Mean field limit for bias voter model on regular trees

TL;DR

This paper analyzes bias voter models on trees and lattices, deriving a mean field limit for the probability of vertices being in a certain state as the degree increases, and showing convergence to consensus under certain conditions.

Contribution

It introduces a mean field limit for bias voter models on trees and lattices and proves convergence to a unanimous state with high rethinking rates.

Findings

Mean field limit for the probability of state 1 as degree grows

Weak convergence to all vertices in state 1 with high rethinking rate

Use of graphical representation and contact process convergence theorem

Abstract

In this paper we are concerned with bias voter models on trees and lattices, where the vertex in state 0 reconsiders its opinion at a larger rate than that of the vertex in state 1. For the process on tree with product measure as initial distribution, we obtain a mean field limit at each moment of the probability that a given vertex is in state 1 as the degree of the tree grows to infinity. Furthermore, for our model on trees and lattices, we show that the process converges weakly to the configuration where all the vertices are in state 1 when the rate at which a vertex in state 0 reconsiders its opinion is sufficiently large. The approach of graphical representation and the complete convergence theorem of contact process are main tools for the proofs of our results.