Sokoban Random Walker Dynamics

• Sokoban random walker is a tracer that navigates a hypercubic lattice filled with obstacles, capable of pushing them to reshape its environment.
• The model exhibits unique transitions with an exact percolation threshold on Bethe lattices (ρ₍c₎ = 1 - 1/(k-1)²) and self-caging behavior on Euclidean grids.
• Generalized pushy dynamics allow multi-obstacle pushes, leading to subdiffusive scaling (e.g., t^(1/3) in 1D and t^(1/4) in 2D) that underscores the interplay between tracer kinetics and geometry.

Searching arXiv for the specified Sokoban random-walk papers to ground the article in the cited literature. The Sokoban random walker is a random walker in a disordered medium of obstacles that can modify its environment by pushing obstacles that block its motion, rather than moving through a fixed obstacle field. In the original Sokoban walk on a hypercubic lattice, each site is initially occupied by an immobile obstacle with probability $\rho$ and empty with probability $1-\rho$, the walker starts at the origin, and at each discrete time step it chooses one of its $2d$ neighboring sites uniformly at random; if the chosen site is empty it hops there, whereas if it is occupied the walker may push that obstacle one lattice spacing further in the same direction provided that the site beyond is empty, with the entire move rejected otherwise (Singh et al., 11 Aug 2025). This local reshaping of the medium places the model between passive transport in static disorder and fully annealed environments. Across recent work, the Sokoban random walker has been used to study exact percolation on trees, the destruction of the classical de Gennes percolation transition in two dimensions, stretched-exponential caging statistics in the Balagurov–Vaks–Donsker–Varadhan (BVDV) universality class, and generalized “pushy” dynamics in which multiple obstacles can be pushed with a resistance penalty (Bonomo et al., 2024, Singh et al., 15 Feb 2026, Bonomo et al., 7 Feb 2026).

1. Model definition and principal variants

In its minimal form, the Sokoban random walker is defined on a $d$-dimensional hypercubic lattice with i.i.d. Bernoulli obstacle occupation of density $\rho$. The walker moves by nearest-neighbor attempts, and the defining kinetic constraint is unilateral pushing without pulling: if the target site is occupied, the obstacle can be shifted by one lattice spacing further in the same direction only if the next site is empty; otherwise the move is rejected (Singh et al., 11 Aug 2025). This is the $N_P=1$ case, where at most one obstacle may be pushed at a time.

A one-dimensional generalization allows the walker to push up to an arbitrary $N_{\rm P}$ number of obstacles in a row, provided the site beyond the $N_{\rm P}$-th obstacle is empty. In that setting, the survival or uncaging probability is defined as

$$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$

where caging occurs when the number of distinct visited sites $\Omega(n)$ saturates (Singh et al., 15 Feb 2026). This formulation makes caging a first-class dynamical observable rather than a by-product of connectivity.

A distinct but related extension is the pushy random walk, formulated on regular lattices in $1-\rho$0 or $1-\rho$1, in continuous time with total attempt rate $1-\rho$2, so that in the absence of obstacles the diffusion constant is $1-\rho$3. If the walker encounters a contiguous column or row of $1-\rho$4 obstacles in the chosen direction, the rate of a successful push-and-hop is

$1-\rho$5

with $1-\rho$6 a push-resistance exponent (Bonomo et al., 7 Feb 2026). The same work states explicitly that this model can be mapped onto a generalized Sokoban in which crates are frictionless and can move in columns or rows, the walker can push $1-\rho$7 crates at once with success probability $1-\rho$8, and crates can be repeatedly pushed.

On the Bethe lattice, the Sokoban rule is specialized to the tree geometry. Every node has coordination number $1-\rho$9, equivalently each non-root node has one parent and $2d$0 children. Obstacles are placed independently on every node except the root with probability $2d$1, and a tracer starts at the unoccupied root. If the tracer chooses an occupied neighboring site, it may push exactly one obstacle into one of its empty children in the same outward direction, chosen at random; no multi-obstacle pushes are allowed, and if all forward neighbors are occupied that branch is permanently blocked (Bonomo et al., 2024).

