This question is related to that one. I ask it here since comments are too short for the extended discussion that was going on there.
I am interested in a very simple interface model. To each $x\in\mathbb{Z}^2$, we associate a random height $h_x\in\mathbb{R}$. Let $\Lambda_N=\{-N,\ldots,N\}^2$. Assume $h_x\equiv 0$ outside $\Lambda_N$. To a pair of neighbouring heights, we associate an energy $0$ if $|h_x-h_y| < 1$ and an energy $+\infty$ otherwise. We then consider the corresponding Gibbs measure. In other words, we put the uniform measure on height configurations satisfying $|h_x-h_y| < 1$ for all pairs of neighbouring vertices, and equal to $0$ outside $\Lambda_N$.
It is an open problem to prove that the variance of $h_0$ diverges as $\log N$, as $N\to\infty$ (actually, it's even open to prove that it diverges at all!).
On the other hand, it is known to hold, if one replaces $+\infty$ by a suitable function diverging outside the interval (fast enough to guarantee existence of the measure, of course). Obviously, one cannot take the limit in the known arguments...
My question: What are quantitative heuristic arguments implying such a claim. By quantitative, I mean that I don't want something like "by analogy with the discrete massless free field", because I already know that ;) . I'd really like a non-rigorous, but mathematical derivation.
Update (April 27, 2014): two colleagues have been able to (rigorously) settle this question in a slightly different geometry (periodic boundary conditions, the spin at the origin forced to be 0). Their preprint can be found here: arXiv:1404.5895. Nevertheless, I'm still intertested in good physical heuristics.
