Segregation in higher dimensions ?

Segregation in 3D