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Universality of critical behaviour in a class of recurrent random walks 

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Published in  Probability Theory and Related Fields. 2004, vol. 130, no. 2, p. 222258  
Abstract  Let X_0=0, X_1, X_2,..., be an aperiodic random walk generated by a sequence xi_1, xi_2,..., of i.i.d. integervalued random variables with common distribution p(.) having zero mean and finite variance. For an Nstep trajectory X=(X_0,X_1,...,X_N) and a monotone convex function V: R^+ > R^+ with V(0)=0, define V(X)= sum_{j=1}^{N1} V(X_j). Further, let I_{N,+}^{a,b} be the set of all nonnegative paths X compatible with the boundary conditions X_0=a, X_N=b. We discuss asymptotic properties of X in I_{N,+}^{a,b} w.r.t. the probability distribution P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{1} exp{lambda V(X)} prod_{i=0}^{N1} p(X_{i+1}X_i) as N > infinity and lambda > 0, Z_{N}^{a,b} being the corresponding normalization. If V(.) grows not faster than polynomially at infinity, define H(lambda) to be the unique solution to the equation lambda H^2 V(H) =1. Our main result reads that as lambda > 0, the typical height of X_{[alpha N]} scales as H(lambda) and the correlations along X decay exponentially on the scale H(lambda)^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(lambda) is proportional to lambda^{1/3} as lambda > 0.  
Keywords  Random walks — Critical behaviour — Universality — Interface — Critical prewetting  
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Citation (ISO format)  HRYNIV, O., VELENIK, Yvan. Universality of critical behaviour in a class of recurrent random walks. In: Probability Theory and Related Fields, 2004, vol. 130, n° 2, p. 222258. doi: 10.1007/s004400040353z https://archiveouverte.unige.ch/unige:6382 