Anticipation decides on lane formation in pedestrian counterflow -- a simulation study
AAnticipation decides on lane formation in pedestrian counterflow –a simulation study
Emilio N.M. Cirillo
E mail: [email protected]
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Universit`a di Roma, via A.Scarpa 16, I–00161, Roma, Italy.
Adrian Muntean
E mail: [email protected]
Department of Mathematics and Computer Science, Karlstad University, Sweden.
Abstract. Human crowds base most of their behavioral decisions upon anticipated states oftheir walking environment. We explore a minimal version of a lattice model to study lanesformation in pedestrian counterflow. Using the concept of horizon depth, our simulationresults suggest that the anticipation effect together with the presence of a small backgroundnoise play an important role in promoting collective behaviors in a counterflow setup. Theseingredients facilitate the formation of seemingly stable lanes and ensure the ergodicity of thesystem.Keywords: Stochastic dynamics, lane formation, anticipation.Appunti: February 18, 2020
1. Introduction
Very much like colloids or bacteria colonies, human crowds can be thought of as many–particle interacting systems. From this perspective, the non–equilibrium statistical mechan-ics becomes the right language to study large crowds coming into play, where highly complexsituations giving rise to interesting collective phenomena, such as free flow, lanes , and grid-lock or jamming (cf. [3, 10, 24, 22], e.g.), may arise.This paper focuses on lane formation in pedestrian counterflow, i.e., a bidirectional pedes-trian movement. The novelty we bring into this context is linked to the ambition to reproduce One of the most efficient transport mechanisms in crowded situations – like those where active colloidalparticles are supposed to cross soft matter solutions – or when pedestrian counterflow in Tokyo’s Shibuyaand Shinjuku railways stations thrives for fluidization – are lanes. An average of 3.5 million people per dayuse the Shinjuku station, making it the busiest station in the world in terms of passenger numbers. Shibuyastation is similarly busy. cm-lane.tex – 18 febbraio 2020 a r X i v : . [ n li n . C G ] F e b he rational behaviour of individuals forming lanes through the means of anticipation, an oldidea that can be traced back at least from Oresme’s time; see [21]. From this perspective,we are in line with some of the statements in [2] and complement existing research on thelane formation topic in pedestrian counterflows. The mechanisms yielding lane formation are still obscure and object of current researchfrom both theoretical and experimental viewpoints. From the crowd management perspec-tives, one strongly believes that controlling in real time the building up and the dissolutionof lanes would be an efficient tool both for organization matters as well as for what concernsthe activity of responsible law enforcement agencies. Regarding lane formation, we refer thereader to [19] for a review from the point of view of self–organization, to [17] for a perspectivefrom the transportation engineering side, as well as to own previous research [11] where weattempted to investigate pedestrian counterflows through heterogeneous domains. Lane for-mation has been observed in many empirical studies (see the discussion in [14, Section 4.1])and it has also been noted that a certain degree of noise favors lane formation, whereas a toolarge noise amplitude lead to a “freezing by heating” effect [15, 16]. It is worth also lookinginto the recent study [18], where lanes are perceived as super–diffusive L´evy walks. Theformation of lanes has been quite well described in the framework of the social force model [13, 14], whereas it has been considered a quite hard phenomenon to be described in theframework of more elementary cellular automata models [7, 8]. Related ideas are reported,for instance, also in [1, 9]. The main drawback of the social force model is that it involvesa large number of parameters. An important breakthrough in this direction can be consid-ered the paper [4] in which the idea of the floor field cellular automaton has been firstlyintroduced. In this model the floor field is constructed dynamically during the evolution ofthe system and allows the coupling between the motion of the particles and a sort of trace left by particles which moved before [19, 23]. The floor field is traditionally made of a static and a dynamic component [4]. More recently a so called anticipation component has alsobeen taken into account [19, 23]. We refer the reader to [12] for an account of anticipationeffects in the context of deterministic dynamical system modeling pedestrian motion. Thestatic floor field is constant in time and not influenced by the presence of other particles, itsimply codes the preferential direction of motion of each particle. The dynamic floor field,inspired by the motion of ants who leave pheromone traces which can be smelled by otherants, evolve with time and codes the trace left by moving pedestrians. The anticipation floorfield allows pedestrian to estimate the route of pedestrians moving in the opposite directionand try to avoid collisions. cm-lane.tex – 18 febbraio 2020 .3. Aim of this research In this framework, we propose an elementary model as well as a different mechanism forlane formation. The main idea behind this mechanism is mildly related to the anticipationfloor field just discussed previously. The aim of our study is to bring evidence that theelementary mechanism yielding lane formation is the pedestrian’s attitude to avoid collisionswith pedestrians moving in the opposite direction, i.e. the anticipation .To keep as simple as possible the modeling level, we use a lattice model approach. Wedefine a discrete time dynamics on a lattice with an exclusion rule, namely, each site can beoccupied by a single particle at time. The formation of lanes at stationarity is studied bymeans of the order parameter proposed in [19]. We demonstrate that, provided the attitudeto avoid collisions is relevant enough, lanes naturally appear in the system. This is in ourview the main mechanism leading to the formation of lanes.The rest of the paper is organized as follows. In Section 2.1 we describe in more detailthe crowd dynamics scenarion we have in mind. The model is presented in Section 2.2.The results of our simulations are discussed in Section 3, whereas our conclusions are finallysummarized in Section 4.
