Type transition of simple random walks on randomly directed regular lattices

Simple random walks on a partially directed version of \(\mathbb{Z}^2\) are considered. More precisely, vertical edges between neighbouring vertices of \(\mathbb{Z}^2\) can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescri...

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Bibliographic Details
Published in:arXiv.org 2014-01
Main Authors: Campanino, Massimo, Petritis, Dimitri
Format: Article
Language:English
Subjects:
Apexes
Graph theory
Horizontal orientation
Lattices
Periodic functions
Perturbation
Random walk
Random walk theory
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Summary:Simple random walks on a partially directed version of \(\mathbb{Z}^2\) are considered. More precisely, vertical edges between neighbouring vertices of \(\mathbb{Z}^2\) can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function, the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of the simple random walk, i.e.\ its being recurrent or transient, and show that there exists a critical value of the decay power, above which the walk is almost surely recurrent and below which is almost surely transient.
ISSN:2331-8422