The dynamics of schelling-type segregation models and a nonlinear graph laplacian variational problem

In this paper we analyze a variant of the famous Schelling segregation model in economics as a dynamical system. This model exhibits, what appears to be, a new clustering mechanism. In particular, we explain why the limiting behavior of the non-locally determined lattice system exhibits a number of pronounced geometric characteristics. Part of our analysis uses a geometrically defined Lyapunov function which we show is essentially the total Laplacian for the associated graph Laplacian. The limit states are minimizers of a natural nonlinear, nonhomogeneous variational problem for the Laplacian, which can also be interpreted as ground state configurations for the lattice gas whose Hamiltonian essentially coincides with our Lyapunov function. Thus we use dynamics to explicitly solve this problem for which there is no known analytic solution. We prove an isoperimetric characterization of the global minimizers on the torus which enables us to explicitly obtain the global minimizers for the graph variational problem. We also provide a geometric characterization of the plethora of local minimizers.