Exponents and front fluctuations in the quenched Kardar-Parisi-Zhang universality class of one- and two- dimensional interfaces

TL;DR

This study uses simulations of a quenched KPZ automaton in 1D and 2D to analyze critical exponents, correlation lengths, and fluctuation PDFs, revealing compatibility with the directed percolation depinning class and non-Gaussian fluctuation distributions.

Contribution

It provides the first direct numerical computation of critical exponents and fluctuation PDFs for the quenched KPZ universality class in multiple dimensions.

Findings

Critical exponents align with directed percolation depinning universality class.

The fluctuation PDFs are non-Gaussian with significant skewness and kurtosis.

The PDFs differ from those of the standard KPZ equation with time-dependent noise.

Abstract

We have simulated an automaton version of the quenched Kardar-Parisi-Zhang (qKPZ) equation in one and two dimensions in order to study the scaling properties of the interface at the depinning transition. Specifically, the $\alpha$, $\beta$, $\theta$, and $\delta$ critical exponents characterizing the surface kinetic roughening and depinning behaviors have been directly computed from the simulations. In addition, by studying the height-difference correlation function in real space, we have also been able to directly compute the dynamic correlation length and its associated dynamic critical exponent $z$. The full sets of scaling exponents are largely compatible with those of the Directed Percolation Depinning universality class for one and two dimensional interfaces. Furthermore, we have computed numerically the probability density function (PDF) of the front fluctuations in the growth regime, finding its asymptotic form in one and two dimensions. While the PDF features strongly non-Gaussian skewness and kurtosis values, it also differs from the PDF of the KPZ equation with time-dependent noise for physical substrate dimensions, both in the central part and at the tails of the distribution.