# Evolution and Steady State of a Long-Range Two-Dimensional Schelling Spin System

@article{Omidvar2018EvolutionAS, title={Evolution and Steady State of a Long-Range Two-Dimensional Schelling Spin System}, author={Hamed Omidvar and Massimo Franceschetti}, journal={ArXiv}, year={2018}, volume={abs/1804.00358} }

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 $\tau$, 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…

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The shape theorem is based on a novel concentration inequality for the spreading time, and provides a precise geometrical description of the process dynamics, and implies that in the steady state the size of the monochromatic region of any agent is exponential with high probability.

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A spin system with long-range interacting particles is considered and it is shown that when particles are placed on the infinite lattice Z2 rather than on a flat torus, any particle is contained in a large monochromatic ball of size exponential in N, w.h.p.

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