Improved Intolerance Intervals and Size Bounds for a Schelling-Type Spin System

We consider a Schelling model of self-organized segregation in an open system that is equivalent to a zero-temperature Ising model with Glauber dynamics, or an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods. Previous work has shown that if the intolerance parameter of the model $\tau\in (\sim 0.488,\sim 0.512) \setminus {1/2}$, then for a sufficiently large neighborhood of interaction $N$, any particle will end up in an exponentially large monochromatic region almost surely. This paper extends the above result to the interval $\tau \in (\sim 0.433,\sim 0.567) \setminus {1/2}$. We also improve the bounds on the size of the monochromatic region by exponential factors in $N$. Finally, we show that 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. The new proof, critically relies on a novel geometric construction related to the formation of the monochromatic region.