Does a Single Zealot Affect an Infinite Group of Voters ?

TL;DR

This paper analyzes how a single zealot influences opinion dynamics in an infinite voter model, revealing dimension-dependent convergence behaviors and the limits of influence in higher dimensions.

Contribution

It provides an exact solution for the magnetization in a voter model with a zealot, highlighting the influence decay with dimension and time.

Findings

In 1d, magnetization decays as t^{-1/2}.

In 2d, magnetization decays as 1/ln(t).

In higher dimensions, the zealot's influence is limited and does not reach all individuals.

Abstract

A method for studying exact properties of a class of {\it inhomogeneous} stochastic many-body systems is developed and presented in the framework of a voter model perturbed by the presence of a ``zealot'', an individual allowed to favour an opinion. We compute exactly the magnetization of this model and find that in one (1d) and two dimensions (2d) it evolves, algebraically ($\sim t^{-1/2}$) in 1d and much slower ($\sim 1/\ln{t}$) in 2d, towards the unanimity state chosen by the zealot. In higher dimensions the stationary magnetization is no longer uniform: the zealot cannot influence all the individuals. Implications to other physical problems are also pointed out.