Theoretical Statistics and Mathematics Unit, ISI Delhi

Random directed forest and the Brownian web

by Rahul Roy, Kumarjit Saha and Anish Sarkar

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.

isid/ms/2014/04 [fulltext]

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