Sokoban Random Walk: From Environment Reshaping to Trapping Transition

Abstract: We study the dynamics of a $\textit{Sokoban random walker}$ 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 a new trapping transition that replaces the classical percolation transition. This transition occurs at a threshold density $\rho_* \approx 0.55$ and separates two qualitatively distinct trapping regimes: a self-trapping regime at low density, where the walker becomes dynamically localized within a self-formed trap, and a pre-existing trapping regime at high density, where confinement arises from the initial arrangement of obstacles.