In this paper, we prove that the bulk of 2-dimensional DLA starting from a long line segment on the x x -axis has a scaling limit to the stationary DLA. The main phenomenological difficulty is the multi-scale, non-monotone interaction of the DLA arms. We overcome this via a coupling scheme between the two processes and an intermediate DLA process with absorbing mesoscopic boundary segments. Our result allows to import results from the more amenable infinite stationary DLA process to the more physical finite aggregations.

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.

In this paper, we consider the threshold-one contact process and the threshold-one voter model w/o spontaneous death on homogeneous trees T d , d ≥ 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.