High dimensional properties of quenched noise growth models

TL;DR

This paper investigates the behavior of quenched noise invasion models in high dimensions, revealing a critical dimension where the interface roughness transitions and demonstrating the fractal nature and anomalous critical behavior of the processes.

Contribution

It introduces the concept of a critical dimension for quenched noise invasion models and analyzes the transition to correlated annealed dynamics in high dimensions.

Findings

Roughness decreases to zero above the critical dimension

At five dimensions, roughness matches that of annealed equations

Processes on Cayley trees exhibit fractal and anomalous critical behavior

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 $\chi_c$ 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.