The $\ell_{\infty}$ Directed Spanning Forest

We study the $\ell_{\infty}$\textit{ directed spanning forest}(DSF), which is a directed forest with vertex set given by a homogeneous Poisson point process such that each Poisson point connects to the nearest Poisson point (in $\ell_{\infty}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the $\ell_{\infty}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two $\ell_{\infty}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the $\ell_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of $\ell_\infty$ balls compel us to develop new argument.