Minority population in the one-dimensional Schelling model of segregation

TL;DR

This paper rigorously analyzes the one-dimensional Schelling segregation model with a minority population, revealing threshold behaviors that differ significantly from the uniform distribution case, advancing understanding of segregation dynamics.

Contribution

It provides the first rigorous analysis of the noiseless one-dimensional Schelling model with a minority population, identifying key threshold phenomena.

Findings

Identifies threshold behaviors in minority segregation dynamics.

Contrasts results with the uniform initial distribution case.

Provides rigorous mathematical analysis of the model.

Abstract

The Schelling model of segregation looks to explain the way in which a population of agents or particles of two types may come to organise itself into large homogeneous clusters, and can be seen as a variant of the Ising model in which the system is subjected to rapid cooling. While the model has been very extensively studied, the unperturbed (noiseless) version has largely resisted rigorous analysis, with most results in the literature pertaining to versions of the model in which noise is introduced into the dynamics so as to make it amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. We rigorously analyse the one-dimensional version of the model in which one of the two types is in the minority, and establish various forms of threshold behaviour. Our results are in sharp contrast with the case when the distribution of the two types is uniform (i.e. each agent has equal chance of being of each type in the initial configuration), which was studied by Brandt, Immorlica, Kamath, and Kleinberg.