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

中文翻译：

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同构树上的一些阈值一吸引力相互作用粒子系统

我们考虑了阈值一接触过程、阈值一选民模型和同质树上具有正自发死亡的阈值一选民模型$\mathbb{T}_d$,$d\ge 2$. 主要受到接触过程的相应论证的启发，我们证明了完全收敛定理对于这三个系统在强生存下成立。当系统弱存活时，完全收敛也可能在某些过渡和/或初始条件下保持。