Localized growth and branching random walks with time correlations.

Abstract

We generalize a model of growth over a disordered environment, to a large class of Itō processes. In particular, we study how the microscopic properties of the noise influence the macroscopic growth rate. The present model can account for growth processes in large dimensions and provides a bed to understand better the tradeoff between exploration and exploitation. An additional mapping to the Schrödinger equation readily provides a set of disorders for which this model can be solved exactly. This mean-field approach exhibits interesting features, such as a freezing transition and an optimal point of growth, which can be studied in detail, and gives yet another explanation for the occurrence of the Zipf law in complex, well-connected systems.