Fluctuation Bounds for the Restricted Solid-on-Solid Model of Surface Growth

TL;DR

The paper rigorously analyzes the RSOS surface growth model in continuous time, proving a universal linear-in-$t$ bound on height fluctuations in all dimensions and, in one dimension, a logarithmic lower bound. It achieves this through a minimal-weight path representation on a disordered Harris lattice and by introducing a dual RSOS process, establishing the distributional identity $f(t,0) \stackrel{d}{=} \min_y f^D(t,y)$. The dual representation enables a Kesten–Hammersley-type argument to show almost-sure linear growth for the dual process, and a detailed renewal/Berry–Esseen analysis yields the logarithmic lower bound for 1D fluctuations. Collectively, the results place RSOS within a first-passage-percolation–inspired framework, provide sharp fluctuation bounds, and open avenues for extensions to related bounded-difference growth models and scaling-limit studies.

Abstract

The restricted solid-on-solid (RSOS) model is a model of continuous-time surface growth characterized by the constraint that adjacent height differences are bounded by a fixed constant. Though the model is conjectured to belong to the KPZ universality class, mathematical progress on the model is very sparse. We study basic properties of the model and establish bounds on surface height fluctuations as a function of time. A linear fluctuation bound is proven for all dimensions, and a logarithmic fluctuation lower bound is given for dimension one. The proofs rely on a new characterization of RSOS as a type of first passage percolation on a disordered lattice, which is of independent interest. The logarithmic lower bound is established by showing equality in distribution to the minimum of a certain dual RSOS process.

Figures (5)

• Figure 1: Example of surface dynamics for the RSOS model when the surface height is updated. Blocks fall and stick to the surface only if adjacent height differences remain bounded by one.
• Figure 2: A random Poisson lattice, with an example lattice path dotted in purple. The lattice path can move to neighboring locations only when it encounters an update. It may also stay at the current location.
• Figure 3: Visualization of accepted updates on the random lattice. The updates are denoted by colored circles, with the colors indicating the new height of the surface. The blue diamond-shaped updates over locations $6,7,8$ indicate the foundation of the orange diamond-shaped update, in the sense of Definition \ref{['def:acceptedupdates']}.
• Figure 4: Visualization of surface height of the dual RSOS process, in $1 + 1$ dimensions.
• Figure 5: The colored updates indicate a pyramid centered at $x = 7$ with height $6$, which corresponds to the accepted updates of an RSOS process. As it corresponds to the accepted updates of an RSOS process, it also has the maximal height among all pyramids centered at $x = 7$ within time $9$. The colors correspond to the layers of the pyramid.

Theorems & Definitions (44)

• Definition 2.1: Continuous-time surface growth model
• Definition 2.2: Path on the random lattice
• Definition 2.3: Lattice depth of an update
• Proposition 2.1
• proof
• Remark 2.1
• Definition 3.1
• Definition 3.2
• Definition 3.3: Foundation of an accepted update
• Lemma 3.1