Theoretical Statistics and Mathematics Unit, ISI Delhi

August 1, 2012 (Wednesday) ,
3:30 PM at Webinar

Speaker:
Kumarjit Saha,
Indian Statistical Institute, Delhi

Title:
Directed Spanning Forest and Brownian web

Abstract of Talk

Consider the $d$ dimensional lattice ${\mathbb Z}^d$ where each vertex
is `open' or `closed' with probability $p$ or $1-p$ respectively. An
open vertex $\mathbf{v} := (\mathbf{v}(1), \mathbf{v}(2), \ldots,
\mathbf{v}(d))$ is connected by an edge to another open vertex $\bw$,
which has the minimum $L_1$ distance among all the open vertices $\bx$
with $\mathbf{x}(d)>\mathbf{v}(d)$. 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 aprropriately scaled, family of its paths converges in
distribution to the Brownian web.