Sublinear variance bounds and the fluctuation order of the continuous-time restricted solid-on-solid (RSOS) surface growth model

The restricted solid-on-solid (RSOS) model of Kim and Kosterlitz is a continuous-time surface growth process on $\mathbb{Z}^d$ in which each site carries an independent rate-one Poisson clock and a proposed unit increment of the height at a site is accepted only if every nearest-neighbor height difference stays bounded by one in absolute value. The model is conjectured to lie in the KPZ universality class, with the height variance at a fixed site believed to grow as $t^{2/3}$ in one dimension, but it appears to be non-integrable and rigorous results are sparse. Sudijono (arXiv 2023; Random Structures & Algorithms 2025) proved $\operatorname{Var} f(t,x) \le t$ in all dimensions — via a representation of the height as a minimal-weight path (a first-passage-percolation-type quantity) on the Poissonian space-time lattice — together with $\liminf_{t\to\infty} \operatorname{Var} f(t,x)/\log t > 0$ in $d=1$ and $\limsup_{t\to\infty} \operatorname{Var} f(t,x) > 0$ in all $d$. These bounds leave open the problem of closing the gap between them: prove a sublinear-in-time variance upper bound ($\operatorname{Var} f(t,x) = o(t)$, the analogue of the Benjamini–Kalai–Schramm improvement of Kesten's linear bound in planar first-passage percolation, equivalently a superconcentration statement in Chatterjee's sense), and establish nontrivial lower bounds in dimensions $d \ge 2$ (the liminf version of the constant-order bound, or the logarithmic bound beyond $d=1$), with full resolution meaning determination of the variance order (conjecturally $t^{2/3}$ in $d=1$). No resolution of these questions is known in the literature to date; the problem remains open.

Background

A continuous-time surface growth model is a random integer-valued height function $f(t,x)$, $t \ge 0$, $x \in \mathbb{Z}^d$, in which every site carries an independent Poisson clock of rate one and, whenever the clock at $x$ rings, the height at $x$ is updated by a local rule that consults the current heights at $x$ and its nearest neighbors. The restricted solid-on-solid (RSOS) model, introduced by Kim and Kosterlitz (1989) as a simple model of vapor deposition, uses the rule: the height at $x$ increases by one if and only if, after the increment, the height difference between $x$ and each nearest neighbor $y$ still has absolute value at most one; equivalently, growth at $x$ is accepted exactly when $f(t-,x) \le \min_{y \sim x} f(t-,y)$, and otherwise the clock ring is discarded. The model is a canonical discrete representative of the KPZ (Kardar–Parisi–Zhang) universality class — the class of random growth processes with slope-dependent growth rates for which one-dimensional members are predicted to have height fluctuations of order $t^{1/3}$ about the growing mean, i.e. variance of order $t^{2/3}$, with spatial correlations on the scale $t^{2/3}$ — and extensive numerical work supports this classification in one dimension, while in dimensions two and higher even the numerics leave the variance exponent unclear.

The basic rigorous observable is the variance of the height at a fixed site, $\operatorname{Var} f(t,x)$, as a function of the growth time $t$, from the flat initial condition $f(0,x) \equiv 0$. Before 2023 almost nothing was proved for the continuous-time model. The closest rigorous results concerned variants and sibling models: Chatterjee (Random Structures & Algorithms 2023) proved superconcentration — variance growing sublinearly in time, of order $t/\log t$ — for two discrete-time variants: an RSOS variant updated at integer times by alternating between the even and odd sublattices of $\mathbb{Z}^d$, choosing uniformly among all admissible heights, and a ballistic-deposition variant updated simultaneously at all sites with random brick heights; his general discrete-time framework shows that superconcentration (variance much smaller than the scale set by the height itself) is equivalent to subroughness (sublinear growth of the expected squared height difference between two fixed sites). Ballistic deposition, the sibling model in which a falling block sticks to the top of the highest neighboring column and which behaves like a last-passage analogue (the counterpart of first-passage percolation in which paths maximize, rather than minimize, total weight), had earlier received a logarithmic variance lower bound and a positive liminf variance statement from Penrose (2008), by an argument inspired by classical planar first-passage percolation (FPP, the model of shortest paths through a lattice with random edge weights).

