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

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 $N\to \mathrm{\infty }$. 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 $N\to \mathrm{\infty }$, 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.