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Hydrodynamic limit of the Schelling model with spontaneous Glauber and Kawasaki dynamics

Florent Barret, Niccolo Torri

Abstract

In the present article we consider the Schelling model, an agent-based model describing a segregation dynamics when we have a cohabitation of two social groups. As for several social models, the behaviour of the Schelling model was analyzed along several directions, notably by exploiting theoretical physics tools and computer simulations. This approach led to conjecture a phase diagram in which either different social groups were segregated in two large clusters or they were mixed. In this article, we describe and analyze a perturbation of the Schelling model as a particle systems model by adding a Glauber and Kawasaki dynamics to the original Schelling dynamics. As far as the authors know, this is the first rigorous mathematical analysis of the perturbed Schelling model. We prove the existence of an hydrodynamic limit described by a reaction-diffusion equation with a discontinuous non-linear reaction term. The existence and uniqueness of the solution is non trivial and the analysis of the limit PDE is interesting in its own. Based on our results, we conjecture, as in other variations of this model, the existence of a phase diagram in which we have a mixed, a segregated and a metastable segregation phase. We also describe how this phase transition can be viewed as a transition between a relevant and irrelevant disorder regime in the model.

Hydrodynamic limit of the Schelling model with spontaneous Glauber and Kawasaki dynamics

Abstract

In the present article we consider the Schelling model, an agent-based model describing a segregation dynamics when we have a cohabitation of two social groups. As for several social models, the behaviour of the Schelling model was analyzed along several directions, notably by exploiting theoretical physics tools and computer simulations. This approach led to conjecture a phase diagram in which either different social groups were segregated in two large clusters or they were mixed. In this article, we describe and analyze a perturbation of the Schelling model as a particle systems model by adding a Glauber and Kawasaki dynamics to the original Schelling dynamics. As far as the authors know, this is the first rigorous mathematical analysis of the perturbed Schelling model. We prove the existence of an hydrodynamic limit described by a reaction-diffusion equation with a discontinuous non-linear reaction term. The existence and uniqueness of the solution is non trivial and the analysis of the limit PDE is interesting in its own. Based on our results, we conjecture, as in other variations of this model, the existence of a phase diagram in which we have a mixed, a segregated and a metastable segregation phase. We also describe how this phase transition can be viewed as a transition between a relevant and irrelevant disorder regime in the model.
Paper Structure (23 sections, 29 theorems, 203 equations, 2 figures)

This paper contains 23 sections, 29 theorems, 203 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Configurations
  4. Infinitesimal generator, construction of the process
  5. Main results
  6. Organisation of the paper
  7. Conjecture on the phase diagram
  8. Relative entropy method
  9. Proof of Theorem \ref{['thm:entropy']}
  10. The current ${\mathcal{J}} _t^N$
  11. Estimates of $V^+$ and $V^-$
  12. Estimate of $V$
  13. Conclusion: Gronwall's inequality \ref{['eq:goalJ_t']}
  14. Estimates on the solutions $(u^N)$ of \ref{['eq:discretPDE']}
  15. Existence and uniqueness of reaction-diffusion PDE
  16. ...and 8 more sections

Key Result

Proposition 2.1

Given an initial configuration $\eta_0$, ${\mathcal{L}} _N$ is the infinitesimal generator of a Feller process, denoted $(\eta^N(t))_{t\geq 0}$.

Figures (2)

  • Figure 1: Representation of $\gamma_{\infty, \beta}(p)$, $p\in[0,1]$, for different values of $\beta$ and a fixed $T=0.3$. The discontinuities of $\gamma_{\infty, \beta}^\prime(p)$ are situated at $p_0(T)$ and $1-p_0(T)$. In \ref{['fig:1']} we have that $p_0(T)=0.3<p^\ell\approx 0.3076$ and the unique stable equilibrium point is $p^c=\frac{1}{2}$. In \ref{['fig:2']} we have that $p^\ell=0.25<p_0(T)=0.3<p^m\approx 0.3535$, so that $p^c=\frac{1}{2}$ is stable, while $p^\ell$ and $p^r$ are metastable equilibrium. In \ref{['fig:3']} we have that $p_0(T)=0.3>p^m\approx 0.2401$ and $p^\ell$ and $p^r$ become stable while $p^c$ is metastable. Finally in \ref{['fig:4']} we illustrated the limit case with $\beta=0$, and the two stable equilibriums are $p^\ell=0$ and $p^r=1$. In this case, the local minimum $p^c$ degenerates into the segment $[p_0(T), 1-p_0(T)]$.
  • Figure 2: Representation of the different phases of the system as function of the parameter $p_0(T)\in [0, \frac{1}{2}]$. When $T$ is close to $0$ or $1$ ($p_0(T)\in (0, p^\ell)$) we do not have segregation (red parts) and typical configurations are provided by a mixing of $0$ and $1$. If $T$ is close to $1/2$ ($p_0(T)\in (p^m,\frac{1}{2})$) we have segregation (green parts): a very large part of the configuration are composed of $0$ or $1$. We have also intermediate values of $T$ ($p_0(T)\in (p^\ell,p^m)$) for which the segregation is metastable (yellow parts).

Theorems & Definitions (65)

  • Definition 2.1
  • Proposition 2.1
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Remark 3.3
  • Remark 3.4
  • Remark 4.1
  • Theorem 4.1
  • ...and 55 more