These definitions share a common feature: the environment is neither strictly static nor globally refreshed. The obstacle field is reshaped locally and irreversibly or quasi-irreversibly by the tracer’s own history. This suggests that the correct organizing principles are geometric and kinetic simultaneously, rather than purely percolative.

2. Exact percolation on the Bethe lattice

The Bethe-lattice formulation admits an exact treatment of escape to infinity. Let $2d$2 denote the probability that the Sokoban eventually escapes to infinity. Because the root has $2d$3 statistically identical branches, one introduces the branch-failure probability

$2d$4

where $2d$5 is the escape probability conditioned on the first node of a branch being empty and $2d$6 is the corresponding quantity when that first node is occupied. The escape probability is then

$2d$7

A branch whose first node is empty satisfies

$2d$8

because its $2d$9 children independently succeed with probability $d$0 (Bonomo et al., 2024).

For an occupied first node, failure occurs either because all $d$1 children are occupied, so the obstacle cannot be pushed, or because the accessible sub-branches do not lead to infinity after the push. After a binomial expansion, the exact recursion for $d$2 is

$d$3

Using the identity quoted in the source and substituting $d$4 and $d$5 yields a compact system, from which one obtains a single implicit equation for $d$6 (Bonomo et al., 2024).

Although that implicit equation is not solved in elementary closed form, the critical density is extracted exactly by sending $d$7. The resulting threshold is

$d$8

Near $d$9 one has $\rho$0, implying

$\rho$1

The transition is therefore second order with mean-field order-parameter exponent $\rho$2 (Bonomo et al., 2024).

The same work compares this with standard site percolation on the Bethe lattice, for which

$\rho$3

Since

$\rho$4

pushing raises the critical obstacle density. The stated interpretation is that each push effectively uses two successive empty sites to clear a path (Bonomo et al., 2024).

This exact solution establishes that on a loopless graph, local environment reshaping can facilitate percolation in the conventional sense of escape to infinity. The result is technically important because it isolates geometry: the enhancement occurs not because pushing is globally powerful, but because on a tree the branching frontier is never closed by loops.

3. Euclidean lattices and the loss of the classical percolation transition

The behavior on Euclidean lattices is sharply different. In the standard ant-in-a-labyrinth problem on the two-dimensional square lattice, the empty sites percolate for obstacle density below the de Gennes threshold, quoted as $\rho$5 in $\rho$6, so a passive walker can escape to infinity (Singh et al., 11 Aug 2025). The Sokoban walker, however, does not preserve this percolative picture.

For the infinite two-dimensional square lattice, numerical and analytical arguments reported in the Bethe-lattice study show that no finite $\rho$7 permits unbounded escape of the Sokoban. The mechanism given there is self-caging: as the walker advances it piles obstacles onto its perimeter, and in $\rho$8 the perimeter of the visited region grows only like the square-root of its area, so the frontier eventually becomes doubly lined by obstacles that cannot be pushed in pairs (Bonomo et al., 2024). In the later hypercubic-lattice treatment, this is restated as the destruction of the de Gennes percolation transition: once pushing is allowed, even minimally, one finds numerically and can argue physically that for any $\rho$9 the walker eventually becomes caged with probability $N_P=1$0, so that

$N_P=1$1

The explanation given is that pushing lets the walker dynamically seal off passages that would otherwise lead to infinity (Singh et al., 11 Aug 2025).

This loss of percolation is one of the central conceptual features of the Sokoban random walker. In static disorder, enhanced mobility is usually associated with lower trapping likelihood. Here the opposite occurs in two dimensions: the ability to modify the environment destroys the infinite connected route that would have existed for a passive tracer. The contrast with the Bethe lattice is therefore not a minor lattice effect but a difference in frontier growth. On the tree, the frontier grows linearly with the number of visited sites and never gets fully blocked; on the square lattice, slower-growing frontiers allow self-generated jamming (Bonomo et al., 2024).

A plausible implication is that the relevant transition in finite-dimensional Sokoban dynamics is not a static connectivity transition of empty sites. The two-dimensional work explicitly replaces that viewpoint with a trapping-based description, in which late-time behavior is controlled by caging and rare large cages rather than by infinite-cluster accessibility (Singh et al., 11 Aug 2025).

