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

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.