Sokoban random walk: A trapping perspective

We study caging/trapping in Sokoban-type models, featuring a random walker moving through a disordered medium of obstacles and capable of pushing some obstacles blocking its path. In one-dimension, we allow the walker to push up to an arbitrary N_{P} number of obstacles. For N_{P}≫1, we use large-deviation theory to show that the survival probability to remain uncaged exhibits crossover from an exponential decay with time at intermediate times to a stretched-exponential decay at long times, with an exponent 1/3 independent of N_{P}. The long-time exponent matches the Balagurov-Vaks-Donsker-Varadhan (BVDV) theory of the classical trapping problem, while the exponential decay is qualitatively distinct from the Rosenstock's intermediate-time theory for classical trapping. Similarly, in two dimensions, numerical simulations reveal that both the Sokoban model and its generalized version exhibit long-time stretched-exponential relaxation with exponent 1/2, again consistent with the BVDV theory. Finally, in two dimensions, we find that the mean trap size is nonmonotonic in ρ: it is small at both low and high densities, but reaches a peak at a characteristic density ρ_{*}. We estimate ρ_{*}≈0.55 for the Sokoban model and ρ_{*}≈0.675 for the generalized Sokoban model.