On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph

Abstract

We investigate the infinite-dimensional limit of nonequilibrium surface growth by numerically integrating stochastic growth equations on a fully connected graph. In particular, we study the Edwards-Wilkinson (EW), Kardar-Parisi-Zhang (KPZ), and tensionless KPZ (TKPZ) equations. Using a network discretization adapted to the all-to-all interaction topology, we analyze the global roughness, height-fluctuation statistics, time power spectra, and two-time correlations. For the EW equation, we obtain an exact expression for the roughness that matches the numerical simulations and shows that the interface becomes flat as $N \to \infty$. We also compute analytically the time power spectrum, show that height fluctuations are Gaussian, and derive an explicit expression for the two-time height autocorrelation function, indicating that the aging behavior is trivial. For the KPZ equation, finite-size and strong-coupling effects can cause deviations from EW behavior at moderate system sizes $N$, often accompanied by numerical instabilities; however, these differences disappear as $N$ increases. In the large-$N$ limit, KPZ dynamics converges to EW behavior, as the four observables analyzed exhibit identical scaling properties. Overall, our results indicate that on a fully connected graph the KPZ nonlinearity is irrelevant as $N\to\infty$, leading to EW-like dynamics with asymptotically flat interfaces. These findings are interpreted in the context of the upper critical dimension of the KPZ universality class.

Figures (11)

• Figure 1: Log-log plot of the squared roughness $w^2(t)$ as a function of time $t$ for the EW equation. Data correspond to different combinations of the parameters $\nu$ and $D$ (see legend). The system size is $N=100$ in all cases. Dashed black lines indicate the theoretical prediction given by Eq. \ref{['eq:rugosidad_EW']}.
• Figure 2: Histogram of the rescaled height fluctuations $\chi$ [see Eq. \ref{['eq:chi']}] for the EW equation $N=100$, $\nu=1$ and $D=1$. The inset shows a magnification of the central region in the interval $-1.5<\chi<1.5$. The solid purple line corresponds to a Gaussian distribution.
• Figure 3: Time power spectrum $S(\omega)$ [see Eq. \ref{['eq:power_spectra']}] for the EW equation. Data are shown for different values of $\nu$ and $D$ (see legend), with system size $N=100$. The solid black line represents the $S(\omega)\sim\omega^{-2}$ scaling.
• Figure 4: Spatially averaged two-time height autocorrelation function, rescaled by the squared roughness, $\overline{C_t(t,t_0)}/w^2(t_0)$, as a function of the time difference $t-t_0$ for the EW equation, shown for different waiting times $t_0=\{10^{-3},2\cdot10^{-3},3\cdot10^{-3},4\cdot10^{-3},6\cdot10^{-3},8\cdot10^{-3},10^{-2},5\cdot10^{-2}\}$, which appear bottom to top in the inset. Parameters are $N=100$, $\nu=1$, and $D=1$. The solid black line corresponds to the theoretical prediction given by Eq. \ref{['eq:aging_pred_EW']}. Inset: Same data shown without rescaling by the squared roughness $w^2(t_0)$.
• Figure 5: Numerical stability of simulations of the KPZ equation using the CI scheme, for several values of $\lambda$ and system sizes $N$, using time steps $\Delta t$ as indicated in the legends. Parameters are $\nu=1$, $D=1$, $N_{\mathrm{steps}}=10^6$, and $c=0.01$. Each grid cell shows the percentage of time steps for which the gradient defined in Eq. \ref{['eq:discretizacion2']} exceeds $1/c$ and therefore the control function effectively caps this gradient. Uncolored cells correspond to $0\%$.