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A Monotonicity Property for Once Reinforced Biased Random Walk on \(\mathbb {Z}^d\)

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Sojourns in Probability Theory and Statistical Physics - III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 300))

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Abstract

We study once-reinforced biased random walk on \(\mathbb {Z}^d\). We prove that for sufficiently large bias, the speed \(v(\beta )\) is monotone decreasing in the reinforcement parameter \(\beta \) in the region \([0,\beta _0]\), where \(\beta _0\) is a small parameter depending on the underlying bias. This result is analogous to results on Galton–Watson trees obtained by Collevecchio and the authors.

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Acknowledgements

This research was supported under Australian Research Council’s Discovery Programme (Future Fellowship project number FT160100166). DK is grateful to the University of Auckland for their hospitality and to the Ecole Polytechnique Fédérale de Lausanne (EPFL) to which he was affiliated to at the time this work was partly done.

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Authors and Affiliations

  1. School of Mathematics and Statistics, The University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia

    Mark Holmes

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Daniel Kious

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  1. Mark Holmes
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  2. Daniel Kious
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Correspondence to Daniel Kious .

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Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

Dedicated to Chuck Newman after 70 years of life and 50 years of massive contributions to probability. Bring on NYU Waiheke!

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Holmes, M., Kious, D. (2019). A Monotonicity Property for Once Reinforced Biased Random Walk on \(\mathbb {Z}^d\). In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - III. Springer Proceedings in Mathematics & Statistics, vol 300. Springer, Singapore. https://doi.org/10.1007/978-981-15-0302-3_10

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