Localized growth and branching random walks with time correlations

Area of Science:

• Complex Systems
• Stochastic Processes
• Statistical Physics

Background:

• Understanding growth dynamics in disordered environments is crucial for various scientific fields.
• Existing models often struggle with high-dimensional systems and complex noise interactions.
• The interplay between exploration and exploitation in growth processes remains a key challenge.

Purpose of the Study:

• To generalize a growth model to a broad class of Itō processes.
• To investigate the influence of microscopic noise properties on macroscopic growth rates.
• To provide a framework for understanding growth in high dimensions and the exploration-exploitation tradeoff.

Main Methods:

• Generalization of a growth model to Itō processes.
• Analysis of the relationship between microscopic noise and macroscopic growth.
• Mapping the model to the Schrödinger equation for exact solvability.
• Application of a mean-field approach.

Main Results:

• The model successfully accounts for growth processes in large dimensions.
• A freezing transition and an optimal growth point were identified.
• The model offers an explanation for the Zipf law in complex systems.
• Exact solutions are obtainable for specific disorder types via Schrödinger equation mapping.

Conclusions:

• The generalized model provides a robust framework for studying growth in disordered systems.
• Microscopic noise characteristics significantly dictate macroscopic growth behavior.
• The model offers insights into fundamental principles governing complex systems, including the Zipf law.
• This work bridges stochastic processes and statistical physics, with implications for diverse scientific domains.