Wetting transitions for a random line in long-range potential

Abstract

We consider a restricted Solid-on-Solid interface in $\Bbb{Z}_{+}$, subject to a potential $V\left( n\right) $ behaving at infinity like $-\mathrm{w}/n^{2}$. Whenever there is a wetting transition as $b_{0}\equiv \exp V\left( 0\right) $ is varied, we prove the following results for the density of returns $m\left( b_{0}\right) $ to the origin: if $\mathrm{w}<-3/8$, then $m\left( b_{0}\right) $ has a jump at $b_{0}^{c}$; if $-3/8<\mathrm{w}<1/8$, then $m\left( b_{0}\right) \sim \left( b_{0}^{c}-b_{0}\right) ^{\theta /\left( 1-\theta \right) }$ where $\theta =1-\frac{\sqrt{1-8\mathrm{w}}}{2}$; if $\mathrm{w}>1/8$, there is no wetting transition.