Evolution and Limiting Configuration of a LongRange SchellingType Spin System
Abstract
We consider a longrange 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 "pstable." Finally, unstable particles that remained pstable 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 selforganized segregation in an open system, a zerotemperature 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 pstable. 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 $N$.
 Publication:

arXiv eprints
 Pub Date:
 April 2018
 arXiv:
 arXiv:1804.00358
 Bibcode:
 2018arXiv180400358O
 Keywords:

 Mathematics  Probability;
 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Social and Information Networks;
 Mathematical Physics
 EPrint:
 arXiv admin note: text overlap with arXiv:1811.10677 (arXiv:1811.10677 is an extension of this work by the same authors and these works share many parts.)