Macroscopic shape of the (2+1)-dimensional solid-on-solid surface with wall pinning in the delocalized phase: two-level concentration, pinning-dependent height shift, and level-line scaling limits

The (2+1)-dimensional solid-on-solid (SOS) surface above an impenetrable wall, with an attractive pinning reward $h$ per site at height zero, undergoes a wetting transition at a critical point $h_w(\beta)$, identified for large $\beta$ as $h_w(\beta)=\log(e^{4\beta}/(e^{4\beta}-1))$: for $h>h_w$ the surface is localized, while for $0\le h\le h_w$ it is delocalized and lifted to logarithmic height by entropic repulsion. For the unpinned surface ($h=0$) a full shape theorem is known: concentration on the two adjacent levels $\lfloor(1/4\beta)\log N\rfloor$ and one below, unique macroscopic level lines with explicit limiting shapes, and cube-root fluctuations. Feldheim–Yang (2023) extended the typical-height result to all subcritical $h\in(0,h_w)$ and reduced the conjectured critical height $\lfloor(1/6\beta)\log N\rfloor$ (a conjecture of Lacoin) to a bound on isolated zeros; Chen–Gheissari–Lubetzky (2024) then proved that critical height with a sharp $+1/3$ correction. This record targets the surviving open core: the macroscopic shape description of the pinned surface throughout the delocalized phase. Subcritically, prove two-level concentration on $\lfloor(1/4\beta)\log N\rfloor-c(h)$ and the level below, with the pinning-induced shift $c(h)$ (expected to diverge as $h\uparrow h_w$), together with level-line structure and fluctuations analogous to the $h=0$ theory; at criticality, prove the corresponding critical shape theorem: unique macroscopic level set, limiting shape, fluctuation bounds, and the dominant level for every system size.

Background

The solid-on-solid (SOS) model is a classical random-surface model of a crystal interface: on the box $\Lambda_N=[[1,N]]^2\subset\mathbb{Z}^2$ one assigns an integer height $\phi(x)$ to each site $x$, with height $0$ on the outer boundary, and the probability of a configuration is proportional to $\exp(-\beta H_N(\phi))$, where $\beta>0$ is the inverse temperature and $H_N(\phi)=\sum_{x\sim y}|\phi(x)-\phi(y)|$ sums absolute height differences over nearest-neighbor pairs (boundary sites included, at height $0$). Constraining the heights to be nonnegative, $\phi\ge 0$, models an impenetrable wall (floor) under the surface. In $2+1$ dimensions (a two-dimensional surface over a two-dimensional substrate) the unconstrained surface undergoes a roughening transition in $\beta$ — rigid at large $\beta$, logarithmically rough at small $\beta$ — and the wall problem is studied at large $\beta$.

Even with no energetic preference for height, the wall has a subtle effect: the constraint $\phi\ge 0$ removes configurations with deep negative excursions and pushes the surface up to logarithmic height, a phenomenon called entropic repulsion. Bricmont, El Mellouki and Fröhlich (1986) bounded the average height between constant multiples of $(1/\beta)\log N$; Caputo, Lubetzky, Martinelli, Sly and Toninelli (2014) proved that for $\beta\ge 1$ the height of almost all sites concentrates at $H=\lfloor(1/4\beta)\log N\rfloor$ with order-one fluctuations, and in a companion paper (JEMS 2016) gave the full macroscopic shape: with high probability (probability tending to $1$ as $N\to\infty$) the height of almost every site takes one of the two adjacent values $H$ or $H-1$; for each height level there is a unique macroscopic level line (the contour of the level set, i.e. the boundary separating sites at height at least that level from sites below it); along sequences of system sizes whose fractional parts of $(1/4\beta)\log N$ converge to a noncritical value, the rescaled macroscopic level lines converge to explicit convex shapes with flat boundary components; and the highest macroscopic level line fluctuates by $N^{1/3+o(1)}$ along the flat part (cube-root fluctuations; a matching lower bound of order $N^{1/3}$ was recently proved by Caddeo, Kim and Lubetzky 2024).

