Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Random directed forest and the Brownian web

Rahul Roy, Kumarjit Saha, and Anish Sarkar

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Abstract

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 $\mathbf{u}:=(\mathbf{u}(1),\mathbf{u}(2),\ldots,\mathbf{u}(d))$ is connected by an edge to another open vertex which has the minimum $L_{1}$ distance among all the open vertices $\mathbf{x}$ with $\mathbf{x}(d)>\mathbf{u}(d)$. 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 $d\geq4$. In addition, for $d=2$, we show that when properly scaled, the family of its paths converges in distribution to the Brownian web.

Résumé

Nous considérons le réseau $\mathbb{Z}^{d}$ dont les sommets sont ouverts ou fermés, respectivement avec probabilité $p$ et $1-p$. Chaque sommet ouvert $\mathbf{u}=(\mathbf{u}(1),\mathbf{u}(2),\dots,\mathbf{u}(d))$ est connecté par une arête au sommet ouvert $\mathbf{x}$ le plus proche de lui, pour la distance $L_{1}$, et satisfaisant $\mathbf{x}(d)>\mathbf{u}(d)$. Nous montrons que le graphe aléatoire résultant est presque sûrement un arbre pour $d=2$ et $3$, et qu’il est une collection infinie d’arbres disjoints pour $d\geq4$. De plus, pour $d=2$, nous montrons que la famille de ses trajectoires correctement renormalisées converge en loi vers la toile Brownienne.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1106-1143.

Dates
Received: 8 January 2014
Revised: 25 February 2015
Accepted: 25 February 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723513

Digital Object Identifier
doi:10.1214/15-AIHP672

Mathematical Reviews number (MathSciNet)
MR3531702

Zentralblatt MATH identifier
1375.60038

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Markov chain Random walk Directed spanning forest Brownian web

Citation

Roy, Rahul; Saha, Kumarjit; Sarkar, Anish. Random directed forest and the Brownian web. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1106--1143. doi:10.1214/15-AIHP672. https://projecteuclid.org/euclid.aihp/1469723513


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