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Random walks on intersecting geometries.

Reza Sepehrinia1, Abbas Ali Saberi1,2, Hor Dashti-Naserabadi3

  • 1Department of Physics, University of Tehran, P. O. Box 14395-547, Tehran, Iran.

Physical Review. E
|October 3, 2019
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Summary
This summary is machine-generated.

We analyzed random walks on a crossing geometry. At long times, diffusion on the plane dominates, but a small drift can direct walkers to the lines.

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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.