Layering transitions, exact wetting point, and critical behavior of the two-dimensional Solid-on-Solid wetting model beyond the low-temperature regime

The two-dimensional Solid-on-Solid (SOS) model above a wall — an integer height field $\varphi:\mathbb{Z}^2\to\mathbb{Z}$ with energy $\beta\sum_{x\sim y}|\varphi(x)-\varphi(y)|$ and a reward $h$ per wall contact — undergoes a wetting transition at $h_w(\beta)$ and, in the localized phase, an infinite sequence of first-order layering transitions $h_n^*(\beta)\downarrow h_w(\beta)$ at which the typical interface height drops by one. Lacoin's two-part work established this picture for sufficiently large $\beta$: the exact critical point $h_w(\beta)=\log(e^{4\beta}/(e^{4\beta}-1))$, the sharp critical behavior of the free energy, the existence and sharp asymptotic location of the layering transitions, and the associated percolating Gibbs states. This resolved the layering-existence conjectures posed as future work in the first of the two papers. Three linked questions constitute the present problem. Questions (i) and (ii) extend the low-temperature picture just described beyond the regime in which it is proved, while question (iii) is a quantitative refinement within that regime: (i) does the layering picture persist at moderate and small $\beta$, where contour and cluster expansions diverge, or does it fail below a computable threshold — and what replaces it; (ii) does the exact formula for $h_w(\beta)$, proven only for $\beta>\beta_1\in(\log 2,\log 3)$, hold at all temperatures, and what is the correct critical behavior of the free energy there; (iii) do the peak-probability constants $\alpha_1(\beta),\alpha_2(\beta)$ governing the sharp asymptotics admit explicit values. Continuous-height (gradient-field) interfaces are outside the scope of this problem.

Background

The model. The SOS model is a simplified description of a phase-separating interface in three dimensions (e.g. the Dobrushin interface of the low-temperature Ising model): the interface is the graph of an integer-valued height field $\varphi:\mathbb{Z}^2\to\mathbb{Z}$, discarding overhangs and bubbles. With inverse temperature $\beta>0$, each unit height difference between neighboring sites costs $\beta$, so contours (level lines of the height function) of length $\ell$ carry weight $e^{-\beta\ell}$ (a Peierls-type weight). In the wetting variant, the interface is constrained to stay above a hard wall ($\varphi\ge 0$) and receives a reward $h$ per contact, i.e. per site with $\varphi(x)=0$; the free energy $f(\beta,h)=\lim_{N\to\infty}N^{-2}\log Z_{N,\beta}^h$ is convex and non-decreasing in $h$, and $\partial_h f$ is the asymptotic contact fraction.

Wetting. Chalker (1982) proved that for every $\beta>0$ there is a critical reward $h_w(\beta)>0$ separating a delocalized phase $h<h_w(\beta)$ — where the contact fraction vanishes and the interface is pushed away from the wall by entropic repulsion (in a box of side $N$ it floats at height of order $(4\beta)^{-1}\log N$) — from a localized phase $h>h_w(\beta)$ with positive contact fraction, and obtained the bounds $\log(e^{4\beta}/(e^{4\beta}-1))\le h_w(\beta)\le\log(16(e^{\beta}+1)/(e^{\beta}-1))$.

Layering. Physics predicts that as $h\downarrow h_w(\beta)$ the typical height does not decrease continuously but by unit steps: at first-order layering transitions $h_n^*(\beta)\downarrow h_w(\beta)$ the typical height drops from $n$ to $n-1$. A Gibbs state (DLR/Dobrushin–Lanford–Russo equilibrium measure) is said to percolate at level $n$ when the level set $\{x:\varphi(x)=n\}$ has a unique infinite connected component of positive density while all connected components of $\{x:\varphi(x)>n\}$ and $\{x:\varphi(x)<n\}$ are finite; a first-order transition corresponds to a discontinuity of $\partial_h f$. Alexander–Dunlop–Miracle-Solé (2011) proved localization at level $n$ and analyticity of $f$ on suitable intervals, but only in ranges requiring $\beta\ge c\log n$.

Lacoin's low-temperature resolution. Part I (Comm. Math. Phys. 362 (2018) 1007–1048) identifies the wetting point exactly: $h_w(\beta)=\log(e^{4\beta}/(e^{4\beta}-1))$ for $\beta$ beyond the contour-summability threshold $\beta_1\in(\log 2,\log 3)$ (the condition that dilute contour sums converge), and proves the sharp critical behavior $\bar f(\beta,u):=f(\beta,h_w(\beta)+u)-f(\beta)\sim F(\beta,u)$ for large $\beta$, where $J:=e^{-2\beta}$ and $F(\beta,u)=\max_{n\ge 0}(\alpha_1J^{2n}u-\frac{2\alpha_2(J^3-J^4)}{1-J^3}J^{3n})$ is piecewise affine with angular points $u_n=(\frac{2\alpha_2}{\alpha_1(1+J)})J^{n+2}$ forming a geometric sequence. The constants are peak probabilities of the free (unpinned) infinite-volume SOS measure $\mathbf{P}_\beta$: a column of height $n$ above a flat level is surrounded by contours of total length $4n$, hence has probability of order $e^{-4\beta n}$, and $\alpha_1(\beta)=\lim_{n\to\infty}e^{4\beta n}\mathbf{P}_\beta[\varphi(0)\ge n]$, $\alpha_2(\beta)=\lim_{n\to\infty}e^{6\beta n}\mathbf{P}_\beta[\min(\varphi(0),\varphi(1))\ge n]$ exist and are positive, but their values are not computed. Part II (J. École Polytechnique — Math. 7 (2020) 1–62) proves the full layering conjecture for $\beta$ sufficiently large: $f$ is $C^\infty$ off a decreasing sequence $(u_n^*(\beta))$ with $u_n^*\sim u_n$ sharply (in fact $e^{2\beta n}u_n^*\to 2\alpha_2J^2/(\alpha_1(1+J))$), non-differentiable at each $u_n^*$, with a unique finite-mean translation-invariant Gibbs state per interval percolating at level $n$, coexistence of extremal percolating states at transition points, and exponential decay of correlations. All of these results live in the rigid low-temperature regime where the contour representation converges and cluster expansions (convergent perturbative expansions of the partition function over dilute polymers/contours) apply, with rigidity resting on the correlation decay of Brandenberger–Wayne (1982).

