Analytical calculation for the percolation crossover in deterministic partially self-avoiding walks in...

Area of Science:

• Statistical Physics
• Complex Systems
• Dynamical Processes

Background:

• Understanding walker behavior in disordered media is crucial for modeling complex systems.
• Self-avoiding walks are fundamental models in statistical physics, but real-world systems often exhibit partial self-avoidance.

Purpose of the Study:

• To analytically determine the conditions under which a partially self-avoiding walker can fully explore a disordered system.
• To identify the critical memory length required for complete exploration and analyze the transition behavior.

Main Methods:

• Analytical calculation of the probability of visiting all points using a deterministic, partially self-avoiding walk.
• Validation of analytical results through Monte Carlo simulations.
• Investigation of the critical memory threshold and system exploration dynamics.

Main Results:

• Derived an analytical expression for the probability of visiting all N points, P{N}(mu)=(1-2{-mu}){N-mu-1}.
• Identified a critical memory threshold, mu{1}=lnN/ln2.
• Demonstrated that memory of order log(N) is sufficient for full system exploration, with a sharp transition observed for large systems.

Conclusions:

• A walker with partial self-avoidance requires only logarithmic memory (log base 2 of N) to fully explore a one-dimensional disordered system.
• The study reveals a sharp transition in exploration behavior around the critical memory length, indicating efficient exploration strategies.
• Findings have implications for understanding transport and diffusion in complex, disordered environments.