Lévy random walks with fluctuating step number and multiscale behavior

Area of Science:

• Statistical Physics
• Complex Systems Analysis
• Probability Theory

Background:

• Random walks are fundamental models in physics and mathematics.
• Fluctuations in step number and length introduce complexities beyond standard models.
• Power-law and stable distributions are crucial for describing phenomena with heavy tails.

Purpose of the Study:

• To investigate random walks with variable step numbers and lengths governed by stable or power-law distributions.
• To derive and characterize the resulting limiting distributions.
• To compare theoretical findings with simulations of self-organized critical systems.

Main Methods:

• Mathematical analysis of random walks in n dimensions.
• Application of the Lévy-Gnedenko generalization of the central limit theorem.
• Investigation of correlated and uncorrelated step numbers.
• Derivation of infinitely divisible limiting distributions and K distributions.

Main Results:

• Identified new classes of limiting distributions for random walks with fluctuating step numbers and power-law step lengths.
• Demonstrated ultraslow convergence to these distributions.
• Showed that finite step numbers introduce an inner scale, modifying behavior.
• Unified K distributions, stable distributions, and power-law tails under a single framework.

Conclusions:

• The derived distributions offer a unified description for various phenomena, including high/low frequency cascades and self-organized criticality.
• The study provides a theoretical foundation for understanding complex systems exhibiting anomalous diffusion.
• Findings have implications for modeling diverse physical and biological processes.