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Universality of critical behaviour in a class of recurrent random walks
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  • Published: 29 April 2004

Universality of critical behaviour in a class of recurrent random walks

  • O. Hryniv1 &
  • Y. Velenik2 

Probability Theory and Related Fields volume 130, pages 222–258 (2004)Cite this article

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  • 14 Citations

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Abstract.

Let X0=0, X1, X2,.. be an aperiodic random walk generated by a sequence ξ1, ξ2,... of i.i.d. integer-valued random variables with common distribution p(·) having zero mean and finite variance. For anN-step trajectory and a monotone convex functionV: withV(0)=0, define Further, let be the set of all non-negative paths compatible with the boundary conditionsX0=a, X N =b. We discuss asymptotic properties of under the probability distribution N→∞ and λ→0, Za,bN,+,λ being the corresponding normalization. If V(·) grows not faster than polynomially at infinity, define H(λ) to be the unique solution to the equation Our main result reads that as λ→0, the typical height of X[α, N] scales as H(λ) and the correlations along decay exponentially on the scale H(λ)2. Using a suitable blocking argument, we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3/2. In the particular case of linear potential V(·), the characteristic length H(λ) is proportional to λ-1/3 as λ→0.

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Authors and Affiliations

  1. Statistical Laboratory, DPMMS, University of Cambridge, Cambridge, CB3 0WB, UK

    O. Hryniv

  2. Laboratoire de Mathématiques Raphaël Salem, UMR-CNRS 6085, Université de Rouen, 76821, Mont Saint Aignan, France

    Y. Velenik

Authors
  1. O. Hryniv
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  2. Y. Velenik
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Correspondence to O. Hryniv.

Additional information

Mathematics Subject Classification (2000):60G50, 60K35; 82B27, 82B41

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Hryniv, O., Velenik, Y. Universality of critical behaviour in a class of recurrent random walks. Probab. Theory Relat. Fields 130, 222–258 (2004). https://doi.org/10.1007/s00440-004-0353-z

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  • Received: 16 October 2003

  • Revised: 13 February 2004

  • Published: 29 April 2004

  • Issue Date: October 2004

  • DOI: https://doi.org/10.1007/s00440-004-0353-z

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Key words and phrases:

  • Random walks
  • Critical behaviour
  • Universality
  • Interface
  • Critical prewetting
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