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Biased Random Walks on Galton–Watson Trees

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Branching Random Walks

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2151))

Abstract

This chapter is a brief presentation of the randomly biased random walk on trees in its slow regime. The model has been introduced by Lyons and Pemantle (Ann Probab 20:125–136, 1992), as an extension of Lyons’s deterministically biased random walk on trees (Lyons, Ann Probab 18:931–958, 1990; Lyons, Ann Probab 20:2043–2088, 1992).

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Notes

  1. 1.

    The root \(\varnothing \) is a vertex of the tree, but \(\stackrel{\leftarrow }{\varnothing }\) is not considered as a vertex of the tree.

  2. 2.

    This simple formula tells us that V plays the role of potential: The higher the potential value is on the path {x 1, …, x n }, the harder it is for the biased random walk to reach x.

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  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris, France

    Zhan Shi

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Shi, Z. (2015). Biased Random Walks on Galton–Watson Trees. In: Branching Random Walks. Lecture Notes in Mathematics(), vol 2151. Springer, Cham. https://doi.org/10.1007/978-3-319-25372-5_7

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