Sokoban percolation on the Bethe lattice

Abstract

`With persistence, a drop of water hollows out the stone' goes the ancient Greek proverb. Yet, canonical percolation models do not account for interactions between a moving tracer and its environment. Recently, we have introduced the Sokoban model, which differs from this convention by allowing a tracer to push single obstacles that block its path. To test how this newfound ability affects percolation, we hereby consider a Bethe lattice on which obstacles are scattered randomly and ask for the probability that the Sokoban percolates through this lattice, i.e., escapes to infinity. We present an exact solution to this problem and determine the escape probability as a function of obstacle density. Similar to regular percolation, we show that the escape probability undergoes a second-order phase transition. We exactly determine the critical obstacle density at which this transition occurs and show that it is higher than that of a tracer without obstacle-pushing abilities. Our findings assert that pushing facilitates percolation on the Bethe lattice, as intuitively expected. This result, however, sharply contrasts with our previous findings on the 2D square lattice. There, the Sokoban cannot escape $\unicode{x2013}$ not even at densities well below the percolation threshold. We discuss the reasons behind this striking difference, which calls for a deeper and better understanding of percolation in the presence of tracer-media interactions.