The unresolved regime. For $\beta$ at and below the contour-summability threshold no technique is currently known: even the order of the critical behavior of $\bar f$ is unknown there (for $\beta>\beta_1$ the order $\bar f\asymp u^3$ is known). In dimensions $d\ge 3$, where the SOS surface is rigid at every temperature (Bricmont–Fontaine–Lebowitz 1982) and the analogous formula $h_w^{(d)}(\beta)=\log(e^{2d\beta}/(e^{2d\beta}-1))$ is only 'reasonable to hope' for all $\beta$, the same question is open. For continuous-height (gradient-field) interfaces above a wall, a non-trivial wetting transition is known (Caputo–Velenik 2000) but the critical value and critical behavior are not identified.

Problem Statement

Common objective. Complete the phase diagram, in the $(\beta,h)$ plane, of the two-dimensional SOS wetting model above a wall: the location of the wetting point $h_w(\beta)$, the layering structure above it, and the quantitative critical behavior of the free energy — beyond the low-temperature regime in which all currently known results are proved. Three linked questions constitute the problem.

(1) Temperature range of the layering picture. Determine for which $\beta>0$ the layering picture established at large $\beta$ holds: a decreasing sequence $(h_n^*(\beta))_{n\ge 1}$ with $\lim_n h_n^*(\beta)=h_w(\beta)$ such that $f(\beta,\cdot)$ is infinitely differentiable on $\mathbb{R}\setminus(\{h_n^*\}\cup\{h_w(\beta)\})$ and not differentiable at each $h_n^*$, with, on each interval $(h_{n+1}^*,h_n^*)$, a unique finite-mean translation-invariant Gibbs state percolating at level $n$, and at each $h_n^*$ the coexistence of (at least) two extremal states percolating at levels $n-1$ and $n$. In particular: does this hold for every $\beta>0$; and if not, identify the threshold and establish what replaces the layering structure below it — e.g. whether infinitely many discontinuity points of $\partial_h f$ accumulating at $h_w(\beta)$ persist, whether the transitions become continuous, or what the regularity of $f$ near $h_w(\beta)$ is.

(2) Critical point and critical behavior at moderate temperature. (a) Prove or disprove that $h_w(\beta)=\log(e^{4\beta}/(e^{4\beta}-1))$ for every $\beta>0$; the lower bound $\log(e^{4\beta}/(e^{4\beta}-1))\le h_w(\beta)$ holds for all $\beta$ (Chalker) and the formula is proved for $\beta>\beta_1\in(\log 2,\log 3)$, so the open range is $0<\beta\le\beta_1$. (b) Determine the critical behavior of $\bar f(\beta,u)=f(\beta,h_w(\beta)+u)-f(\beta)$ as $u\downarrow 0$ for $\beta$ outside the cluster-expansion regime: for $\beta>\beta_1$ the order $\bar f(\beta,u)\asymp u^3$ is known, and for $\beta$ large the sharp form $\bar f\sim F(\beta,u)$ with $F$ as in the background; below the contour-summability threshold not even the order is known.

(3) Explicit constants. Derive explicit closed-form values, or a sharp analytic characterization (e.g. a rigorously convergent series representation with error bounds), of $\alpha_1(\beta)=\lim_{n\to\infty}e^{4\beta n}\mathbf{P}_\beta[\varphi(0)\ge n]$ and $\alpha_2(\beta)=\lim_{n\to\infty}e^{6\beta n}\mathbf{P}_\beta[\min(\varphi(0),\varphi(1))\ge n]$, which determine the sharp critical asymptotics and the layering-point locations in the low-temperature regime.

Partial progress — e.g. component (1) or (2) established on a temperature range strictly larger than the low-temperature regime in which the cited results are proved (for (2a), for some $0<\beta\le\beta_1$), or an explicit value or rigorous characterization in (3) — constitutes progress toward the objective, which is itself resolved only when (1), (2) and (3) are all settled.

References

1. Hubert Lacoin, 'Wetting and layering for Solid-on-Solid I: Identification of the wetting point and critical behavior', Communications in Mathematical Physics 362(3):1007–1048, 2018. DOI: 10.1007/s00220-018-3162-4. https://doi.org/10.1007/s00220-018-3162-4 (arXiv:1703.06162)
2. Hubert Lacoin, 'Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy', Journal de l'École Polytechnique — Mathématiques 7:1–62, 2020. DOI: 10.5802/jep.110. https://doi.org/10.5802/jep.110 (arXiv:1712.03736)
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