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Monotonicity for excited random walk in high dimensions
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  • Open Access
  • Published: 07 April 2009

Monotonicity for excited random walk in high dimensions

  • Remco van der Hofstad1 &
  • Mark Holmes2 

Probability Theory and Related Fields volume 147, pages 333–348 (2010)Cite this article

  • 372 Accesses

  • 16 Citations

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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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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Remco van der Hofstad

  2. Department of Statistics, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Mark Holmes

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  1. Remco van der Hofstad
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Correspondence to Remco van der Hofstad.

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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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  • Received: 30 May 2008

  • Revised: 16 March 2009

  • Published: 07 April 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0215-9

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Mathematics Subject Classification (2000)

  • 60K35
  • 60K37
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