Flips reveal the universal impact of memory on random explorations

Area of Science:

• Physics
• Mathematics
• Complex Systems

Background:

• Random walk theory is crucial for understanding phenomena like diffusion and foraging.
• Existing models often assume memoryless (Markovian) dynamics, which are unrealistic for many real-world systems.
• Real-world systems frequently exhibit memory effects, making analysis of space exploration challenging.

Purpose of the Study:

• To introduce a novel metric, the 'flip probability,' for quantifying space exploration in random walks.
• To demonstrate the universal applicability of this metric across different walker dynamics, including memory-dependent ones.
• To analyze the impact of memory on exploration and identify its underlying mechanisms.

Main Methods:

• Definition of the 'flip' event in a one-dimensional random walk.
• Theoretical analysis of the flip probability for both Markovian and non-Markovian walkers.
• Computational simulations of non-Markovian models.
• Analysis of experimental datasets.

Main Results:

• The flip probability universally obeys the law πn ∝ 1/n, irrespective of the walker's dynamics (memoryless or not).
• This universality highlights the significant and consistent impact of memory on exploration.
• The mechanism behind this universality was identified and extended to higher dimensions and fractal domains.

Conclusions:

• The flip probability provides a universal measure for space exploration, effectively quantifying the impact of memory.
• This finding offers a new tool for analyzing complex systems with history-dependent dynamics.
• The universality of the flip probability simplifies the study of exploration across diverse scientific fields.