Sokoban Random Walk: From Environment Reshaping to Trapping Crossover

We study the dynamics of a Sokoban random walker moving in a disordered medium with obstacle density $\rho$. In contrast to the classic model of de Gennes with static obstacles that exhibits a percolation transition, the Sokoban walker is capable of modifying its environment by pushing a few surrounding obstacles. Surprisingly, even a limited pushing ability leads to a loss of the percolation transition. Through a combination of a rigorous large-deviation calculation and extensive numerical simulations, we demonstrate that the Sokoban model belongs to the Balagurov-Vaks-Donsker-Varadhan trapping universality class. The survival probability that the walker has not yet been trapped inside a cage exhibits stretched-exponential relaxation at late times. Furthermore, using the average trap size as a proxy, we identify the emergence of a dynamical crossover at a density $\rho_* \approx 0.55$ between two qualitatively different trapping mechanisms: a self-trapping mechanism at low density, where the walker becomes dynamically localized within a self-formed trap, and a pre-existing trapping mechanism at high density, where confinement arises from the initial arrangement of obstacles. This crossover is responsible for the loss of the classical percolation transition.