The 2D-directed spanning forest is almost surely a tree

Authors

  • David Coupier,

    Corresponding author
    1. Laboratoire Paul Painlevé, Equipe Probabilités Statistiques, UFR de Mathématiques, Université des Sciences et Technologies Lille 1. Cité scientifique, 59 655 Villeneuve d'Ascq Cedex, France
    • Laboratoire Paul Painlevé, Equipe Probabilités Statistiques, UFR de Mathématiques, Université des Sciences et Technologies Lille 1. Cité scientifique, 59 655 Villeneuve d'Ascq Cedex, France
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  • Viet Chi Tran

    Corresponding author
    1. Laboratoire Paul Painlevé, Equipe Probabilités Statistiques, UFR de Mathématiques, Université des Sciences et Technologies Lille 1. Cité scientifique, 59 655 Villeneuve d'Ascq Cedex, France
    • Laboratoire Paul Painlevé, Equipe Probabilités Statistiques, UFR de Mathématiques, Université des Sciences et Technologies Lille 1. Cité scientifique, 59 655 Villeneuve d'Ascq Cedex, France
    Search for more papers by this author

Abstract

We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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