# Phase transition and power-law coarsening in Ising-doped voter model

## TL;DR

This paper studies a mixed opinion model combining Voter and Ising agents, revealing how a small fraction of Ising agents induces phase transitions and power-law coarsening behaviors.

## Contribution

It introduces a hybrid opinion formation model showing how minimal Ising agent presence drastically alters Voter model dynamics and induces Ising-like coarsening.

## Key findings

- Small Ising fraction causes phase transition in Voter model.
- Power-law coarsening emerges due to Ising agents.
- Ferromagnetic order persists up to a critical temperature.

## Abstract

We examine an opinion formation model, which is a mixture of Voter and Ising agents. Numerical simulations show that even a very small fraction ($\sim 1\%$) of the Ising agents drastically changes the behaviour of the Voter model. The Voter agents act as a medium, which correlates sparsely dispersed Ising agents, and the resulting ferromagnetic ordering persists up to a certain temperature. Upon addition of the Ising agents, a logarithmically slow coarsening of the Voter model ($d=2$), or its active steady state ($d=3$), change into an Ising-type power-law coarsening.