  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.
