Unperturbed Schelling segregation in two or three dimensions

TL;DR

This paper provides the first rigorous analysis of unperturbed Schelling segregation models in two and three dimensions, establishing their phase diagrams and advancing understanding of self-organizing segregation phenomena.

Contribution

It offers the first comprehensive mathematical analysis of unperturbed Schelling models in higher dimensions, extending prior one-dimensional results.

Findings

Established the phase diagram for 2D and 3D Schelling models.

Answered a recent open challenge regarding unperturbed models.

Extended rigorous analysis beyond one-dimensional cases.

Abstract

Schelling's model of segregation, first described in 1969, has become one of the best known models of self-organising behaviour. While Schelling's explicit concern was to understand the mechanisms underlying racial segregation in large cities from a game theoretic perspective, the model should be seen as one of a family, arising in fields as diverse as statistical mechanics, neural networks and the social sciences, and which are concerned with interacting populations situated on network structures. Despite extensive study, however, the (unperturbed) Schelling model has largely resisted rigorous analysis, prior results in the literature generally pertaining to variants of the model in which noise is introduced into the dynamics of the system, the resulting model then being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. A series of recent papers (one by Brandt, Immorlica, Kamath, and Kleinberg, and two by the authors), has seen the first rigorous analysis of the one dimensional version of the unperturbed model. Here we provide the first rigorous analysis of the two and three dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from a recent paper by Brandt, Immorlica, Kamath, and Kleinberg.