2. The model
In this section, we present the crowd dynamics scenario we have in mind. Here we definethe chosen modeling strategy and briefly explain the main observables that will be closelyfollowed in our simulations. These observables are our main tools to explore the internalcoherent crowd structures which are expected to form in pedestrian counterflows.
Our crowd dynamics setup is as follows: Two different types of pedestrians enter a verticalstrip: those moving upward (“red particles”) and those moving downward (“blue particles”).At each time the pedestrians move mostly forward with respect to their preferential direction,but they will have a small probability r , called the background noise , to do something dif-ferent, namely, stepping laterally or even moving backward. This noise mimics the presenceof irregularities (small obstacles) in the strip, or simply, the pedestrian’s loss of visual focusdue to interactions with the surrounding ambient as it is often promoted in environmentalpsychology reports. The reason for such a background noise is also technical. Indeed, themathematical model turns to be a discrete time Markov Chain. In the case r = 0, the modelwould exhibit many absorbing varieties made of those configurations in which particles areperfectly in–lane. In other words, set of configurations in which columns are occupied either cm-lane.tex – 18 febbraio 2020 y red or blue particles would be absorbing varieties of the state space. Considering r > of the model.Pedestrians do not move simultaneously, but sequentially; namely, the new positionreached by a particular pedestrian has to be taken into account when moving the followingone. For this reason, we refer to our model as lattice model rather than cellular automaton.The distinguishing feature we introduce in this context is the idea of horizon : if onepedestrian spots another one moving in the opposite direction in front of him within an apriori fixed distance (i.e., the horizon depth), then she/he will try to step laterally with theprobability h . The correlation between the lateral motion of a particle and the approachingof an opposing one will yield lane formation as we see in Section 3. The model will beexplored by numerical simulations for different values of the parameters. Having in viewthe application to pedestrian motions, the relevant parameter regime is r (cid:28) h (cid:29) The model is very much inspired by the one proposed by the authors in [5, 6]. Thewalking space is chosen to be the strip
Λ = { , . . . , L } × { , . . . , L } ⊂ Z . Each site or cell in Λ can be either empty or occupied by a single particle (hard core repulsion).Each particle is either red or blue . For red particles, the forward direction is downward,whereas for blue particles the forward direction is upward. We let N r and N b be the numberof red and blue particles at the initial time t = 0, respectively. We let N := N r + N b be totalnumber of particles at the initial time and n ( t ) be the total number of particles at time t .Particles are labelled. At each step of the dynamics, we choose sequentially at randomwith uniform probability one of the N particles. If the particle lies in the lattice, then wedisplace it with the probabilities specified below: if the cell where the particle should bemoved to is occupied, then the particle is not moved (hard core repulsion acts accordinglyto the simple exclusion rule). Time is increased by one after N particles have been selectedand possibly moved.We let r ∈ [0 ,
1] be the background noise and h ∈ [0 ,
1] be the lateral move probability .Moreover, given a particle, its horizon is the vertical slab made of the first H ≥ H will be called horizon depth .Either a red particle moves to one of the four neighboring cells with probabilities 1 − r/ r/ r/ r/ H = 0, or the horizon is empty, or the closestparticle in the horizon is red. Otherwise, it moves to one of the four neighboring cells with Ergodicity is lost for instance when an horizontal line with blue particles opposing red particles is formed. cm-lane.tex – 18 febbraio 2020 (cid:45) L (cid:63)(cid:54) L (cid:54)(cid:63)(cid:45)(cid:27) (cid:115)(cid:115) − r/ r/ r/ r/ (cid:54)(cid:63)(cid:45)(cid:27) (cid:115)(cid:115) − h h/ h/ (cid:54)(cid:63)(cid:45)(cid:27) (cid:115)(cid:115) − r/ r/ r/ r/ (cid:54)(cid:63)(cid:45)(cid:27) (cid:115)(cid:115) − h h/ h/ Figure 1: Schematic representation of the model for horizon H = 3. Arrows denote possiblemoves and the related probabilities are reported in the cell.probabilities 1 − h (down), h/ h/ − r/ r/ r/ r/ H = 0, or the horizon is empty,or the closest particle in the horizon is blue. Otherwise, the particle moves to one of the fourneighboring cells with probabilities 1 − h (up), h/ h/ anticipation mechanism . The parameter r is calledbackground noise. The model aims to describe two families of pedestrians one heading downand the other heading up. This is precisely what happens in our model with the red and blueparticles if r = 0. However, when r is positive, different moves are allowed in the system asit happens to real pedestrian crowds in motion – sometimes they displace not following theirprescribed best trajectory, but with random shifts due to external noise, such as sounds,light flashes, images, or obstacles. Note that if r is small, red and blue particles experiencean important downward and, respectively, an upward drift, which becomes smaller when r cm-lane.tex – 18 febbraio 2020 ncreases and finally disappears at r = 1, when the walk becomes perfectly symmetric.We introduced the parameter r not only to mimic random real world shifts, but also fora technical reason. Indeed, in the case H = 0, namely, when the anticipation effect is notconsidered, for r = 0 our model would be completely trivial, indeed, red and blue particleswould move one against the other and eventually would stop each others. A residual trivialmotion will be present only in those columns populated by particles moving all in the samedirection.In the following, we study the model for a wide choice of the parameters; the readershould always keep in mind that the values relevant for pedestrian flow scenarios are r (cid:28) h (cid:29)
0. Indeed, a walker will change his direction of motion only in the presence of anopposing pedestrian (or other obstacle) and in such a case he will do it almost surely.The vertical boundaries are considered as occupied sites, that is to say, reflecting bound-ary conditions are imposed on those vertical boundaries of the strip Λ. On the other hand,the horizontal boundaries are considered filled with empty spots, so that a particle on thefirst row trying to jump up will exit the system and, similarly, a particle on the L –th rowtrying to jump down will exit. Particles which did exit the lattice, when selected for a move,will re–enter the strip with the same horizontal coordinate at row one if own color red, andat row L if the particle is blue, provided the target site is empty. We have not consideredstrictly imposed vertical periodic boundary conditions – the upper and the lower rows of thestrip mimick the presence of doors at the end of the corridor (see, also, Appendix A). Quantitative investigations of the model will be performed by means of the followingobservables. We call upward current at time t , the ratio between the total number of blueparticles which exited the system through the top boundary and time. Similarly, we call downward current at time t , the ratio between the total number of red particles which exitedthe system through the bottom boundary and time. Note that both these currents are definedas positive numbers. The currents will be used to detect the presence of jamming in thesystem. More precisely, since the upward and the downward current will be approximativelyequal in all the simulations, we will focus on the average current , namely, the average betweenthe upward and the downward currents.Additionally, to give a quantitative estimate of the presence of lanes in the system, wedefine a suitable order parameter following closely the ideas proposed in [19] and based ondevelopments from [20] done in the framework of colloidal systems.Fix the time t and consider a particle labelled by k ∈ { , . . . , N } such that it lies in thelattice at time t . Let n r ,k ( t ) the total number of red particles occupying cells belonging tothe same column as the particle k . Furthermore, let n b ,k ( t ) the total number of blue particles cm-lane.tex – 18 febbraio 2020 ccupying cells belonging to the same column as the particle k . Then set φ ( t ) = 1 n ( t ) n ( t ) (cid:88) k =1 (cid:104) n r ,k ( t ) − n b ,k ( t ) n r ,k ( t ) + n b ,k ( t ) (cid:105) . (2.1)Note that in a state in which blue and red particles moved perfectly in separate lanes,the order parameter would be equal to one. For disordered states, we expect φ ( t ) to besmall, though strictly positive. In the next sections, we shall use the expression orderedconfigurations when referring to configurations in which red and blue particle occupy differentcolumns.
