Loss of Percolation Transition in the Presence of Simple Tracer-Media Interactions Ofek Lauber Bonomo1and Shlomi Reuveni1 1School of Chemistry Center for the Physics Chemistry of Living Systems Ratner Institute for Single Molecule Chemistry

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Loss of Percolation Transition in the Presence of Simple Tracer-Media Interactions

Ofek Lauber Bonomo1,∗and Shlomi Reuveni1,†

1School of Chemistry, Center for the Physics & Chemistry of Living Systems, Ratner Institute for Single Molecule Chemistry,

and the Sackler Center for Computational Molecular & Materials Science, Tel Aviv University, 6997801, Tel Aviv, Israel

(Dated: December 27, 2022)

Random motion in disordered media is sensitive to the presence of obstacles which prevent atoms, molecules,

and other particles from moving freely in space. When obstacles are static, a sharp transition between con-

ﬁned motion and free diffusion occurs at a critical obstacle density: the percolation threshold. To test if this

conventional wisdom continues to hold in the presence of simple tracer-media interactions, we introduce the

Sokoban random walk. Akin to the protagonist of an eponymous video game, the Sokoban has some ability to

push away obstacles that block its path. While one expects this will allow the Sokoban to venture further away,

we surprisingly ﬁnd that this is not always the case. Indeed, as it moves — pushing obstacles around — the

Sokoban always conﬁnes itself to a ﬁnite region whose mean size is uniquely determined by the initial obstacle

density. Consequently, the percolation transition is lost. This ﬁnding breaks from the ruling “ant in a labyrinth”

paradigm, vividly illustrating that even weak and localized tracer-media interactions cannot be neglected when

coming to understand transport phenomena.

More than a century after their introduction to the readers of

Nature by Karl Pearson [1], random walks continue to fasci-

nate and draw attention [2, 3]. While initially motivated by the

theory of gambling [4] and ﬁnancial speculation [5], random

walks became important in the natural sciences following the

pioneering works of Einstein [6] Smoluchowski [7] and others

[8] on diffusion of atoms and molecules. In the time following

the publication of these seminal works, random walks were

further established as a versatile modelling tool [9–14], with

applications in physics [15–21], chemistry [23–28], biology,

[29–33] movement ecology [34–39], ﬁnance and economics

[40, 41].

One paradigmatic random walk is the “ant in a labyrinth”

[42], which was introduced by Pierre-Gilles de Gennes, as

a simple model for diffusion in disordered media [43–45].

Consider an ant walking on a two-dimensional square lattice,

where a fraction ρof the lattice sites are occupied with ob-

stacles, and all other sites are empty. Each time unit, the ant

takes a step onto an empty neighbouring site that is chosen

randomly. Given a speciﬁc lattice size and density ρ, one

can ask how does the mean squared displacement (MSD) of

the ant depend on time? When ρis small, i.e., most sites

are empty, the ant’s motion is almost unobstructed. In this

case, the MSD scales linearly with time. On the other ex-

treme, when ρis large, most sites are occupied by obstacles

and the ant’s motion is highly restricted. In this limit the ant

will ﬁnd itself caged in the labyrinth, resulting in an MSD

that saturates asymptotically. As it turns out, for large enough

systems, the transition between restricted motion and free dif-

fusion is sharp, occurring at a critical density 0 <ρc<1 [46].

The basic assumption in de Gennes’ model is that obsta-

cles comprising the media are immobile. Namely, it is as-

sumed that the ant’s motion has no effect on the distribution

of obstacles around it. Yet, this assumption is often violated,

e.g. when considering active particles that burn energy and ex-

ert strong forces on their surroundings. Ironically, the ant —

which can lift many times its body weight — is a quintessen-

tial example of such an active particle. Assuming immobile

FIG. 1. The Sokoban random walk. Panel (a): Laws of motion.

The walk has two feasible moves: (i) it can step into an unoccupied

site; (ii) it can step into an occupied site by pushing away an ob-

stacle that occupied it one site forward, in its direction of motion.

Only one obstacle can be pushed at a time. Thus, two occupied sites

in a row create a block. At each time step, the walker chooses be-

tween all feasible moves with equal probability. Panel (b): Initial

conﬁguration of a 15 ×15 arena. Arenas are generated by randomly

distributing obstacles, such that each site has probability ρto be oc-

cupied. White squares indicate unoccupied sites. Black and gray

squares indicate obstacles (these are identical, but distinguished here

for clarity). Panel (c): A possible trajectory of the Sokoban random

walk. Gray squares indicate obstacles that were pushed during the

course of the walk.

obstacles is thus fair when considering a hydrogen atom dif-

fusing in a solid, but the validity of this assumption is ques-

tionable for animals and microorganisms plowing their way

through crowded environments. Validity is also questionable

on the microscopic scale where one may encounter immobile

obstacles that cannot be nudged by thermal ﬂuctuations (on

relevant time scales), but may nevertheless be pushed around

by active particles in the media. Yet, to date, little is known

arXiv:2210.04343v2 [cond-mat.stat-mech] 24 Dec 2022

on if and how transport properties are affected as a result.

The model.—We introduce a minimalist model to show that

tracer-media interactions from the type mentioned above re-

sult in a drastic, qualitative, change of transport properties. To

this end, we consider a random walker that has some ability

to push away obstacles that block its path. We imagine an

n×nsquare arena where a fraction ρof the available sites are

occupied by obstacles. Taking nto be odd, we place a ran-

dom walker at the center of this arena. The random walk then

takes place according to the following rules which are illus-

trated in Fig. 1a. The walker can move into an unoccupied

neighbouring site, placed horizontally or vertically relative to

its position. In addition, even when a site is occupied by an

obstacle, the walker can move into this site while pushing the

obstacle one site forward, in its direction of motion. Yet, this

can only be done provided that the next site (in the direction

of motion) is vacant. Thus, the walker cannot push more than

one obstacle at a time. Finally, at each time step, the walker

chooses between all feasible moves with equal probability.

