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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.

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Thanks for sharing this. I would like to ask if the temperature is relevant in this problem. I mean if I would like to try an argument using cluster expansion for the second moment, in the regime of low temperatures it could this have any relevance for the problem ? –  Leandro Oct 7 '11 at 21:07
No, temperature should be irrelevant (energy being always $0$ or $\infty$). I (and quite a few others) have tried various approaches to this problem, but it still resists ;) . Actually, there are very few tools to deal with such systems of purely entropic nature (and many interesting such systems). –  Yvan Velenik Oct 8 '11 at 7:43
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What about numerical simulation? It should be possible to construct a Monte-Carlo simulation, evaluate it is for various values of $N$, and extrapolate (note: the extrapolation may involve accepting the heuristic arguments that you know).

I'm familar with work like Ground-State Roughness of the Disordered Substrate and Flux Lines in two dimensions, C. Zeng, A.A. Middleton, and Y. Shapir, Phys. Rev. Letts. 77, 3204 (1996) which extrapolate from finite size numerical simulations, but involve quenched disorder, and thus not directly applicable (this paper does refer to H. W. J. Blote and H. H. J. Hilhorst J. Phys. A 15, L631 (1982) in reference to logarithmically rough interfaces, but I haven't read that paper).

You may also want to check 'S. F. Edwards and D. R. Wilkinson. Proc. Roy. Soc. London A 381, 17(1982) and other literature related to the "restricted solid on solid" model for deposition/interface growth.

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Well, I wouldn't be surprised if this has been done long ago (maybe for a variant of this model). There is actually no doubt that the result is as claimed above. The question is really about good heuristic arguments about why this should be true (this might in turn lead to a rigorous proof of this fact, which is what I'm after). Unfortunately, numerical simulations do not lead to any sort of real understanding... By the way, some colleagues have made substantial progress about this problem (in a slightly different setting), which I'll describe once their preprint is made available. –  Yvan Velenik Oct 30 '13 at 17:54
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