4. Survival probability and the BVDV trapping universality class

The long-time statistics of caging are described through survival probabilities. In one dimension, the generalized Sokoban model with pushing capacity $N_P=1$2 admits a large-deviation treatment. For $N_P=1$3, the survival probability exhibits an intermediate-time exponential regime for $N_P=1$4:

$N_P=1$5

Defining $N_P=1$6, one equivalently has

$N_P=1$7

For $N_P=1$8, the same analysis yields the stretched-exponential law

$N_P=1$9

with exponent $N_{\rm P}$0 independent of $N_{\rm P}$1 (Singh et al., 15 Feb 2026).

A closely related derivation for the minimal Sokoban walk in one dimension writes the long-time survival as

$N_{\rm P}$2

with pre-exponential $N_{\rm P}$3 up to logarithmic factors, again giving a $N_{\rm P}$4 stretched exponential at long times (Singh et al., 11 Aug 2025). Both treatments identify the same asymptotic exponent and both connect it to BVDV theory.

The reference point is the classical reactive-trapping result

$N_{\rm P}$5

For $N_{\rm P}$6, the exponent is $N_{\rm P}$7; for $N_{\rm P}$8, it is $N_{\rm P}$9 (Singh et al., 15 Feb 2026). The two-dimensional Sokoban simulations indeed show a late-time $N_{\rm P}$0-stretched law when time is rescaled as $N_{\rm P}$1:

$N_{\rm P}$2

for both the Sokoban model and the generalized Sokoban model (Singh et al., 15 Feb 2026). Likewise, the hypercubic-lattice study reports

$N_{\rm P}$3

in $N_{\rm P}$4, and summarizes the general form as

$N_{\rm P}$5

with a renormalized numerical prefactor relative to reactive trapping (Singh et al., 11 Aug 2025).

These results place the Sokoban random walker in the BVDV universality class despite the fact that the medium is being reshaped. The stated reason is that the leading rare-void mechanism is unaltered: the tail of the cage-size distribution is dominated by large holes in which little or no pushing occurs (Singh et al., 11 Aug 2025). This is a nontrivial point. The dynamics are locally non-passive, yet the asymptotic stretching exponent remains the one associated with rare-region trapping.

The intermediate-time regime is, however, not identical to classical trapping. In one dimension the large-$N_{\rm P}$6 Sokoban model has an exponential decay at moderate times that is qualitatively distinct from Rosenstock’s intermediate-time theory (Singh et al., 15 Feb 2026). This separation between asymptotic universality and pre-asymptotic specificity is a defining feature of the topic.

5. Trap geometry, trap-size statistics, and the trapping transition in two dimensions

In two dimensions, the trapping time $N_{\rm P}$7 is accompanied by a finite connected region of empty sites enclosed by the dynamics, the cage. The trap size $N_{\rm P}$8 is defined as the number of empty sites in that region, or equivalently, in one formulation, as the total number of distinct sites visited by the walker at trapping time (Singh et al., 15 Feb 2026, Singh et al., 11 Aug 2025). The average trap size $N_{\rm P}$9 becomes the key observable for identifying a new transition that replaces the lost percolation threshold.

Simulations on the square lattice show that $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$0 is nonmonotonic in $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$1. At high density, $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$2, the empty region around the origin is small and $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$3; lowering $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$4 enlarges pre-existing voids and $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$5 grows. Below a crossover density $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$6, dynamically self-created cages dominate, and $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$7 decreases again as $$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$8 (Singh et al., 15 Feb 2026). A finer characterization places the maximum at

$$S(n|N_{\rm P})=\mathrm{Prob}[\text{walker not yet caged by time }n] =\langle 1_{n<n_T}\rangle_{(\text{disorder}+\text{moves})},$$9

from simulation scans with $\Omega(n)$0 (Singh et al., 15 Feb 2026). The hypercubic-lattice study emphasizes the same turnover at

$\Omega(n)$1

for the original two-dimensional Sokoban walk and interprets it as a smooth trapping transition separating two qualitatively distinct regimes (Singh et al., 11 Aug 2025).

