Self-avoiding walk on a square lattice with correlated vacancies

Area of Science:

• Statistical Physics
• Complex Systems
• Computational Physics

Background:

• Percolation theory describes random networks, but real systems exhibit spatial correlations.
• Self-avoiding walks (SAWs) model polymer chains and diffusion in disordered media.
• The Ising model introduces spatial correlations crucial for understanding metric space properties.

Purpose of the Study:

• To investigate the behavior of self-avoiding walks on a site-diluted correlated percolation lattice.
• To test Flory's mean-field relation for generalized exponents in correlated systems.
• To extract the diffusivity parameter (κ) from Schramm-Loewner evolution theory.

Main Methods:

• Utilizing the Ising model to generate spatial correlations.
• Employing an enriched Rosenbluth Monte Carlo method for simulation.
• Applying perturbative Fokker-Planck-like equations and winding angle analysis.

Main Results:

• Exponents of the self-avoiding walk agree with Flory's approximation at the critical Ising system.
• Off-critical Ising systems reveal a relationship between fractal dimension and correlation length: D_{F}^{SAW}(T)-D_{F}^{SAW}(T_{c})∼1/sqrt[ξ(T)].
• The diffusivity parameter (κ) was successfully extracted using winding angle analysis.

Conclusions:

• Flory's mean-field relation provides a valid approximation for correlated percolation systems.
• The fractal dimension of the walker's path is directly influenced by the correlation length of the host Ising system.
• This work bridges statistical physics models with polymer physics concepts in correlated environments.