Growth over time-correlated disorder: a spectral approach to Mean-field

We generalize a model of growth over a disordered environment, to a large class of It\=o 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 trade-off between exploration and exploitation. An additional mapping to the Schr\"ordinger 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, that can be studied in details, and gives yet another explanation for the occurrence of the $\textit{Zipf law}$ in complex, well-connected systems.