Wetting transitions for a random line in long-range potential

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 $m\left( b_{0}\right) \sim \left( b_{0}^{c}-b_{0}\right) ^{θ/\left( 1-θ\right) }$ where $θ=1-\frac{\sqrt{1-8\mathrm{w}}}{2}$; if $\mathrm{w}>1/8$, there is no wetting transition.

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