For the continuous-time asynchronous RSOS model itself, Sudijono (arXiv 2023, published in Random Structures & Algorithms 2025) established the first bounds: (i) $\operatorname{Var} f(t,x) \le t$ in all spatial dimensions; (ii) $\liminf_{t\to\infty} \operatorname{Var} f(t,x)/\log t > 0$ in one dimension; (iii) $\limsup_{t\to\infty} \operatorname{Var} f(t,x) > 0$ in all dimensions. The upper bound rests on a pathwise identity of independent interest: the RSOS height equals the minimal weight $W(\gamma)$ over nearest-neighbor paths $\gamma$ through the space-time Poisson point process of clock rings, where $W(\gamma)$ counts the Poisson points lying on $\gamma$ — so RSOS is an analogue of first-passage percolation on a disordered lattice, just as ballistic deposition is an analogue of last-passage percolation — combined with the standard Poincaré inequality on Poisson space (a variance bound for functionals of a Poisson point process). The lower bound rests on an equality in distribution between the one-dimensional RSOS height and the minimum of a dual RSOS process, following Penrose's strategy; in $d \ge 2$ the dual interface evolves like the Eden growth model (a random cluster-growth process whose deterministic asymptotic shape is known, though sharp fluctuation bounds are not), which suggests but does not yield an extension. The author also reports that the model appears to be non-integrable: its one-dimensional slope process takes values in $\{-1,0,1\}$ rather than the $\{\pm 1\}$ of the TASEP (totally asymmetric simple exclusion process) / single-step representation, so the exact KPZ asymptotics available for solvable models are out of reach. Between the logarithmic lower bound and the linear upper bound in $d=1$, and between the limsup-only statement and the linear upper bound in $d \ge 2$, the variance-order question is the central open quantitative problem left by the source, and the source explicitly poses the sublinear upper bound as the goal mirroring the development of variance bounds in planar FPP (Kesten's linear bound, the Benjamini–Kalai–Schramm sublinear improvement, and recent multiscale concentration results of Basu–Sidoravicius–Sly for rotationally invariant FPP).

Problem Statement

For the continuous-time restricted solid-on-solid (RSOS) surface growth model on $\mathbb{Z}^d$ with unit height restriction — rate-one independent Poisson clocks, the height at $x$ incremented by one exactly when $f(t-,x) \le \min_{y \sim x} f(t-,y)$, so that all nearest-neighbor height differences remain bounded by one — started from the flat initial condition $f(0,x) \equiv 0$, determine the order of growth of the height variance $\operatorname{Var} f(t,x)$ at a fixed site as $t \to \infty$. The common objective is the fluctuation scale of the RSOS interface; the currently open ends of the same question are: (1) the upper end — prove a sublinear-in-time variance upper bound, i.e. exhibit $g(t) = o(t)$ with $\operatorname{Var} f(t,x) \le g(t)$ for all large $t$ (a logarithmic improvement in the spirit of Benjamini–Kalai–Schramm, a polynomial improvement, or better; any technique is admissible, including an extension of Chatterjee's superconcentration/subroughness framework — superconcentration meaning variance growing sublinearly in time, subroughness meaning sublinear growth of the expected squared height difference between two fixed sites — to asynchronous continuous-time updates); (2) the lower end — prove a nontrivial lower bound in dimensions $d \ge 2$: either $\liminf_{t\to\infty} \operatorname{Var} f(t,x) > 0$ (bounded away from zero at all large times, not merely along a subsequence as is currently known in all dimensions via the limsup result), or the logarithmic lower bound $\liminf_{t\to\infty} \operatorname{Var} f(t,x)/\log t > 0$ beyond $d = 1$; and (3) the complete answer — pin the growth order of $\operatorname{Var} f(t,x)$ up to constants (or identify the exponent), in one dimension confirming or refuting the KPZ (Kardar–Parisi–Zhang universality) conjecture that the variance grows like $t^{2/3}$, where in $d \ge 2$ even the exponent is unsettled. Item (3) is the completion standard for the task — determining the fluctuation order implies the weaker bounds in (1) and (2) — whereas (1) and (2) are intermediate milestones: proving either is a genuine partial resolution but does not by itself complete the task. The model is the Poisson-clock (continuous-time, asynchronous) RSOS model; results for synchronous discrete-time variants or for $k>1$ generalizations of the restriction are neighboring but distinct questions.

References

1. Timothy Sudijono. Fluctuation Bounds for the Restricted Solid-on-Solid Model of Surface Growth. Random Structures & Algorithms 66(3), 2025. DOI: 10.1002/rsa.70004; arXiv:2304.07160. https://arxiv.org/abs/2304.07160
2. Jin Min Kim and J. M. Kosterlitz. Growth in a restricted solid-on-solid model. Physical Review Letters 62(19):2289-2292, 1989. DOI: 10.1103/PhysRevLett.62.2289. https://link.aps.org/doi/10.1103/PhysRevLett.62.2289
3. Sourav Chatterjee. Superconcentration in surface growth. Random Structures & Algorithms 62(2):304-334, 2023. DOI: 10.1002/rsa.21108; arXiv:2103.09199. https://arxiv.org/abs/2103.09199
4. Mathew D. Penrose. Growth and roughness of the interface for ballistic deposition. Journal of Statistical Physics 131(2):247-268, 2008. DOI: 10.1007/s10955-008-9507-1. https://link.springer.com/article/10.1007/s10955-008-9507-1
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8. Pablo Groisman, Alejandro F. Ramírez, Santiago Saglietti, and Sebastián Zaninovich. GUE fluctuations near the axis in one-sided ballistic deposition. arXiv:2504.15837, 2025. https://arxiv.org/abs/2504.15837
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