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Type Transition of Simple Random Walks on Randomly Directed Regular Lattices

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Massimo Campanino  and
Dimitri Petritis
Massimo Campanino*
Affiliation:
Università degli Studi di Bologna
Dimitri Petritis*
Affiliation:
Université de Rennes 1
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, I-40126 Bologna, Italy. Email address: massimo.campanino@unibo.it
∗∗ Postal address: Institut de Recherche Mathématique, Université de Rennes I, Campus de Beaulieu, F-35042 Rennes Cedex, France. Email address: dimitri.petritis@univ-rennes1.fr
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Abstract

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Simple random walks on a partially directed version of Z2 are considered. More precisely, vertical edges between neighbouring vertices of Z2 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 simple random walk that is recurrent or transient, and show that there exists a critical value of the decay power, above which it is almost surely recurrent and below which it is almost surely transient.

Keywords

Markov chainrandom environmentrecurrence criteriarandom graphdirected graph

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1065 - 1080
Copyright
© Applied Probability Trust 

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