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.
Article PDF
Similar content being viewed by others
References
Abraham, D.B., Smith, E.R.: An exactly Solved Model with a Wetting Transition. J. Statist. Phys. 43 (3/4), 621–643 (1986)
Billingsley, P.: Convergence of probability measures. Wiley, 1968
Billingsley, P.: Convergence of probability measures, 2nd edition. Wiley, 1999
Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4 (3), 480–485 (1976)
de Bruijn, N.G.: Asymptotic methods in analysis. North-Holland, 1958
Gnedenko, B.V.: The theory of probability. Chelsea, 1962
Kaigh, W.D.: An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4 (1), 115–121 (1976)
Lindvall, T.: Lectures on the coupling method. Wiley, 1992
Louchard, G.: Kac’s formula, Levy’s local time and Brownian excursion. J. Appl. Probab. 21 (3), 479–499 (1984)
Prähofer, M., Spohn, H.: Scale Invariance of the PNG Droplet and the Airy Process. J. Statist. Phys. 108 (5/6), 1071–1106 (2002)
Velenik, Y.: Entropic Repulsion of an Interface in an External Field. to appear in Probab. Theory Relat. Fields (2004); DOI 10.1007/s00440-003-0328-5
Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton, NJ, 1941, pp. 406
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000):60G50, 60K35; 82B27, 82B41
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0353-z