Random walks on intersecting geometries

Area of Science:

• Statistical Mechanics
• Probability Theory
• Mathematical Physics

Background:

• Random walks are fundamental models in statistical mechanics.
• Understanding walker behavior in complex geometries is crucial.
• Previous studies often focused on simpler lattice structures.

Purpose of the Study:

• To analytically study simple symmetric random walks on a novel crossing geometry.
• To calculate the probability density of a walker's position over time.
• To investigate the influence of geometry and drift on walker behavior.

Main Methods:

• Developed an analytical approach for random walks on a plane square lattice intersected by multiple lines meeting at the origin.
• Exactly calculated the probability density as a function of time.
• Performed extensive simulations to validate analytical predictions.

Main Results:

• Determined that plane diffusion eventually dominates walker behavior for large times (t > t_c ∝ n_l^2).
• Showed that a small drift perturbation can bias the walker towards the line geometry.
• Confirmed analytical predictions with simulation results.

Conclusions:

• The behavior of random walks in this crossing geometry is predictable and analytically tractable.
• A subtle drift can significantly alter long-time walker distribution.
• The analytical method is adaptable to other complex geometries with a common origin.