Fluid limit of threshold voter models on tori

TL;DR

This paper derives a fluid limit for threshold voter models on high-dimensional tori, revealing a phase transition in the proportion of vertices in state 1 based on initial density p.

Contribution

It establishes a fluid limit for the model on tori as dimension increases, highlighting a phase transition phenomenon depending on initial density p.

Findings

Fluid limit derived for high-dimensional tori

Phase transition at p=1/2 in the fluid limit

Proportion of vertices in state 1 exhibits distinct behaviors based on p

Abstract

In this paper, we are concerned with threshold voter models on tori. Assuming that the initial distribution of the process is product measure with density p, we obtain a fluid limit of the proportion of vertices in state 1 as the dimension of the torus grows to infinity. The fluid limit performs a phase transition phenomenon from p < 1/2 to p > 1/2.