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

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 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 ${\tau \in (\tau^*,1-\tau^*) \setminus \{1/2\}}$ where ${\tau^* \approx 0.488}$, and when the size of the neighborhood of interaction $N$ is sufficiently large, any particle is contained in a large "monochromatic region" of size exponential in $N$, almost surely. When particles are placed on the infinite lattice $\mathbb{Z}^2$ rather than on a flat torus, for the values of $\tau$ mentioned above, sufficiently large $N$, and after a sufficiently long evolution time, any particle is contained in a large monochromatic region of size exponential in $N$, almost surely.