The Sokoban Random Walk: A Trapping Perspective

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 $ρ$: it is small at both low and high densities, but reaches a peak at a characteristic density $ρ_*$. We estimate $ρ_* \approx 0.55$ for the Sokoban model and $ρ_* \approx 0.675$ for the generalized Sokoban model.

Figures (15)

• Figure 1: The dynamical rules governing the time evolution of a $N_{\rm P}$-Sokoban walker with $N_{\rm P}=1$. The walker is shown in red, obstacles in gray, and vacant sites in white. As shown in the top panel, a jump to a nearest-neighbor site is allowed if the target site is vacant. If the target site is occupied, the walker can still move by pushing the obstacle to the next-nearest site, provided that this site is vacant; see the middle panel. If this condition is not satisfied, as shown in the bottom panel, the attempted jump is unsuccessful.
• Figure 2: Trapping by caging of a Sokoban walker in two dimensions, starting from the origin. Panel (a) displays the final configuration in a $50\times 50$ window, covering the ranges $55–105$ along the $x$-axis and $15–65$ along the $y$-axis, from a representative realization at $\rho=0.4$. The Sokoban walker (shown in red) is trapped inside a cage (shown in blue) formed by surrounding obstacles (shown in gray), and its position in the figure is $(80,40)$ (the walker initially is at the origin, outside the displayed window). Panel (b) shows a magnified view of a part of the configuration responsible for trapping, with the trap size $A_{\rm T}=6$. In panel (c), we show the same part at the initial time and see that the trap does not exist initially. The Sokoban walker at some later time enters this region and modifies it into a trap in panel (b) through successive interaction with the obstacles. Finally, in panel (d), we have plotted the number of distinct visited sites, $\Omega(n)$, as a function of time $n$. This number is saturated for all $n \geq n _{\rm T}$ with $n _{\rm T} = 6 \times 10^3$.
• Figure 3: Schematics of the one-dimensional $N_{\rm P}$-Sokoban model with $N_{\rm P} = 2$. Panel (a) shows a realization of the walker and obstacles at the initial time. The walker, located at the origin, is shown in red and obstacles are shown in gray and green. They are identical, but distinguished here to illustrate that the gray ones will be pushed by the walker. As time evolves, the gray obstacles on both sides of the origin are pushed until time $n=n_{\rm T}$ in panel (b). Here, the pushed (gray) obstacles on each side lie on consecutive sites in front of the green obstacle. Beyond this point, the walker cannot push the obstacles due to its finite $(N_{\rm p}=2)$ pushing capacity. Consequently, in every realization, the Sokoban walker is confined inside a finite interval $[-L_1, L_2]$ throughout its time evolution with $L_1$ and $L_2$ determined by the positions of the $(N_{\rm P}+1)$-th obstacles on the $x<0$ and $x>0$ sides, respectively, see Eq. \ref{['hagqo']}. These obstacles are shown in green in this figure. For the chosen realization in this figure, we have $L_1=4$ and $L_2=5$ determined by the positions of the green obstacles, $-Y_{O_{3}}^{-}=-7$ and $Y_{O_{3}}^{+}=8$, in either direction using Eq. \ref{['hagqo']}.
• Figure 4: Left Panel: A schematic illustration of the trajectory of the random walker, where it reaches $x = -L_1$ at the $m$-th time step and subsequently reaches $x = L_2$ at a later time step $n_{\rm T}$. For the shown trajectory, $L_1=3$, $L_2=4$, $m=7$ and $n_{\rm T} = 25$. We have broken the trajectory in two parts: from $0$ to $m$ shown in red and from $m$ to $n_{\rm T}$ shown in blue. Right Panel: Here we have shown an example of the second type of trajectories where the walker first reaches $x = L_2$ at the $m$-th time step and then reaches $x = -L_1$ at $n_{\rm T}$ time step. For this trajectory, $L_1=3$, $L_2=4$, $m=6$ and $n_{\rm T}=18$.
• Figure 5: Parametric-plot of $\langle A _{\rm T}|N_{\rm P} \rangle$ and $\langle n _{\rm T}| N_{\rm P} \rangle$ parametrised by $\rho$, illustrating their scaling-change behavior for the $N_{\rm P}$-Sokoban walker with $N_{\rm P}=0$ (left) and $N_{\rm P}=1$ (right). In both panels, we plot density in the bottom axis while the averages $\langle A _{\rm T}|N_{\rm P} \rangle$ and $\langle n _{\rm T} |N_{\rm P} \rangle$ are plotted in the left and top axes respectively. The dashed lines are our theoretical results for small and high values of $\rho$ given in Eq. \ref{['bhuqm']}.