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I have encountered a physical system where the microstates are described by a vector $$k = \left[k_1, k_2, \ldots, k_n\right]$$ where all the $k_i$ are strictly positive integers smaller than some $k_{max}$. The associated energy for a microstate $k$ is given by $$a \sum_{i=1}^{n}k_i + b \sum_{i=1}^{n-1}\left|k_i-k_{i+1}\right|.$$ I am also interested in the case where the energy is given by $$a \sum_{i=1}^{n}k_i + b \sum_{i=1}^{n-1}\left(k_i-k_{i+1}\right)^2.$$

I was wondering whether such systems have studied before, and in particular whether something is known about the partition function of the system where $k_{max}\rightarrow \infty$. For a finite $k_{max}$ the partition function can be calculated numerically by using the transfer matrix method. One can approximate the partition function of the $k_{max} \rightarrow \infty$ case by taking $k_{max}$ sufficiently large, but I am hoping to get something more concrete than this.

I am primarily interested in the 1D open boundary case, but periodic boundary conditions and potential natural generalisations to higher dimensional lattices would also be of interest. One can also consider the case where the $k_i$ are now strictly positive real numbers instead.

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The limiting case $k_{\rm max}\to\infty$ is well known. When the interaction is $\sum_{k=1}^{n-1} |k_i - k_{i-1}|$, this is known as the (one-dimensional) SOS model; when the interaction is $\sum_{k=1}^{n-1} (k_i - k_{i-1})^2$, this is known as the (one-dimensional) discrete (or integer-valued) Gaussian free field (GFF).

The versions with continuous spins are respectively known as the continuous SOS model and the Gaussian free field (or, sometimes, as the harmonic crystal) .

In all cases, the spins are usually not supposed positive (that is, they take values in $\mathbb{Z}$ or $\mathbb{R}$). However, taking them positive is a common variant, useful to model an interface above a wall.

In the variant with the positivity constraint, the term $a\sum_{i=1}k_i$ (with $a>0$) models a layer of unstable phase above a wall (the height of the layer above site $i$ being given by $k_i$).

In dimension $1$, the fact that the spins take discrete or continuous values, and the particular form of the interaction potential (that is, SOS or GFF or any other reasonable one) play very little role. The behavior of this system (when $n\to\infty$) is well understood when $a$ is small: the width of the unstable layer if of order $a^{-1/3}$ and the correlation length of order $a^{-2/3}$. In fact, much more precise information is available: after a scaling by $a^{1/3}$ vertically and $a^{2/3}$ horizontally, the distribution of the (linear interpolation of) the function $i\mapsto k_i$ converges (as $n\to\infty$ and $a=a(n)\to 0$ not too fast) to the distribution of the trajectories of a stationary Ferrari-Spohn diffusion. This is proved in this paper. A substantially older paper relying on exact computation (via transfer matrix) rather than probabilistic tools, and obtaining weaker results is this one. Of course, exact computations are more sensitive to details, so the latter paper applies only to the continuous SOS model, while the former applies to a very general class of interactions (and, actually, to a very general class of confining potentials, not necessarily linear in the $k_i$, although this in general changes the relevant scaling).

For finite $k_{\rm max}$, the results would be essentially the same as long as $k_{\rm max} \gg a^{-1/3}$. When $k_{\rm max} \ll a^{-1/3}$, I'd expect that the confining potential plays no role. This should be easy to prove.

The higher-dimensional analogues have also been studied, although the results are far from being as complete.


As mentioned above, this is often used to model a layer of unstable phase above a hard wall. Interestingly, in the case where $a=\lambda/n$, where $n$ is the linear size of the system, the above-mentioned convergence of the rescaled interface to a Ferrari-Spohn diffusion was even proved in a much more complex setting, namely the 2-dimensional Ising model in an external magnetic field; see this paper.

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