Probability Theory and Related Fields

, Volume 130, Issue 2, pp 222–258

Universality of critical behaviour in a class of recurrent random walks


DOI: 10.1007/s00440-004-0353-z

Cite this article as:
Hryniv, O. & Velenik, Y. Probab. Theory Relat. Fields (2004) 130: 222. doi:10.1007/s00440-004-0353-z


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 Open image in new window and a monotone convex functionV: Open image in new window withV(0)=0, define Open image in new window Further, let Open image in new window be the set of all non-negative paths Open image in new window compatible with the boundary conditionsX0=a, XN=b. We discuss asymptotic properties of Open image in new window under the probability distribution Open image in new windowN→∞ 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 Open image in new window Our main result reads that as λ→0, the typical height of X, N] scales as H(λ) and the correlations along Open image in new window 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.

Key words and phrases:

Random walks Critical behaviour Universality Interface Critical prewetting 

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Statistical Laboratory, DPMMSUniversity of CambridgeCambridgeUK
  2. 2.Laboratoire de Mathématiques Raphaël Salem, UMR-CNRS 6085Université de RouenMont Saint AignanFrance

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