Percolation on sites visited by continuous random walks in a simple cubic lattice

Abstract

We investigate the percolation on sites visited by random walks with fixed step lengths in a simple cubic lattice, where the random walker moves in continuous space. Using the Newman–Ziff algorithm combined with finite-size scaling analysis, we calculate the percolation threshold and critical exponents 𝜈, 𝛽, and 𝛾 for various step lengths. Our results reveal that the values of these exponents depend on the step length 𝑙. Specifically, for 2 ≤ 𝑙 ≤ 3, the critical exponents align with those of the percolation models based on discrete random walks in three dimensions, and gradually transform to the values of the ordinary three-dimensional site percolation as 𝑙 increases. We analyze that these changes occur because the correlation function varies with the step length 𝑙. Moreover, we confirm that the hyperscaling relation 𝜈𝑑 = 2𝛽 + 𝛾 is valid, despite the variation in the critical exponents.