Asymptotic fate and universal height statistics of the 2D-to-1D dimensional crossover in Kardar-Parisi-Zhang growth on strongly rectangular substrates

Carrasco and Oliveira [Phys. Rev. E 109, L042102 (2024)] demonstrated by extensive simulations that two-dimensional Kardar-Parisi-Zhang (KPZ) growth on strongly elongated substrates ($L_y \gg L_x$) undergoes a 2D-to-1D dimensional crossover: the roughness crosses from $W \sim t^{\beta_{2D}}$ to $W \sim t^{1/3}$ at a crossover time $t_c \sim L_x^{z_{2D}}$, the one-point height distributions cross to Tracy-Widom GOE or GUE depending on geometry, and a new universal 'single-side-saturation' line height distribution, with the magnitude of the skewness and the excess kurtosis both approximately 0.11, appears along the saturated direction. On this basis the authors conjectured the long-time fate of any 2D KPZ system with finite $L_x$: cylindrical systems (finite $L_x$, indefinitely growing $L_y$) should end in the 1D circular KPZ subclass (Tracy-Widom GUE), and flat systems with finite $L_x$ and $L_y \to \infty$ in the 1D flat subclass (Tracy-Widom GOE). The problem posed here is the analytic establishment of that crossover scenario, in the two mandatory components enumerated below: (1) prove or refute the long-time fate conjecture at the level of the KPZ universality class, deriving, for a positive resolution, the crossover time $t_c$ and the scaling form $F_c$ with its two asymptotic regimes; and (2) derive the universal statistics of the single-side-saturation regime. No analytic derivation of these results is currently available.

Background

The Kardar-Parisi-Zhang (KPZ) equation, $\partial_t h = \nu \nabla^2 h + (\lambda/2)(\nabla h)^2 + \eta$ with Gaussian white noise $\eta$, is the minimal stochastic description of a growing interface height field $h(\mathbf{x},t)$ with local slope-dependent growth; it defines a strong-coupling universality class spanning interface growth, directed polymers, and stochastic transport (Kardar, Parisi and Zhang, 1986). The standard measure of fluctuations is the global roughness $W = \langle (h - \bar h)^2 \rangle^{1/2}$, where the bar and brackets denote spatial and configurational averages. While the lateral correlation length $\xi(t)$ is smaller than the substrate, the interface is in a growth regime with $W \sim t^{\beta}$; once $\xi$ spans the substrate, flat systems enter a steady state regime with saturated roughness $W_s \sim L^{\alpha}$ (the Family-Vicsek finite-size scaling form). In $1+1$ dimensions the KPZ class is exactly solved: $\beta = 1/3$, and the scaled one-point height distribution is a Tracy-Widom distribution from random-matrix theory — the Gaussian orthogonal ensemble (GOE) for a flat substrate of fixed length, and the Gaussian unitary ensemble (GUE) for curved interfaces whose length grows faster than $\xi$, as in growth from a circular seed (Prähofer and Spohn, 2000); the analogous circular geometry in $2+1$ dimensions, a ring growing radially in the plane, was studied by Carrasco and Oliveira (2019). In $2+1$ dimensions no exact solution exists: the growth exponent $\beta_{2D} \approx 0.2414$ is known from simulations (with $\beta_d = 7/(8d+13)$ proposed as exact in all dimensions; Oliveira, 2022), and the asymptotic growth-regime height distributions for flat, cylindrical ($L_x$ fixed, $L_y \sim t$), and spherical ($L \sim t$) geometries are likewise known only numerically (Carrasco and Oliveira, 2023; Oliveira, 2022).

Two-dimensional KPZ growth has almost always been studied on square substrates $L_x = L_y$, where only $L_x$ and $\xi$ are relevant lengths. Carrasco and Oliveira (2024) instead simulated three KPZ-class lattice models (the restricted solid-on-solid model, which forbids deposition creating height differences larger than one lattice spacing with a neighbor; the single-step model, which deposits on sites whose height differs by one from all neighbors; and the etching model, which raises the chosen site's lower neighbors to its own height and then deposits there) on strongly elongated substrates, in flat and cylindrical geometries, with periodic boundary conditions in both substrate directions; in flat geometry $L_x$ and $L_y$ stay fixed during growth, while in the cylindrical geometry $L_x$ is fixed and $L_y$ is enlarged steadily in time ($L_y = L_0 + \omega t$), so the steady state is never reached. When $\xi$ reaches the narrow side $L_x$ while $L_y \gg L_x$, they found: (i) a dimensional crossover of the roughness from $W \sim t^{\beta_{2D}}$ to $W \sim t^{\beta_{1D}}$ at $t_c \sim L_x^{z_{2D}}$ (with $z = \alpha/\beta$ the dynamic exponent), following the universal scaling form $W \simeq A^{1/2} L_x^{\alpha_{2D}} F_c(|\lambda|A^{1/2}t/L_x^{z_{2D}})$, where $A$ and $\lambda$ are model-dependent amplitude parameters; (ii) growth-regime height distributions crossing over from the 2D flat (respectively cylindrical) distributions to the Tracy-Widom GOE (respectively GUE) distribution; (iii) an asymptotic growth velocity interpolating as $v_\infty(L_x) \simeq v_\infty^{(2D)} - c\lambda A L_x^{-(2-2\alpha_{2D})}$ with $c \approx 0.18$; (iv) in the steady state of flat systems a continuous family of universal height distributions and saturation roughness $W_s^2 \simeq A L_x^{\alpha_{2D}} L_y^{\alpha_{2D}} G(L_y/L_x)$, interpolating between the 2D and 1D behaviors; and (v) height profiles measured along the saturated $x$-direction ('line height distributions') converging to a new universal distribution whose skewness magnitude $|S|$ (third cumulant divided by $W^3$) and excess kurtosis $K$ (fourth cumulant divided by $W^4$) both saturate at approximately 0.11, fitting no known 1D or 2D KPZ distribution — the 'single-side-saturation' regime.

