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February 2022 Tree builder random walk: Recurrence, transience and ballisticity
Giulio Iacobelli, Rodrigo Ribeiro, Glauco Valle, Leonel Zuaznábar
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Bernoulli 28(1): 150-180 (February 2022). DOI: 10.3150/21-BEJ1337

Abstract

The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of Markovian and non-Markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or transient. If s is odd, the walker is ballistic, thus transient. If s is even, the walker’s behavior can be explained from local properties of the growing tree and it can be either null recurrent or it gets trapped on some limited part of the growing tree.

Funding Statement

R.R. is supported by The Stochastic Models of Disordered and Complex Systems (NC120062) supported by the Millenium Scientific Initiative of the Ministry of Science and Technology (Chile). G.V. is supported by CNPq grant 308006/2018-6, Universal CNPq project 421383/2016-0 and FAPERJ grant E-26/203.048/2016. L.Z. is supported by PNPD/CAPES grant 88882.315481/2013-01.

Acknowledgements

We thank the referee for carefully reading our manuscript and for the valuable comments which helped us improve the paper.

Citation

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Giulio Iacobelli. Rodrigo Ribeiro. Glauco Valle. Leonel Zuaznábar. "Tree builder random walk: Recurrence, transience and ballisticity." Bernoulli 28 (1) 150 - 180, February 2022. https://doi.org/10.3150/21-BEJ1337

Information

Received: 1 April 2020; Revised: 1 February 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1337

Keywords: Ballisticity , random environment , Random trees , Random walks , recurrence , transience

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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