Evolution and Steady State of a Long-Range Two-Dimensional Schelling Spin System
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We consider a long-range interacting particle system in which binary particles are located at the integer points of a flat torus. Based on the interactions with other particles in its "neighborhood" and on the value of a common intolerance threshold τ, every particle decides whether to change its state after an independent and exponentially distributed waiting time. This is equivalent to a Schelling model of self-organized segregation in an open system, a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. We first prove a shape theorem for the spread of the "affected" nodes during the process dynamics. Second, we show that when the process stops, for all τ∈ (τ^*,1-τ^*) ∖{1/2} where τ^* ≈ 0.488, and when the size of the neighborhood of interaction N is sufficiently large, every particle is contained in a large "monochromatic region" of size exponential in N, almost surely. When particles are placed on the infinite lattice Z^2 rather than on a flat torus, for the values of τ mentioned above, sufficiently large N, and after a sufficiently long evolution time, every particle is contained in a large monochromatic region of size exponential in N, almost surely.
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