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

• Mark Holmes
• Daniel Kious
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 300)

## 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

## Notes

### 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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