  We discuss the behavior of bounded slope quenched noise invasion models in
high dimensions. We first observe that the roughness of such a steady state
interface is generated by the combination of the roughness of the invasion
process $\chi_c$ and the roughness of the underlying interface dynamics. In
high enough dimension we argue that $\chi_c$ decreases to zero. This defines a
critical dimension for the problem, over which it reduces to the correlated
annealed dynamics, which we show to have the same roughness as the annealed
equation at five dimensions. We argue that on the Cayley tree with one
additional height coordinate the associated processes are fractal. The critical
behavior is anomalous due to strong effects of rare events. Numerical
simulations of the model on a Cayley tree and high dimensional lattices support
those theoretical predictions.
