Mark Pollicott and Howard Weiss
The Dynamics of Schelling-Type Segregation Models and a Nonlinear 
Graph Laplacian Variational Problem
(1156K, pdf)

ABSTRACT.  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 non-linear, non-homogeneous 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.