Kardar-Parisi-Zhang universality class in (2+1) dimensions: Universal geometry-dependent distributions and finite-time corrections

Abstract

The dynamical regimes of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class are investigated in d = 2 + 1 by extensive simulations considering flat and curved geometries. Geometry-dependent universal distributions, different from their Tracy-Widom counterpart in onedimension, were found. Distributions exhibit finite-time corrections hallmarked by a shift in the mean decaying as t −β , where β is the growth exponent. Our results support a generalization of the ansatz h = v∞t + (Γt) β χ + η + ζt −β to higher dimensions, where v∞, Γ, ζ and η are nonuniversal quantities whereas β and χ are universal and the last one depends on the surface geometry. Generalized Gumbel distributions provide very good fits of the distributions in at least four orders of magnitude around the peak, which can be used for comparisons with experiments. Our numerical results call for analytical approaches and experimental realizations of KPZ class in two-dimensional systems. where ξ is a white noise of mean zero and amplitude √ D. Despite of its original conception for evolving interfaces, the KPZ equation has also found its place in others important physical systems [4].A great advance in the theoretical understanding of the KPZ universality class has begun at early 2000s with the seminal works of Johansson [5] and Prähofer and Spohn [6] presenting analytical asymptotic solutions of some models in the KPZ class. These solutions link the height's stochastic fluctuations to universal distributions [7] of the random matrix theory. Inspired in these exact results, the ansatzwith the exactly known growth exponent β = 1/3, was conjectured as describing the asymptotic interface fluctuations of any model belonging to KPZ class in d = 1 + 1 [4,8]. In this equation, s λ = sgn(λ), while the asymptotic velocity v ∞ and Γ are model dependent parameters and χ is a universal random variable with time-independent distribution given by the Gaussian orthogonal ensemble (GOE) for flat geometries [5,6] where η and ζ are non-universal. The correction η introduces a shift in the distribution of the scaled height q = h−v∞t s λ (Γt) β in relation to the asymptotic distributions. The hallmark of this correction, a shift in the mean vanishing as q − χ ∼ t −1/3 , has been verified in the crystal liquid experiments [2] and computer simulations of several models [9,10,16]. To our knowledge, only two exceptions have been reported. In the first one, Ferrari and Frings [15] analyzed the partially asymmetric simple exclusion process and found a specific value of the asymmetry parameter where there is no correction up order O(t −2/3 ). Off-lattice simulations of an Eden model consistent with a decay t −2/3 were reported [11], but a subsequent analysis showed that the unusual behavior is an artifact of low precision estimates of v ∞ and a long crossover to the scaling law t −1/3 [16].In contrast to the deep understanding of the KPZ class in d = 1 + 1, essentially no exact results are available in d = 2 + 1, the most important dimension for applications [...

A KPZ Cocktail-Shaken, not Stirred...

“…Here, we perform a 2+1 KPZ Euler integration on expanding substrates with Ω = 10, complementing our earlier efforts on the 3d pt-pt SHE with multiplicative noise. Results are indicated here in Figure 4, with insets demonstrating that the variance, found to be 0.326, is dead-on, the skewness and kurtosis well within accepted values [84,85,86,89], while subleading corrections render extraction of the universal mean a bit more challenging. Nevertheless, successive fit lines monotonically approach a well-defined boundary (i.e., envelope), with intercepts converging to ξ 2 ≈-2.32, a quite decent value.…”

“…Halpin-Healy [84] simulated 3d extremal paths connecting far corners of a cube filled with exponentially distributed site energies; the underlying pt-pt energy fluctuation limit distribution was measured to possess universal variance, skewness and kurtosis: 0.291, 0.331, and 0.212, respectively. Later authors [86], studying three distinct 3d pt-pt KPZ growth models, among them single-step (SSC) and on-lattice version of Takeuchi Eden D, find values in the range s 2 ≈ 0.32-0.34 and k 2 ≈ 0.20-0.22, providing additional evidence for these characteristic, universal quantities. Tucked away in his doctoral dissertation [87]- Table 7.2, Prähofer presciently records s 2 =0.323(5) and k 2 =0.21(4) for his own early 3d pt-pt PNG simulation.…”

“…Importantly, the present KPZ Renaissance has inspired a renewed interest in the 2+1 KPZ problem, as well as other dimension-dependent issues, such as the existence of a finite UCD, and using Wilsonian renormalization group methods, hopes of elucidating the full KPZ phase diagram [83]. Universality of the 2+1 KPZ Class and its associated limit distributions (i.e., higher-dimensional analogs of TW-GUE, TW-GOE and BR-F 0 ) have been well-characterized [84,85,86,87], along with universal spatial and temporal correlators [88,89], supplemented by secondary KPZ "patch" PDFs [88,90]. On the mathematical side, the transmission has shifted into hyperdrive; a rapidly accelerating moveable feast-e.g., [91,92,93,94,95,96,97,98,99,100].…”

Roughening transition and universality of single step growth models in (2+1)-dimensions

“…There also exist some known results that can additionally be tested, which will be the purpose of our future work. Most notably, it is known that (2+1)-dimensional KPZ interfaces display one-point height fluctuations described by a (generalized) Tracy-Widom probability distribution function [45][46][47], which should hold for p 0.25 < and be falsified for p 0.25 > .…”

Macroscopic Response to Microscopic Intrinsic Noise in Three-Dimensional Fisher Fronts

“…2. Indeed, 2D KPZ behavior characterized by α ≃ 0.39 and β ≃ 0.24 [32] provides a much poorer description of the numerical data; see the respective dot-dashed lines in Figs. 2(a) and 2(b).…”