Wetting transitions for a random line in long-range potential

Abstract

We consider a restricted Solid-on-Solid interface in ${\mathbb{Z}}_{+}$, 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 $m\left(b_0\right)\sim {\left(b_0^c-b_0\right)}^{\hat{I}¸/\left(1-\hat{I}¸\right)}$ where $\hat{I}¸=1-\frac{\sqrt{1-8\mathrm{w}}}{2}$; if $\mathrm{w}>1/8$, there is no wetting transition.

72 pages, 3 figures