Random directed forest and the Brownian web

Article Type

Research Article

Publication Title

Annales de l'institut Henri Poincare (B) Probability and Statistics

Abstract

Consider the d dimensional lattice Zd where each vertex is open or closed with probability p or 1 - p respectively. An open vertex u := (u(1), u(2),...,u(d)) is connected by an edge to another open vertex which has the minimum L1 distance among all the open vertices x with x(d) > u(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 ≥4. In addition, for d = 2, we show that when properly scaled, the family of its paths converges in distribution to the Brownian web.

First Page

1106

Last Page

1143

DOI

10.1214/15-AIHP672

Publication Date

8-1-2016

Comments

Open Access; Green Open Access

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