Above $\Omega(n)$2, trapping is dominated by pre-existing cages: the walker’s cage is essentially a large connected void in the initial obstacle configuration, and pushing only slightly modifies its boundary (Singh et al., 11 Aug 2025). Below $\Omega(n)$3, self-trapping dominates: although extensive empty regions exist, the walker can encounter finite clusters of obstacles that it dynamically rearranges into its own cage, and as $\Omega(n)$4 decreases further such manipulable clusters become rarer and smaller (Singh et al., 11 Aug 2025). The trapping-perspective study states the same dichotomy more compactly: at $\Omega(n)$5 trapping is dominated by pre-existing cages, whereas at $\Omega(n)$6 it arises by the walker reorganizing surrounding obstacles into a self-trap (Singh et al., 15 Feb 2026).

This turnover is distinct from the de Gennes threshold. It is not a statement about infinite-cluster connectivity of empty sites, but about the dominant origin of finite cages. That distinction matters conceptually: a finite-density system may be subcritical in the sense of ultimate caging for every $\Omega(n)$7, yet still exhibit a sharp change in the mechanism and scale of trapping. The use of $\Omega(n)$8 as a proxy captures exactly that restructuring of the dynamics (Singh et al., 11 Aug 2025).

6. Generalized pushy dynamics, diffusion and subdiffusion, and open directions

The pushy random walk broadens the Sokoban idea by allowing multiple-obstacle pushes with resistance exponent $\Omega(n)$9. In one dimension, the walker progressively carves out an obstacle-free cavity of length $1-\rho$00, with a crust of thickness $1-\rho$01 on either side satisfying the mass-balance relation

$1-\rho$02

For $1-\rho$03, the cavity length obeys

$1-\rho$04

whose solution is

$1-\rho$05

More generally,

$1-\rho$06

so the one-dimensional mean-squared displacement scales as

$1-\rho$07

The resulting dynamics are subdiffusive because pushing a larger and larger block of obstacles becomes progressively harder (Bonomo et al., 7 Feb 2026).

In two dimensions, the corresponding circular-cavity picture uses a disk of radius $1-\rho$08 surrounded by an annular crust of thickness $1-\rho$09, with

$1-\rho$10

For $1-\rho$11,

$1-\rho$12

and therefore

$1-\rho$13

In general, resistance $1-\rho$14 gives $1-\rho$15 (Bonomo et al., 7 Feb 2026).

The same work reports a diffusion–subdiffusion transition in $1-\rho$16. At low obstacle density the crust remains full of holes, the walker eventually escapes the disk, and ordinary diffusion occurs with effective diffusion coefficient $1-\rho$17; as $1-\rho$18 one finds $1-\rho$19. At $1-\rho$20 the crust becomes “impenetrable” and the walker remains confined to a slowly growing disk $1-\rho$21. Numerically, for $1-\rho$22,

$1-\rho$23

The model is presented explicitly as a generalized Sokoban in which taking $1-\rho$24 recovers the single-crate limit of standard Sokoban, since $1-\rho$25 becomes always blocked (Bonomo et al., 7 Feb 2026).

These generalized results do not replace the minimal Sokoban findings; rather, they clarify the role of pushing capacity and resistance. They show that allowing multiple-obstacle pushes can restore large-distance penetration at finite density and produce a diffusion–localization crossover absent from the strictly single-push two-dimensional Sokoban description. This suggests that the single-push constraint is a decisive kinetic ingredient in the loss of percolation on Euclidean lattices.

Several open directions are stated explicitly in the Bethe-lattice work: studying Sokoban percolation on hypercubic lattices in $1-\rho$26, fractal graphs, small-world and random networks; allowing multi-obstacle pushes or obstacle rearrangements of finite range; and investigating dynamical quantities such as first-passage times and sub-diffusion exponents in the pushy regime (Bonomo et al., 2024). Taken together with the trapping and pushy-random-walk studies, these directions define the present research frontier: understanding how tracer–media interactions, local kinetic constraints, and underlying geometry jointly determine whether reshaping promotes transport, induces self-caging, or drives subdiffusive confinement.