The Sokoban Random Walk: A Trapping Perspective

TL;DR

This paper investigates trapping phenomena in Sokoban-like models, revealing universal long-time decay behaviors and a nonmonotonic trap size dependence on obstacle density through theoretical analysis and numerical simulations.

Contribution

It introduces a trapping perspective to Sokoban models, deriving universal decay exponents and analyzing trap size behavior, extending classical trapping theory to these disordered systems.

Findings

Long-time decay follows a stretched-exponential form with exponent 1/3 in 1D and 1/2 in 2D.

Survival probability exhibits crossover from exponential to stretched-exponential decay.

Mean trap size peaks at a characteristic obstacle density, indicating nonmonotonic behavior.

Abstract

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 $N_{\rm P}$ number of obstacles. For $N_{\rm P}\gg 1$, 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 $N_{\rm P}$. 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.