Interface fluctuations for deposition on enlarging flat substrates

Abstract

We investigate solid-on-solid models that belong to the Kardar-Parisi-Zhang (KPZ) universality class on substrates that expand laterally at a constant rate by duplication of columns. Despite the null global curvature, we show that all investigated models have asymptotic height distributions and spatial covariances in agreement with those expected for the KPZ subclass for curved surfaces. In 1 + 1 dimensions, the height distribution and covariance are given by the GUE Tracy-Widom distribution and the Airy 2 process instead of the GOE and Airy 1 foreseen for flat interfaces. These results imply that when the KPZ class splits into curved and flat subclasses, as conventionally considered, the expanding substrate may play a role equivalent to, or perhaps more important than, the global curvature. Moreover, the translational invariance of the interfaces evolving on growing domains allowed us to accurately determine, in 2 + 1 dimensions, the analog of the GUE Tracy-Widom distribution for height distribution and that of the Airy 2 process for spatial covariance. Temporal covariance is also calculated and shown to be universal in each dimension and in each of the two subclasses. A logarithmic correction associated with the duplication of columns is observed and theoretically elucidated. Finally, crossover between regimes with fixed-size and enlarging substrates is also investigated.

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.…”

“…On a practical level, this results in thousand-fold less payoff, given present-day CPU capabilities and equivalent run times; that is, probablilities down to 10 −6 vs 10 −9 , for TW-GUE vs. GOE, respectively. In any case, we report here results for KPZ Euler that rely neither on polar coords, nor a constrained pt-pt Monte Carlo, but rather upon an interesting numerical trick, built upon expanding substrates [89]. We refer the reader there for technical details, but the basic idea is to perform the simulation in the flat geometry, starting with a tiny substrate of system size L = L 0 , but increasing the size of the substrate by Ω sites per unit time, stochastically duplicating column heights of randomly chosen points.…”

“…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].…”

Eden model with nonlocal growth rules and kinetic roughening in biological systems

“…Particular interest is given to the rough interfaces of compact and spherical patterns observed in bacterial colonies [4][5][6][7][8] and clusters of normal [9][10][11] and tumor [12,13] cells grown on culture under controlled experimental conditions. The Eden model [14] is a benchmark of stochastic processes, in the important class of growth models on expanding substrates [15][16][17][18][19][20][21], which forms radial clusters with irregular (fractal) borders. In this model, new cells are irreversibly added at random positions of the neighborhood of previously existent cells.…”

Hallmarks of the Kardar–Parisi–Zhang universality class elicited by scanning probe microscopy

“…Summed up the observations reported in the previous section, it is worth to point out strategies to rid the analysis from this tip induced KPZ traits. There exist other KPZ hallmarks that could be used to confirm the universality as, for example, temporal and spatial correlation functions (see, e.g., [28,48,60,61] and references therein). The former requires a huge amount of data for a reliable analysis and therefore are not currently feasible in experiments with SPM.…”