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 $N_{\mathrm{P}}$ number of obstacles. For $N_{\mathrm{P}}\gg 1$, 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 $N_{\mathrm{P}}$. 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 $\rho$: it is small at both low and high densities, but reaches a peak at a characteristic density ${\rho }_{*}$. We estimate ${\rho }_{*}\approx 0.55$ for the Sokoban model and ${\rho }_{*}\approx 0.675$ for the generalized Sokoban model.

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• Classical statistical mechanics
• Nonequilibrium statistical mechanics
• Percolation
• Random walks