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Aiming to investigate the upper critical dimension, $d_u$, of the KPZ class,in [EPL 103 (2013) 10005] some growth models were numerically analyzed usingCayley trees (CTs) as substrates, as a way to access their behavior in theinfinite-dimensional limit, and some unexpected results were reported:logarithmic roughness scaling, differing for EW and KPZ models (indicating thateven at $d=\infty$ the KPZ nonlinearity is still relevant); beyondasymptotically rough EW surfaces above the upper critical dimension of the EWclass. Motivated by these strange findings, I revisit these growth models hereto show that such results are simple consequences of boundary effects, inherentto systems defined on CTs. In fact, I demonstrate that the anomalous boundaryof the CT leads the growing surfaces to develop curved shapes, which explainsthe 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 theCT, which can be seen as an approximation for the Bethe lattice, smoothsurfaces are found for both EW and KPZ classes, consistently with the behaviorexpected for growing systems in dimensions $d \geqslant d_u$. Interestingfeatures of the 1-pt height fluctuations, such as the possibility ofnon-saturation in the steady state regime, are also discussed for substrates ingeneral.

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