- 02-482 Mark Pollicott and Howard Weiss
- The Dynamics of Schelling-Type Segregation Models and a Nonlinear
Graph Laplacian Variational Problem
Nov 22, 02
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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.