  I report on an extensive numerical investigation of various discrete growth
models describing equilibrium and nonequilibrium interfaces on a substrate of a
finite Bethe lattice. An unusual logarithmic scaling behavior is observed for
the nonequilibrium models describing the scaling structure of the infinite
dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This
gives rise to the classification of different growing processes on the Bethe
lattice in terms of logarithmic scaling exponents which depend on both the
model and the coordination number of the underlying lattice. The equilibrium
growth model also exhibits a logarithmic temporal scaling but with an ordinary
power law scaling behavior with respect to the appropriately defined lattice
size. The results may imply that no finite upper critical dimension exists for
the KPZ equation.
