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Random walk in random scenery and self-intersection local times in dimensions d ≥  5
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  • Published: 15 August 2006

Random walk in random scenery and self-intersection local times in dimensions d ≥  5

  • Amine Asselah1 &
  • Fabienne Castell1 

Probability Theory and Related Fields volume 138, pages 1–32 (2007)Cite this article

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Abstract

Let {S k , k ≥  0} be a symmetric random walk on \({\mathbb Z}^d\), and \(\{\eta(x), x\in {\mathbb Z^d\}}\) an independent random field of centered i.i.d. random variables with tail decay \(P(\eta(x)> t)\approx\exp(-t^{\alpha})\). We consider a random walk in random scenery, that is \(X_n=\eta(S_0)+\dots+\eta(S_n)\). We present asymptotics for the probability, over both randomness, that {X n  > n β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process \(\sum_x l_n^2(x)\), where l n (x) is the number of visits of site x up to time n.

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Authors and Affiliations

  1. L.A.T.P., Université de Provence, 39 Rue Joliot-Curie, 13453, Marseille Cedex 13, France

    Amine Asselah & Fabienne Castell

Authors
  1. Amine Asselah
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  2. Fabienne Castell
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Correspondence to Amine Asselah.

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Asselah, A., Castell, F. Random walk in random scenery and self-intersection local times in dimensions d ≥  5. Probab. Theory Relat. Fields 138, 1–32 (2007). https://doi.org/10.1007/s00440-006-0014-5

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  • Received: 19 January 2006

  • Revised: 19 May 2006

  • Published: 15 August 2006

  • Issue Date: May 2007

  • DOI: https://doi.org/10.1007/s00440-006-0014-5

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Keywords

  • Moderate deviations
  • Self-intersection
  • Local times
  • Random walk
  • Random scenery

AMS Subject Classification Numbers (2000)

  • 60K37
  • 60F10
  • 60J55
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