Directed Spanning Forest for Ginibre Ensemble Is a Tree

In this note, we discuss the Burton-Keane type argument provided by Coupier and Tran to show that the two-dimensional directed spanning forest (DSF) constructed on a homogeneous Poisson point process is connected. We then modify it to show that the DSF constructed on the Ginibre ensemble, which is a strongly correlated point process, is connected a.s.