# Fluctuations of an interface with hammock potential

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

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).