On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917308111257600 |
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| author | Marcos, J. M. Meléndez, J. J. Cuerno, R. Ruiz-Lorenzo, J. J. |
| author_facet | Marcos, J. M. Meléndez, J. J. Cuerno, R. Ruiz-Lorenzo, J. J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02000 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph Marcos, J. M. Meléndez, J. J. Cuerno, R. Ruiz-Lorenzo, J. J. Statistical Mechanics Mathematical Physics 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. |
| title | On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph |
| topic | Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2603.02000 |