Ergodic behaviour of "signed voter models"

TL;DR

This paper investigates the ergodic properties of signed voter models on graphs, extending traditional voter models by incorporating positive and negative edges, and addresses questions about their long-term behavior.

Contribution

It introduces and analyzes the ergodic behavior of signed voter models, providing new insights into their dynamics on locally finite graphs.

Findings

Characterization of ergodic behavior in signed voter models

Conditions under which models exhibit ergodicity or non-ergodicity

Extension of classical voter model results to signed interactions

Abstract

We consider some questions raised by the recent paper of Gantert, L\"owe and Steif (2005) concerning ``signed'' voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed).