Percolation Perspective on Sites Not Visited by a Random Walk in Two Dimensions

TL;DR

This study investigates the percolation properties of sites unvisited by a random walk on a two-dimensional lattice, revealing smooth percolation probability, fractal boundaries, and effective critical exponents that approach theoretical values slowly.

Contribution

The paper provides numerical analysis of vacant site clusters in 2D, showing the absence of a sharp threshold and proposing effective critical exponents with a heuristic explanation.

Findings

Percolation probability decreases smoothly with walk steps

Vacant clusters have fractal boundaries with dimension 4/3

Effective exponents drift slowly towards theoretical values as lattice size increases

Abstract

We consider the percolation problem of sites on an $L\times L$ square lattice with periodic boundary conditions which were unvisited by a random walk of $N=uL^2$ steps, i.e. are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with $u$. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size $L$ is the only large length scale in this problem. The typical mass (number of sites $s$) in the largest cluster is proportional to $L^2$, and the mean mass of the remaining (smaller) clusters is also proportional to $L^2$. The normalized (per site) density $n_s$ of clusters of size (mass) $s$ is proportional to $s^{-\tau}$, while the volume fraction $P_k$ occupied by the $k$th largest cluster scales as $k^{-q}$. We put forward a heuristic argument that $\tau=2$ and $q=1$. However, the numerically measured values are $\tau\approx1.83$ and $q\approx1.20$. We suggest that these are effective exponents that drift towards their asymptotic values with increasing $L$ as slowly as $1/\ln L$ approaches zero.