High dimensional properties of quenched noise growth models

Abstract

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 $χ_c$ and the roughness of the underlying interface dynamics. In high enough dimension we argue that $χ_c$ 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.