# On some threshold-one attractive interacting particle systems on homogeneous trees

## TL;DR

This paper investigates threshold-one contact and voter models on homogeneous trees, proving complete convergence under strong survival and exploring conditions for convergence under weak survival.

## Contribution

It extends the understanding of convergence properties for these models on trees, adapting arguments from ordinary contact processes.

## Key findings

- Complete convergence theorem proven under strong survival
- Convergence may occur under specific conditions during weak survival
- Results apply to models without spontaneous death on homogeneous trees

## Abstract

In this paper, we consider the threshold-one contact process and the threshold-one voter model w/o spontaneous death on homogeneous trees $\mathbb{T}_d$, $d\ge 2$. Mainly inspired by the corresponding arguments for ordinary contact processes, we prove that the complete convergence theorem holds for these three systems under strong survival. When the systems survives weakly, complete convergence may also hold under certain transition and/or initial conditions.