The ${\mathrm{\ell }}_{\mathrm{\infty }}$ Directed Spanning Forest

We study the ${\mathrm{\ell }}_{\mathrm{\infty }}$$\mathit{\text{directed}}\mathit{\text{spanning}}\mathit{\text{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 ${\mathrm{\ell }}_{\mathrm{\infty }}$ distance) with a strictly larger $y$-coordinate. In this paper, we prove that the ${\mathrm{\ell }}_{\mathrm{\infty }}$ DSF is connected and we find optimal estimates on the tail distribution of coalescing time of two ${\mathrm{\ell }}_{\mathrm{\infty }}$ DSF paths. Similar estimates were earlier obtained in \cite{coupier20212d} for the ${\mathrm{\ell }}_2$ (Euclidean) DSF and showed that when properly scaled, it converges in distribution to the Brownian web. The geometry of ${\mathrm{\ell }}_{\mathrm{\infty }}$ balls compel us to develop new argument.