Ergodicity of the voter model with dynamic anti-voter bonds

TL;DR

This paper studies a dynamic variant of the voter model with evolving positive and negative edges, proving ergodicity on countably infinite graphs for all parameters.

Contribution

It introduces a joint spin-bond Markov process with dynamic edge signs and proves its ergodicity on any countably infinite graph.

Findings

The process is ergodic for any simple graph with countably many vertices.

Ergodicity holds for all transition kernels and parameters of edge dynamics.

The model generalizes classical voter dynamics with evolving anti-voter bonds.

Abstract

The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion. In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics.