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Peer-reviewedCC BY 4.0SciPost Physics

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

  • Jesús María Marcos Merino1
  • Juan José Meléndez1
  • Rodolfo Cuerno2
  • Juan J. Ruiz-Lorenzo1
  1. 1Universidad de Extremadura
  2. 2Carlos III University of Madrid
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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 the number of nodes in the graph, N N , tends to infinity. 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 N , often accompanied by numerical instabilities; however, these differences disappear as N N increases. In the large- N 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∞ N → ∞ , 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.

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Cite

BibTeXRIS
BibTeX
@article{merino2026upper,
  title = {{On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph}},
  author = {Jesús María Marcos Merino and Juan José Meléndez and Rodolfo Cuerno and Juan J. Ruiz-Lorenzo},
  journal = {SciPost Physics},
  year = {2026},
  month = oct,
  doi = {10.21468/scipostphys.21.4.087},
  url = {https://scipost.org/10.21468/SciPostPhys.21.4.087},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On the upper critical dimension of the KPZ universality class: KPZ and related equations on a fully connected graph
AU  - Jesús María Marcos Merino
AU  - Juan José Meléndez
AU  - Rodolfo Cuerno
AU  - Juan J. Ruiz-Lorenzo
PY  - 2026
DA  - 2026/10/09
JO  - SciPost Physics
DO  - 10.21468/scipostphys.21.4.087
UR  - https://scipost.org/10.21468/SciPostPhys.21.4.087
AB  - 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 the number of nodes in the graph, N N , tends to infinity. 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 N , often accompanied by numerical instabilities; however, these differences disappear as N N increases. In the large- N 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∞ N → ∞ , 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.
SN  - 2542-4653
ER  -