Abstract
We prove that the drift θ(d, β) for excited random walk in dimension d is monotone in the excitement parameter \({\beta \in [0,1]}\) , when d is sufficiently large. We give an explicit criterion for monotonicity involving random walk Green’s functions, and use rigorous numerical upper bounds provided by Hara (Private communication, 2007) to verify the criterion for d ≥ 9.
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Acknowledgments
The work of RvdH and MH was supported in part by Netherlands Organisation for Scientific Research (NWO). The work of MH was also supported by a FRDF grant from the University of Auckland. The authors thank Itai Benjamini for suggesting this problem to us, Takashi Hara for providing the Green’s functions upper bounds in (5.4), and an anonymous referee for many helpful suggestions that significantly improved the presentation of this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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van der Hofstad, R., Holmes, M. Monotonicity for excited random walk in high dimensions. Probab. Theory Relat. Fields 147, 333–348 (2010). https://doi.org/10.1007/s00440-009-0215-9
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DOI: https://doi.org/10.1007/s00440-009-0215-9