On this numerical basis the authors conjecture the long-time fate of any 2D KPZ system with a finite short side: a cylindrical system (finite $L_x$, $L_y$ increasing indefinitely, so the steady state is never reached) must follow the 1D circular KPZ subclass at long times, with $W \sim t^{1/3}$ and Tracy-Widom GUE statistics, while a flat system with finite $L_x$ and $L_y \to \infty$ must fall asymptotically into the 1D flat KPZ subclass, with $W \sim t^{1/3}$ and Tracy-Widom GOE statistics. They further expect an analogous sequence of dimensional crossovers in higher dimensions ($d$-dimensional flat substrates with lateral sizes $1 \ll L_1 \ll L_2 \ll \cdots \ll L_d$, crossing through $W \sim t^{\beta_{dD}}$ down to $W \sim t^{1/3}$), which they judge numerically very hard to observe, and they propose rectangular substrates as a possible starting point toward the long-sought exact solution of 2D KPZ growth. All of these statements rest on simulations; none currently has an analytic derivation or proof.

Problem Statement

Establish analytically — prove, derive, or refute — the dimensional-crossover scenario conjectured by Carrasco and Oliveira (2024) for KPZ growth on strongly elongated 2D substrates, at the level of the KPZ universality class. A legitimate setting is the $(2+1)$-dimensional KPZ equation with periodic boundary conditions on a strip of finite width $L_x$ — with $L_y$ fixed and large for the flat case, or enlarged linearly in time for the cylindrical case — or any KPZ-class lattice model in the same geometry (with a universality argument, in which case the claim covers 'any 2D KPZ system' as the source states). The components, in the source's own hierarchy, each independently resolvable:

(1) Asymptotic fate (the central conjecture). Prove or refute: (a) any 2D KPZ system in cylindrical geometry — finite $L_x$, with the substrate length $L_y$ increasing indefinitely so that the steady state is never reached — follows at long times the 1D circular KPZ subclass, with $W \sim t^{1/3}$ and the one-point height distribution converging to the Tracy-Widom GUE distribution; and (b) any flat 2D KPZ system with finite $L_x$ and $L_y \to \infty$ converges at long times (in the growth regime, for $t_c \ll t \ll t_s$, where $t_s$ is the saturation time at which the correlation length $\xi$ reaches $L_y$) to the 1D flat KPZ subclass, with $W \sim t^{1/3}$ and Tracy-Widom GOE statistics. A positive resolution of (1) requires establishing both branches (1a) and (1b) together with the derivation of the crossover time $t_c \sim L_x^{z_{2D}}$ and the universal scaling form $W \simeq A^{1/2} L_x^{\alpha_{2D}} F_c(|\lambda|A^{1/2} t/L_x^{z_{2D}})$ with its two asymptotic regimes; establishing (1a) and (1b) without this derivation is not a complete positive resolution. Identifying the effective one-dimensional dynamics (including any renormalization of the KPZ nonlinearity and amplitude parameters by the saturated transverse modes), if that is the mechanism, is a desirable additional step.

(2) Universal statistics of the single-side-saturation regime. Derive analytically the new universal line height distribution of the saturated direction — whose simulated skewness and excess kurtosis both take the value $\approx 0.11$ across KPZ models and geometries — either as a closed form, an exact characterization, or an identification within a broader analytic theory of KPZ height statistics.

The source's suggestion that rectangular substrates may open a route to exact solutions of 2D KPZ growth, and its outlook that an analogous sequence of temporal crossovers is expected in higher dimensions ($d$-dimensional flat substrates with $1 \ll L_1 \ll L_2 \ll \cdots \ll L_d$, which the source itself judges difficult to observe in simulations already for $d = 3$), are background context rather than required results of this problem. The steady-state scaling function $G(L_y/L_x)$ and the velocity-correction amplitude $c \approx 0.18$ are numerically established results of the source rather than open analytic targets of this problem. Refutations are admissible: a demonstrated 2D KPZ-class system with finite $L_x$ whose long-time growth-regime statistics do not converge to the relevant stated 1D subclass refutes the corresponding branch of (1), and a refutation of either branch (1a) or (1b) resolves the fate conjecture negatively.

References

1. Ismael S. S. Carrasco and Tiago J. Oliveira, 'Dimensional crossover in Kardar-Parisi-Zhang growth', Physical Review E 109, L042102 (2024). DOI: 10.1103/PhysRevE.109.L042102, https://doi.org/10.1103/PhysRevE.109.L042102 (arXiv:2404.19516, https://arxiv.org/abs/2404.19516)
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