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Let $\Lambda_n = \{ -n,\ldots,n\}^2$ and let $\{X_i\}_{i\in \Lambda_n}$ be the family of $\mathbb{R}$-valued of random variables with the density proportional to

$\exp(-\sum_{i\sim j} |X_i - X_j|),$

where $i\sim j$ spans over $\Lambda_{n+1}$, $\sim$ denotes the neigbourhood in $Z^2$ and we assume that for $i \in \Lambda_{n+1} \setminus \Lambda_n$ we have $X_i =0 $ (boundary conditions). This model is known as the solid-on-solid (SOS) model.

My questions are:

  1. Does anybody know if there exists a proof of a upperbound for the height of this model?

  2. Does anybody know a "simple proof" of any upperbound? (I would be happy to have a weak estimate if it can be proved easily).

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