Majority Rule Dynamics in Finite Dimensions

TL;DR

This paper studies how a majority rule opinion model in finite dimensions evolves over time, leading to consensus through domain coarsening, with insights into the role of geometrical organization on long-term dynamics.

Contribution

It introduces a detailed analysis of the long-time behavior and geometrical organization in a finite-dimensional majority rule model, highlighting the impact on consensus formation.

Findings

Consensus is achieved through domain coarsening.

Two distinct time scales govern the approach to consensus.

Geometrical organization influences long-time kinetics.

Abstract

We investigate the long-time behavior of a majority rule opinion dynamics model in finite spatial dimensions. Each site of the system is endowed with a two-state spin variable that evolves by majority rule. In a single update event, a group of spins with a fixed (odd) size is specified and all members of the group adopt the local majority state. Repeated application of this update step leads to a coarsening mosaic of spin domains and ultimate consensus in a finite system. The approach to consensus is governed by two disparate time scales, with the longer time scale arising from realizations in which spins organize into coherent single-opinion bands. The consequences of this geometrical organization on the long-time kinetics are explored.