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.
Keywords
- Once-reinforced random walk
- Reinforced random walk
- Large bias
- Coupling
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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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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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