Sokoban Random Walk: From Environment Reshaping to Trapping Crossover

TL;DR

The paper analyzes a Sokoban random walker that can push a limited number of obstacles, showing that environmental reshaping destroys the classical percolation transition and that the trapping dynamics fall into the Balagurov–Vaks–Donsker–Varadhan universality class. In 1D, the survival probability exhibits a BVDV-type stretched exponential with exponent $1/3$, while in 2D the exponent is $1/2$ with a density-dependent prefactor; a dynamical crossover occurs around $\rho_* \approx 0.55$ between self-trapping and pre-existing trapping. The results are robust to variations in pushing rules and are supported by rigorous large-deviation analysis and extensive simulations, with implications for experimental realizations in active matter and robotics.

Abstract

We study the dynamics of a Sokoban random walker moving in a disordered medium with obstacle density $ρ$. 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 $ρ_* \approx 0.55$ 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.

Figures (5)

• Figure 1: (a) The initial configuration of the obstacles in two dimensions with walker shown in red and the obstacles shown in gray and green. They are identical, but distinguished here to illustrate that the green ones will be pushed by the walker. (b) This represents the trapped scenario where, the green obstacles have been pushed and the walker cannot visit any new lattice site further. The trap size $A_{\rm T} = 21$ is demonstrated in blue.
• Figure 2: Long-time relaxation of the survival probability as a function of time in one (left panel) and two (right panel) dimensions. The left panel shows a comparison between numerical simulations (colored symbols) and the theoretical result given in Eq. \ref{['sok-surv-eq-35']} (dashed lines). The right panel, on the other hand, shows the survival probability as a function of $\eta = n / \langle n_{\rm T} \rangle$. Here the dashed lines are the fits to the simulation data given in Eq. \ref{['ahbvci']} (shown by symbols).
• Figure 3: Nonmonotonic behavior of $\langle A_{\rm T} \rangle$ as a function of $\rho$ for a two-dimensional Sokoban walker using numerical simulations (red symbols). The turnover density is $\rho _{*} \approx 0.55$, marking a crossover from the low-density self-trapping to the pre-existing traps at high density.
• Figure S1: Survival probability $S(n)$ for the one-dimensional $N_{\rm P}$-Sokoban model for three different values of $N_{\rm P}$. The obstacle density is fixed to $\rho = 0.1$ for all three cases. The red symbols denote the simulation data, while the dashed line represents the stretched-exponential form in Eq. \ref{['SM-eq-1']} associated with the BVDV universality.
• Figure S2: Left Panel: Survival probability $S(\eta)$ vs $\eta = n / \langle n_{\rm T} \rangle$ for the two-dimensional generalized Sokoban model for two different values of density. The colored symbols represent the simulation data, while the dashed lines are fit to the data corresponding to BVDV form in Eq. \ref{['SM-eq-2']}. Right Panel: Dependence of the mean trap size $\langle A_{\rm T} \rangle$ on obstacle density $\rho$. The curve is nonmonotonic, with $\langle A_{\rm T} \rangle$ attaining its maximum value of approximately $60$ at $\rho_* \approx 0.675$. This peak reflects the crossover between pre-existing trapping at high densities and self-trapping at lower densities, similar to the behavior observed in the original Sokoban model.