Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth

TL;DR

This paper conducts a detailed numerical study of non-universal parameters and corrections in KPZ interface growth models in 1+1 dimensions, examining their effects on universality and scaling laws.

Contribution

It provides new insights into how non-universal parameters influence KPZ scaling, especially in curved geometries, and clarifies the role of corrections in the universality class.

Findings

a0Gamma from the first cumulant aligns well with the KPZ ansatz.

a0Higher cumulants suggest possible violations of the generalized ansatz.

a0A crossover to expected scaling behavior was observed in the Eden model.

Abstract

We present a comprehensive numerical investigation of non-universal parameters and corrections related to interface fluctuations of models belonging to the Kardar-Parisi-Zhang (KPZ) universality class, in d=1+1, for both flat and curved geometries. We analyzed two classes of models. In the isotropic models the non-universal parameters are uniform along the surface, whereas in the anisotropic growth they vary. In the latter case, that produces curved surfaces, the statistics must be computed independently along fixed directions. The ansatz h = v t + (\Gamma t)^{1/3} \chi + \eta, where \chi is a Tracy-Widom (geometry-dependent) distribution and \eta is a time-independent correction, is probed. Our numerical analysis shows that the non-universal parameter \Gamma determined through the first cumulant leads to a very good accordance with the extended KPZ ansatz for all investigated models in contrast with the estimates of \Gamma obtained from higher order cumulants that indicate a violation of the generalized ansatz for some of the studied models. We associate the discrepancies to corrections of unknown nature, which hampers an accurate estimation of \Gamma at finite times. The discrepancies in \Gamma via different approaches are relatively small but sufficient to modify the scaling law t^{-1/3} that characterize the finite-time corrections due to \eta. Among the investigated models, we have revisited an off-lattice Eden model that supposedly disobeyed the shift in the mean scaling as t^{-1/3} and showed that there is a crossover to the expected regime. We have found model-dependent (non-universal) corrections for cumulants of order n > 1. All investigated models are consistent with a further term of order t^{-1/3} in the KPZ ansatz.