Percolation Clusters of Sites Not Visited by a Random Walk in Two Dimensions

  Amit Federbush  ,  Yacov Kantor  
Tel-Aviv University

We consider a random walk (RW) of uM steps on a square lattice of M=L×L sites with periodic boundaries, and study the percolation of sites unvisited by the RW. The d-dimensional version of this problem (on a hypercubic lattice) at d≥3 has a sharp percolation threshold threshold at some uc: for an infinite lattice, there is a spanning cluster if u<uc and there is no such cluster otherwise [1,2].
However, d=2 is the lower critical dimension of this problem and it has no percolation threshold, and the percolation (spanning) probability Π(L,u) converges to a smooth function of u as L→∞ [1].

We numerically study two-dimensional lattices of linear sizes ranging up to L=4096. We find that the clusters are not fractal, but they have fractal boundaries with fractal dimension 4/3. The lattice size L is the only large length scale
in this problem, with the typical mass of the largest cluster as well as the mean mass of the remaining (smaller) clusters being both proportional to L2The normalized (per site) density ns of clusters of size (mass) s is proportional to s [3], while the volume fraction Pk occupied by the kth largest cluster scales as k-q. We propose a heuristic scaling argument that τ=2 and q=1. This prediction slightly differs from the numerically measured values of τ≈1.83 and q≈1.20. We notice a very slow dependence of these exponents on L, and suggest that their effective values drift towards their asymptotic values with increasing L as slowly as 1/lnL approaches zero.

[1] Y. Kantor and M. Kardar, Phys. Rev. E 100, 022125 (2019).

[2] T. Abete, A. de Candia, D. Lairez, and A. Coniglio, Phys. Rev. Lett. 93, 228301 (2004).

[3] D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd ed. (Taylor and Francis, London, UK, 1991).