Percolation perspective on sites not visited by a random walk in two dimensions

Area of Science:

• Statistical Physics
• Percolation Theory
• Computational Physics

Background:

• Investigating site percolation on L×L square lattices with periodic boundary conditions.
• Focusing on sites unvisited by a random walk of N=uL^2 steps, termed vacant sites.

Purpose of the Study:

• Analyze the percolation behavior of vacant sites in a 2D lattice.
• Characterize the properties of vacant clusters, including their fractal dimensions and size distributions.
• Compare simulation results with theoretical predictions for percolation exponents.

Main Methods:

• Numerical simulations on L×L square lattices with periodic boundary conditions.
• Analysis of spanning probability, cluster size distributions, and fractal dimensions.
• Calculation of percolation exponents τ and q.

Main Results:

• The percolation probability is a smooth function of u, decreasing monotonically, without a sharp threshold.
• Vacant clusters are non-fractal but possess fractal boundaries with a dimension of 4/3.
• Cluster size distribution follows n_s ∝ s^{-τ} with τ≈1.83, and cluster volume fraction P_k ∝ k^{-q} with q≈1.20.

Conclusions:

• The 2D percolation problem of vacant sites differs significantly from higher-dimensional counterparts.
• The observed exponents τ and q are likely effective, slowly converging to asymptotic values with increasing lattice size L.
• Fractal boundaries of vacant clusters are a key characteristic of this 2D system.