Adding a pinning (wetting) reward — an attraction of the surface to the wall — changes the measure to

$$\mathbb{P}^{\beta,h}_N(\phi)\propto\exp\left(-\beta H_N(\phi)+h\cdot|\{x\in\Lambda_N:\phi(x)=0\}|\right),\qquad \phi\ge 0,$$

with pinning parameter $h\ge 0$: each site touching the wall collects the reward $h$. Chalker (1982) showed that there is a critical wetting point $h_w(\beta)>0$ separating a delocalized phase $h<h_w$, in which the surface still escapes to logarithmic height and a vanishing fraction of sites touch the wall, from a localized phase $h>h_w$ in which the surface stays at a bounded height. Lacoin (2018, 2020) proved, for sufficiently large $\beta$, that $h_w(\beta)=\log(e^{4\beta}/(e^{4\beta}-1))$, that the free-energy excess is of order $u^3$ at $h=h_w+u$, and that the localized phase exhibits an infinite sequence of first-order layering transitions (discontinuous jumps of the typical height between integer levels, accumulating at $h_w$ from above).

Feldheim and Yang (2023) initiated the fine description of the delocalized phase with pinning. They proved (for $\beta\ge 1$) that for every fixed subcritical $h\in(0,h_w)$ the surface still concentrates at $H=\lfloor(1/4\beta)\log N\rfloor$ with constant-order fluctuations, via two-sided tail bounds; and at criticality they proved that the number of non-isolated zero sites — sites at height $0$ having a nearest neighbor also at height $0$, denoted $q_{2+}$, as opposed to isolated zeros $q_1$, which are sites at height $0$ all of whose neighbors are at least $1$ — is at most of order $N$ with high probability. They further showed that the conjectured critical typical height $H_w=\lfloor(1/6\beta)\log N\rfloor$ (a conjecture of H. Lacoin) would follow if the number of isolated zeros were at most of order $N^{4/3}$; heuristically, at criticality the pinning reward exactly cancels the entropy of $1\times 1$ downward spikes, so the height is governed by $2\times 1$ spikes, whose rate $e^{-6\beta H_w}$ replaces $e^{-4\beta H}$ in the entropic-repulsion balance and changes the coefficient from $1/4\beta$ to $1/6\beta$. Chen, Gheissari and Lubetzky (2024) then proved this conjecture: for $\beta$ large enough, at $h=h_w$ the surface is still delocalized, and with high probability all but $(C/\beta)N^2$ sites have height in the two adjacent levels $\{h^*_N-1,h^*_N\}$ with $h^*_N=\lfloor(1/6\beta)\log N+\frac13\rfloor$, the $\frac13$ refinement arising from toothlike $2\times 1$ spikes with one extra block appended below a bottom vertex; they also identify the single dominant level for a subset of $N$ of natural density at least $0.6$, and prove that connected sets of sites above $h^*_N$ have only logarithmic size.

What remains open — the subject of this problem — is the shape-level description of the pinned surface throughout the delocalized phase $0<h\le h_w$. For subcritical $h$, the two-level concentration around $H-c(h)$ with the pinning-induced shift $c(h)$ and the level-line structure are not proved at all: the state of the art is the order-one fluctuation window of Feldheim–Yang. At criticality, beyond the two-level set $\{h^*_N-1,h^*_N\}$ of Chen–Gheissari–Lubetzky, the level-line structure is open; those authors explicitly state, as an expectation rather than a theorem, that there should be a unique macroscopic connected component of the sites at these heights whose scaling limit is a Wulff shape (the deterministic equilibrium crystal shape minimizing anisotropic surface tension, expected to describe the limiting macroscopic contour).

Problem Statement

Fix the inverse temperature $\beta$ in the low-temperature regime of the existing theory ($\beta\ge 1$ for the subcritical component, as in Feldheim–Yang; $\beta$ large enough for the critical component, as in Chen–Gheissari–Lubetzky). Consider the $(2+1)$-dimensional SOS surface above the wall with pinning on $\Lambda_N=[[1,N]]^2\subset\mathbb{Z}^2$:

$$\mathbb{P}^{\beta,h}_N(\phi)=\frac{1}{Z^{\beta,h}_N}\exp\left(-\beta\sum_{x\sim y}|\phi(x)-\phi(y)|+h\,\bigl|\{x\in\Lambda_N:\phi(x)=0\}\bigr|\right),\qquad \phi:\Lambda_N\to\mathbb{Z}_{\ge 0},\ \ \phi=0\ \text{on}\ \mathbb{Z}^2\setminus\Lambda_N,$$

