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 $\nu$, $\beta$, and $\gamma$ for various step lengths. Our results reveal that the values of these exponents depend on the step length $l$. Specifically, for $2\le l\le 3$, 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 $\nu d=2\beta +\gamma$ is valid, despite the variation in the critical exponents.