The two-dimensional ${\ell }_{\infty }$ Directed Spanning Forest

Abstract

We investigate the ${\ell }_{\infty }$ \textit{ directed spanning forest} (DSF), a directed forest whose vertex set is given by a homogeneous Poisson point process, in which each Poisson point connects to the nearest Poisson point, measured in ${\ell }_{\infty }$ distance, with a strictly larger $y$- coordinate. In this paper, we prove that the two-dimensional ${\ell }_{\infty }$ DSF is connected and obtain an optimal estimate on the tail decay of coalescing time of two ${\ell }_{\infty }$ DSF paths. Similar estimates were earlier obtained in \cite{MR4203342} for the ${\ell }_2$ (Euclidean) DSF and in \cite{garcia2025directed} for the ${\ell }_p$ DSF with $p\in [1,\infty ]$ in two-dimension. The geometry of ${\ell }_{\infty }$ balls compels us to develop new arguments.

22 pages, 4 figures