Evolution and Limiting Configuration of a Long-Range Schelling-Type Spin System

We consider a long-range interacting particle system in which binary particles -- whose initial states are chosen uniformly at random -- are located at the nodes of a flat torus $(\mathbb{Z}/h\mathbb{Z})^2$. Each node of the torus is connected to all the nodes located in an $l_\infty$-ball of radius $w$ in the toroidal space centered at itself and we assume that $h$ is exponentially larger than $w^2$. Based on the states of the neighboring particles and on the value of a common intolerance threshold $\tau$, every particle is labeled "stable," or "unstable." Every unstable particle that can become stable by flipping its state is labeled "p-stable." Finally, unstable particles that remained p-stable for a random, independent and identically distributed waiting time, flip their state and become stable. When the waiting times have an exponential distribution and $\tau \le 1/2$, this model 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 spreading of the "affected" nodes of a given state -- namely nodes on which a particle of a given state would be p-stable. As $w \rightarrow \infty$, this spreading starts with high probability (w.h.p.) from any $l_\infty$-ball in the torus having radius $w/2$ and containing only affected nodes, and continues for a time that is at least exponential in the cardinalilty of the neighborhood of interaction $N = (2w+1)^2$. Second, we show that when the process reaches a limiting configuration and no more state changes occur, for all ${\tau \in (\tau^*,1-\tau^*) \setminus \{1/2\}}$ where ${\tau^* \approx 0.488}$, w.h.p. any particle is contained in a large "monochromatic ball" of cardinality exponential in $N$.