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

In the present article we consider the Schelling model, an agent-based modeldescribing a segregation dynamics when we have a cohabitation of two socialgroups. As for several social models, the behaviour of the Schelling model wasanalyzed along several directions, notably by exploiting theoretical physicstools and computer simulations. This approach led to conjecture a phase diagramin which either different social groups were segregated in two large clustersor they were mixed. In this article, we describe and analyze a perturbation ofthe Schelling model as a particle systems model by adding a Glauber andKawasaki dynamics to the original Schelling dynamics. As far as the authorsknow, this is the first rigorous mathematical analysis of the perturbedSchelling model. We prove the existence of an hydrodynamic limit described by areaction-diffusion equation with a discontinuous non-linear reaction term. Theexistence and uniqueness of the solution is non trivial and the analysis of thelimit PDE is interesting in its own. Based on our results, we conjecture, as inother variations of this model, the existence of a phase diagram in which wehave a mixed, a segregated and a metastable segregation phase. We also describehow this phase transition can be viewed as a transition between a relevant andirrelevant disorder regime in the model.