, Volume 130, Issue 2, pp 222-258

Universality of critical behaviour in a class of recurrent random walks

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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 conditionsX 0=a, X N =b. We discuss asymptotic properties of under the probability distribution N→∞ and λ→0, Z a,b N,+,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 , 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.

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