The model presented herein is inspired by the video game

Sokoban (Japanese for warehouse keeper), which was created

in 1981 by Hiroyuki Imabayashi. The premise of the game

is simple: The player, playing as the keeper, pushes boxes

around in a warehouse, in attempt to transport them to marked

storage locations. The rules of the game are similar to the rules

of the walk presented in Fig. 1. While being fairly simple

to play, solving Sokoban puzzles turns out to be a difﬁcult

computational task. It was ﬁrst proved to be NP-hard [47] and

was later shown to be PSPACE-complete [48].

An illustration of a trajectory of the Sokoban random walk

is given in panels (b) and (c) of Fig. 1. The initial conﬁgu-

ration of the arena is given in panel (b), and the trajectory of

the walk is illustrated in panel (c). Note that a simple random

walk, i.e., one that cannot push obstacles that stand in its path,

would have actually been caged by the initial conﬁguration of

the arena. In contrast, the Sokoban was able to escape this

cage by pushing some of the obstacles surrounding it (high-

lighted in gray). More generally, we expect that the ability to

push obstacles will enable the Sokoban random walk to ven-

ture further away from its initial position when compared to

a simple random walk without this pushing ability (“ant in a

labyrinth”).

Monte Carlo simulations.—To test this hypothesis, we sim-

ulate the Sokoban and simple random walks, for a large num-

ber of randomly generated and sufﬁciently large arenas, so

as to completely avoid boundary effects. In Fig. 2a we plot

the mean squared displacement, given by MSD(t) = hr2(t)i,

where r(t)is the Euclidean distance to the initial position at

time t, and h·i indicates an ensemble average over all gen-

erated walks. Plots are made for the Sokoban (purple) and

simple (yellow) random walks at three different obstacle den-

sities. As expected, in the long time limit, the MSD of the

Sokoban is signiﬁcantly higher compared to the MSD of the

simple random walk. As a result, the Sokoban explores larger

portions of the arena as illustrated by the trajectories given in

Fig. 2b. Further illustration of the Sokoban walk is provided

Start

25 Sites

ρ = 0.50

a b

SOKOBAN

1 100 104106

100

1000

104

105

106

Time

MSD

Simple

ρ = 0.50

ρ = 0.45

ρ = 0.55

SOKOBAN

Simple

FIG. 2. Sokoban vs. simple random walk. Panel (a): Mean squared

displacement (MSD) of a simple random walk (yellow) and the

Sokoban random walk (purple) as a function of time, for three dif-

ferent obstacle densities: ρ=0.45,0.5,0.55. In the long time limit,

the MSD of the Sokoban is orders of magnitude higher compared to

the MSD of the simple random walk. Panel (b): Trajectories of the

simple (yellow) and Sokoban (purple) random walks, starting in an

identical arena with ρ=0.5. The difference in MSD is evident.

in a supplementary video (SV1.avi).

All densities in Fig. 2 were taken to be above the 2D site-

percolation threshold, namely, the critical density ρc'0.407

[49], above which the simple random walk eventually be-

comes restricted (caged). The fact that for these densities the

Sokoban was able to explore larger portions of the arena, hints

that the critical density for this walk should be higher than, or

equal to, the percolation threshold; allowing the Sokoban to

roam unbounded when the density of obstacles drops below

ρc. However, when simulating the Sokoban for ρ<ρc, we

surprisingly ﬁnd that its MSD still saturates in the long time

limit. An example is given in panels (a-c) of Fig. 3, where

we take ρ=0.4, and present a time evolution of a typical

Sokoban trajectory. Snapshots are taken for t=105,106and

107. For t>

∼107the walk does not visit new sites, indicating

it is indeed conﬁned (see Fig. S1 [50]).

Further evidence that the Sokoban random walk dynami-

cally conﬁnes itself at densities lower than the percolation

threshold comes from extensive numerical simulations that

we perform for this system. Deﬁning the exploration radius

r∞=limt→∞pMSD(t), i.e., the level at which the square root

of the MSD saturates, we plot this quantity as a function of

ρfor the Sokoban and simple random walks (Fig. 3d). As

expected, for the simple random walk r∞diverges when ρap-

proaches ρc'0.407 from above; indicating the existence of a

critical density beyond which the walk is no longer conﬁned.

However, for the Sokoban random walk we ﬁnd that r∞is ﬁ-

nite for all values of ρsampled.

The results presented in panels (a-d) of Fig. 3 assert that the

critical density of the Sokoban random walk cannot be higher

than the percolation threshold. Thus, if such critical density

even exists, it must be lower than the percolation threshold.

Alternatively, it is possible that the Sokoban random walk

does not have a critical density and that this walk dynami-

cally conﬁnes itself at every positive obstacle density ρ>0.

One way to try and ﬁnd out will be to simulate this system for

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LossofPercolationTransitioninthePresenceofSimpleTracer-MediaInteractionsOfekLauberBonomo1;andShlomiReuveni1;1SchoolofChemistry,CenterforthePhysics&ChemistryofLivingSystems,RatnerInstituteforSingleMoleculeChemistry,andtheSacklerCenterforComputationalMolecular&MaterialsScience,TelAvivUniversity,6997...

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Loss of Percolation Transition in the Presence of Simple Tracer-Media Interactions Ofek Lauber Bonomo1and Shlomi Reuveni1 1School of Chemistry Center for the Physics Chemistry of Living Systems Ratner Institute for Single Molecule Chemistry

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