Surface growth on treelike lattices and the upper critical dimension of the KPZ class

TL;DR

This paper revisits growth models on treelike lattices to clarify the upper critical dimension of the KPZ class, revealing that previous anomalous results were due to boundary effects and confirming expected behaviors in high dimensions.

Contribution

The study demonstrates that boundary effects on Cayley trees cause anomalous surface behaviors, clarifying the true nature of the KPZ upper critical dimension and correcting prior misinterpretations.

Findings

Boundary effects cause curved surface shapes on Cayley trees.

Height fluctuations at the central site show smooth surfaces in high dimensions.

Non-saturation of height fluctuations possible in steady state regimes.

Abstract

Aiming to investigate the upper critical dimension, $d_u$, of the KPZ class, in [EPL 103 (2013) 10005] some growth models were numerically analyzed using Cayley trees (CTs) as substrates, as a way to access their behavior in the infinite-dimensional limit, and some unexpected results were reported: logarithmic roughness scaling, differing for EW and KPZ models (indicating that even at $d=\infty$ the KPZ nonlinearity is still relevant); beyond asymptotically rough EW surfaces above the upper critical dimension of the EW class. Motivated by these strange findings, I revisit these growth models here to show that such results are simple consequences of boundary effects, inherent to systems defined on CTs. In fact, I demonstrate that the anomalous boundary of the CT leads the growing surfaces to develop curved shapes, which explains the strange behaviors previously found for these systems, once the global "roughness" were analyzed for non-flat surfaces in the study above. Importantly, by measuring the height fluctuations at the central site of the CT, which can be seen as an approximation for the Bethe lattice, smooth surfaces are found for both EW and KPZ classes, consistently with the behavior expected for growing systems in dimensions $d \geqslant d_u$. Interesting features of the 1-pt height fluctuations, such as the possibility of non-saturation in the steady state regime, are also discussed for substrates in general.