Ordering dynamics of the multi-state voter model

TL;DR

This paper generalizes the voter model to multiple states, deriving analytical expressions for key dynamics and comparing them with numerical results, thereby expanding understanding of ordering processes in complex systems.

Contribution

It introduces a multi-state voter model, providing analytical formulas for exit probability and consensus time for any number of states, extending classical binary results.

Findings

Derived mean-field expressions for arbitrary number of states.

Numerical comparison between multi-state and binary voter models.

Insights into the influence of the number of states on ordering dynamics.

Abstract

The voter model is a paradigm of ordering dynamics. At each time step, a random node is selected and copies the state of one of its neighbors. Traditionally, this state has been considered as a binary variable. Here, we relax this assumption and address the case in which the number of states is a parameter that can assume any value, from 2 to \infty, in the thermodynamic limit. We derive mean-field analytical expressions for the exit probability and the consensus time for the case of an arbitrary number of states. We then perform a numerical study of the model in low dimensional lattices, comparing the case of multiple states with the usual binary voter model. Our work generalizes the well-known results for the voter model, and sheds light on the role of the so far almost neglected parameter accounting for the number of states.