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

Abstract

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 τ ∈ (∼ 0.488, ∼ 0.512) \ {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 τ ∈ (∼ 0.433, ∼ 0.567) \ {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 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, 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.