Abstract

Universality of Critical Behaviour in a Class of Recurrent Random Walks O. Hryniv and Y. Velenik Probab. Theory Relat. Fields 130, 222-258 (2004). Let $X_0=0, X_1, X_2, \ldots$, be an aperiodic random walk generated by a sequence $x_1, x_2, \ldots$, of i.i.d. integer-valued random variables with common distribution $p(\,\cdot\,)$ having zero mean and finite variance. For an $N$-step trajectory $\textbf{X}=(X_0,X_1,\ldots,X_N)$ and a monotone convex function $V: \mathbb{R}^+\to\mathbb{R}^+$ with $V(0)=0$, define $\textbf{V}(\textbf{X})=V(|X_1|)+\cdots+V(|X_{N-1}|)$. Further, let $I_{N,+}^{a,b}$ be the set of all non-negative paths $\textbf{X}$ compatible with the boundary conditions $X_0=a$, $X_N=b$. We discuss asymptotic properties of $\textbf{X}$ in $I_{N,+}^{a,b}$ w.r.t. the probability distribution
\[ \textbf{P}_N^{a,b}(\textbf{X}) = (\textbf{Z}_N^{a,b})^{-1} \exp\{-\lambda \textbf{V}(\textbf{X})\} p(X_1-X_0)p(X_2-X_1)\cdots p(X_N-X_{N-1}), \] as $N\to\infty$ and $\lambda\to 0$, $\textbf{Z}_N^{a,b}$ being the corresponding normalization. If $V(\,\cdot\,)$ 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 $\lambda\to 0$, the typical height of $X_{[\alpha N]}$ scales as $H(\lambda)$ and the correlations along $\textbf{X}$ 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 $V(\,\cdot\,)$, the characteristic length $H(\lambda)$ is proportional to $\lambda^{-1/3}$ as $\lambda\to 0$. Key words: random walk, effective interface model, critical prewetting, wetting, critical behaviour, universality. Files: PDF file, Published version, bibtex