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

TL;DR

The paper investigates the quenched KPZ (qKPZ) universality class at depinning in $d=1$ and $d=2$ using automaton simulations, obtaining a full set of directly measured critical exponents ($F_c$, $ heta$, $eta$, $ u$, $z$, $eta$, $ ext{and } abla ext{ exponents}$) and confirming Family-Vicsek scaling through the height-difference correlation $C_2(r,t)$. It additionally characterizes the universal PDF of front fluctuations in the growth regime, finding strong non-Gaussian features and dimension-specific tail behavior that differ from KPZ with time-dependent noise. The results support the identification of qKPZ at depinning with the directed-percolation depinning (DPD) universality class and provide high-precision exponents and PDF shapes, highlighting finite-size effects and the robustness of automaton models for disordered-interface dynamics. The work contributes precise benchmarks for DP-related universality, clarifies the scaling relations (e.g., $eta+ heta=1$, $eta z= abla$), and offers accessible data for further comparative studies. Overall, the study advances understanding of quenched-disordered interfacial growth and the universal statistics governing depinning phenomena.

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 $α$, $β$, $θ$, and $δ$ 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.

Figures (10)

• Figure 1: Steady-state front velocity vs. $F$ for different values of $L$ as indicated in the legends, for (a) $d = 1$ and (b) $d=2$. The solid lines correspond to $F\sim(F-F_c)^{\theta}$ for the values of $F_c$ and $\theta$ indicated in the corresponding panel.
• Figure 2: Average front vs. time for different values of $L$, as indicated in the legends, for (a) $d = 1$ and (b) $d=2$. The straight solid lines correspond to $\overline{h}(t)\sim t^{1-\delta}$ computed using the value of $\delta$ obtained for the largest system size in each case, see legends and Table \ref{['tab:exponents1D']}.
• Figure 3: Squared roughness vs. time for different values of $L$, as indicated in the legends, for (a) $d = 1$ and (b) $d=2$. The straight solid lines correspond to $w(t)\sim t^{\beta}$ using the value of $\beta$ obtained for the largest system size in each case, see legends and Table \ref{['tab:exponents1D']}.
• Figure 4: Logarithmic plot of the saturation squared roughness $w^2_\text{sat}$ vs. $L$ for the 1D (main panel) and 2D (inset) models. According to Eq. \ref{['eq:FV']}, the slope of the best-fit straight line in the plot equals 2$\alpha$, with $\alpha$ as indicated in each legend.
• Figure 5: Correlation length computed from Eq. \ref{['eq:a_scaling']} with $a = 0.9$, for different system sizes $L$ as indicated in the legends, (a) $d = 1$ and (b) $d = 2$. The straight lines correspond to $\xi(t) \sim t^{1/z}$ with $z$ as indicated in the legend, which corresponds to the largest system size in each case, see Table \ref{['tab:exponents1D']}. In both figures, the insets plot the $C_2(L/2,t)$ function vs. $\xi(t)$; these plots yield straight lines with slope 2$\alpha$, see the text.