The voter model chordal interface in two dimensions

Consider the voter model on a box of side length $L$ (in the triangular lattice) with boundary votes fixed forever as type 0 or type 1 on two different halves of the boundary. Motivated by analogous questions in percolation, we study several geometric objects at stationarity, as $L\rightarrow \infty$. One is the interface between the (large -- i.e., boundary connected) 0-cluster and 1-cluster. Another is the set of large "coalescing classes" determined by the coalescing walk process dual to the voter model.