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

TL;DR

This paper analyzes a long-range Schelling-type spin system on a torus, proving shape theorems for affected nodes and showing that the system reaches large monochromatic regions under certain conditions.

Contribution

It introduces a shape theorem for the spread of affected nodes and characterizes the limiting monochromatic configurations in a long-range Schelling model.

Findings

High probability spreading of affected nodes from initial conditions

Existence of large monochromatic regions in the limit

Model equivalence to Ising and cellular automaton dynamics

Abstract

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 $N$.