where the sum runs over nearest-neighbor pairs $x\sim y$ of $\mathbb{Z}^2$ with at least one endpoint in $\Lambda_N$ (sites outside $\Lambda_N$ have height $0$). Let $h_w(\beta)$ be the critical wetting point of this measure, i.e. the free-energy critical point

$$h_w(\beta)=\sup\{h\ge 0:\ f(\beta,h)=f(\beta,0)\},\qquad f(\beta,h)=\lim_{N\to\infty}N^{-2}\log Z^{\beta,h}_N,$$

whose existence and positivity for every $\beta>0$ go back to Chalker (separating the delocalized phase $h<h_w$ from the localized phase $h>h_w$); for large $\beta$ one has Lacoin's explicit identification $h_w(\beta)=\log\left(\frac{e^{4\beta}}{e^{4\beta}-1}\right)$. The problem is to extend the macroscopic shape theory of the unpinned surface (Caputo–Lubetzky–Martinelli–Sly–Toninelli, JEMS 2016) to this pinned measure throughout the delocalized phase, namely the following two linked components. Both components, and every deliverable listed under them, are mandatory and jointly constitute the target, so that establishing only one of the two components, or only a proper subset of the listed deliverables, does not by itself solve the problem; a rigorous refutation of the kind admitted in the final paragraph below, however, is also a complete answer.

(1) Subcritical regime. For each fixed $h\in(0,h_w(\beta))$, prove that, with high probability as $N\to\infty$, the height of almost all sites concentrates on two adjacent integer levels

$$\lfloor (4\beta)^{-1}\log N\rfloor-c(h)\quad\text{and}\quad \lfloor (4\beta)^{-1}\log N\rfloor-c(h)-1$$

for an integer shift $c(h)\ge 0$, with quantified exceptional sets comparable to the $h=0$ theory; determine $c(h)$, and in particular prove or refute the expected divergence $c(h)\to\infty$ as $h\uparrow h_w$. Derive the corresponding structure of the macroscopic level lines: a unique macroscopic level line for each relevant height, limiting shapes along sequences of $N$ with convergent fractional parts of $(4\beta)^{-1}\log N$, and fluctuation bounds of the predicted cube-root order $N^{1/3+o(1)}$ for the highest macroscopic level line.

(2) Critical regime. At $h=h_w(\beta)$ — where the typical levels $h^*_N=\lfloor(1/6\beta)\log N+\frac13\rfloor$ and $h^*_N-1$ are now known to carry all but $(C/\beta)N^2$ sites for $\beta$ large — prove the analogous shape theorem: with high probability the two-level set $\{x:\phi(x)\in\{h^*_N-1,h^*_N\}\}$ is occupied by a unique macroscopic connected component that contains all but $o(N^2)$ of the sites of that set, with its rescaled boundary converging to an explicit limiting shape (conjecturally of Wulff type, with a flat component along the box boundary), fluctuation bounds for the macroscopic level lines (predicted cube-root order along the flat part), and identification of the single dominant level among $h^*_N-1$ and $h^*_N$ for every $N$ (currently settled only on a subset of natural density at least $0.6$).

A rigorous refutation is an admissible answer: a proof that for some $\beta$ and some $h\in(0,h_w]$ the surface does not concentrate on two adjacent levels, or that the level-line picture differs qualitatively from the $h=0$ theory (for example through an additional intermediate macroscopic scale or regime), together with a derivation of the correct macroscopic behavior. Statements restricted to what is already known — the order-one fluctuation window at subcritical $h$, or the two-level set $\{h^*_N-1,h^*_N\}$ at criticality — do not by themselves solve the problem.

References

1. Naomi Feldheim and Shangjie Yang, Typical height of the (2+1)-D Solid-on-Solid surface with pinning above a wall in the delocalized phase, Stochastic Processes and their Applications 165 (2023) 168–182, DOI 10.1016/j.spa.2023.08.009, arXiv:2301.09197, https://arxiv.org/abs/2301.09197
2. Joseph Chen, Reza Gheissari, and Eyal Lubetzky, Critical wetting in the (2+1)D solid-on-solid model, Electronic Communications in Probability 29 (2024), DOI 10.1214/24-ECP628, arXiv:2406.12040, https://arxiv.org/abs/2406.12040
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