# The Schelling model on $\mathbb{Z}$

## TL;DR

This paper rigorously analyzes a Schelling model on the integer lattice, revealing how different movement rules and distributions influence long-term segregation patterns and dynamics.

## Contribution

It introduces a formal framework for the Schelling model on $\\mathbb{Z}$, exploring how various dynamics affect asymptotic behavior and extending to multiple agent types.

## Key findings

- Behavior differs with bounded vs. unbounded moving distributions
- Lazy agents' dynamics lead to different equilibrium states
- The model exhibits phase transitions depending on movement rules

## Abstract

A version of the Schelling model on $\mathbb{Z}$ is defined, where two types of agents are allocated on the sites. An agent prefers to be surrounded by other agents of its own type, and may choose to move if this is not the case. It then sends a request to an agent of opposite type chosen according to some given moving distribution and, if the move is beneficial for both agents, they swap location. We show that certain choices in the dynamics are crucial for the properties of the model. In particular, the model exhibits different asymptotic behavior depending on whether the moving distribution has bounded or unbounded support. Furthermore, the behavior changes if the agents are lazy in the sense that they only swap location if this strictly improves their situation. Generalizations to a version that includes multiple types are discussed. The work provides a rigorous analysis of so called Kawasaki dynamics on an infinite structure with local interactions.