Publications and Preprints

Consider the $d$ dimensional lattice $\Z^d$ where each vertex is \textit{open} or \textit{closed} with probability $p$ or $1-p$ respectively. An open vertex $\bu := (\bu(1), \bu(2),\dotsc,\bu(d))$ is connected by an edge to another open vertex which has the minimum $L_1$ distance among all the open vertices with $\bx(d)>\bu(d)$. It is shown that this random graph is a tree almost surely for $d=2$ and $3$ and it is an infinite collection of disjoint trees for $d\geq 4$. In addition for $d=2$, we show that when properly scaled, the family of its paths converge in distribution to the Brownian web.