3. Numerical simulations
We simulate the model introduced in Section 2.2 posed on the strip with side lengths L = 50 and L = 100. We fix a parameter ρ , called density , and the total number ofparticles will then be N = ρL L . The numbers of red and blue particles, N r and N b , willdiffer at most by one and will be such that N r + N b = N . The values of H , h , r , and ρ willbe specified both in the forthcoming discussion of the numerical results and in the captionof the figures.All simulations are run for 8 × time steps: remember that at each time step N particlesare randomly selected for motion. The order parameter is computed by averaging its valueeach 10 time steps starting from the thermalization time 10 . The currents are computedby applying the definition (given in Section 2.3) at the end of the simulation. The computedobservables are very stable and the statistical errors are not significative, hence they are notreported in the pictures.Aiming to a good vizualization of the effects, the results will be presented by means of twodifferent kind of graphs: configuration pictures and scatter plots. In configuration pictures,each point represents the position of a particle: red points stand for red particles and bluepoints stand for blue particles. In scatter plots, either the current or the order parameterare reported for approximatively 20 ×
20 different values of the considered parameters evenlyspaced in the intervals specified in the graphs. No data interpolation is performed, eachmeasured value corresponds to a square pixel in the picture. The colors shown in scatterplots are adapted to picture data, but in all the pictures blue corresponds to the half ofthe maximum value in the plot. Moreover, for the values below such half value we use graytones, whereas for the values above it, we use the following brilliant colors: magenta, red,orange, yellow, and green. cm-lane.tex – 18 febbraio 2020 s we already pointed out in Section 2.3, the order parameter φ is a positive numberclose to one for ordered configurations. On the other hand, it is not clear how small such aparameter will be for disordered configurations. Particularly, we cannot infer that φ will beclose to zero. Indeed, cf. (2.1), φ is defined as a sum of positive numbers, so that fluctuationsin random configurations will add up and not cancel. Mainly for this reason, we perform afirst study of the system for a wide choice of the parameter r . Obviously, we expect thatfor an r not small, the system will be essentially disordered. In this way, we will give aquantitative measure of the values that the order parameter φ should exhibit for disorderedstates.In Figure 2, we plot the average current in a scatter plot as a function of the backgroundnoise r and the density ρ for different choices of the horizon depth H and of the lateral moveprobability h . As expected, the highest values of the average current are found for low valuesof the background noise. Indeed, as already noted, when r is small, particles experience animportant forward drift. Nevertheless, seeing the left diagram in the picture, we realize thatwhen H = 0 for some intermediate value of the density (focus, for instance, to the case ρ = 0 . r small, becomes important if r is mildlyincreased, but it eventually becomes zero for even larger values of the background noise.This effect is due to the fact that, for small values of the background noise, the dynamicsis trapped in blocked (clogged) configurations, whereas a larger value of randomness in thedynamics helps particles to avoid blocking opponents restoring the global current to not zerovalues. d e n s i t y d e n s i t y d e n s i t y Figure 2: Scatter plot of the average current in the plane r – ρ for r ∈ [0 . , ρ ∈ [0 . , . H = 0 (left), H = 5 and h = 0 . H = 5 and h = 0 . r , in absence of the anticipation cm-lane.tex – 18 febbraio 2020 igure 3: From the left to the right it is depicted the final configuration of the simulationsfor the cases H = 0, r = 0 .
1, and ρ = 0 .
15 (first graph), H = 0, r = 0 .
5, and ρ = 0 . H = 0, r = 0 .
7, and ρ = 0 .
15 (third graph), H = 5, h = 0 . r = 0 .
5, and ρ = 0 .
15 (fourth graph), H = 5, h = 0 . r = 0 .
5, and ρ = 0 .
15 (fifth graph), and H = 5, h = 0 . r = 0 .
8, and ρ = 0 .
45 (sixth graph).mechanism, allows to avoid blocking configurations. In this respect, randomness favorstransport. It is interesting to remark that a similar phenomenon was found by the authorsin [5, Figures 6.14 and 6.15], where it was remarked that the so called residence time it is anot monotonic function of the lateral displacement probability. We recall that the residencetime was defined in loc. cit. as the typical time that a particle started at one side of thestrip needs to cross the whole strip and exit from the opposite boundary. Consequently, theresidence time and the current are closely related quantities.Another interesting phenomenon can be observed comparing the second, the fourth andthe fifth panels in Figure 3. In these three cases, the values of background noise and densityare not changed, but the anticipation effect is introduced and the lateral move probability ischanged. The pictures show that adding the anticipation mechanism with a sufficiently largelateral move probability blocking configurations can be avoided. The fact that anticipationhelps transport in a wide region of the parameter space is also evident from the currentgraphs in Figure 2, but the configurations reported in Figure 3 provide a striking evidence.The sixth configuration in Figure 3 shows that for very large values of the density, even alarge lateral move probability is not sufficient to the restore current, and consequently, thedynamics is eventually trapped in a blocked configuration.In Figure 4, we finally come to the main target of this section, namely, the graph of theorder parameter in the plane r – ρ . The scatter plot is reported for the same cases consideredin Figure 2. In all the cases, we do not expect lane formation, due to the rather high valuesof the background noise considered in the pictures. Indeed, the graphs in Figure 4 show cm-lane.tex – 18 febbraio 2020 d e n s i t y d e n s i t y d e n s i t y Figure 4: Scatter plot of the order parameter φ in the plane r – ρ for the same case consideredin Figure 2.that the order parameter is approximatively constant in the whole region considered in thesimulations. Moreover, the stationary value of the order parameter ranges between 0 . .
3. Hence, in the sequel of our discussion we will consider such a value as the referencepoint of the order parameter for completely disordered configurations. The small islands inthe central and right panel corresponding to higher values of the order parameter can beneglected since they are observed in correspondence of blocked configurations. The relativehigh value of φ is just a random value depending on the random initial condition, for instancein the case illustrated in the sixth panel in Figure 3. This is due to the fact that, in thefinal configuration, many red particles remained blocked outside the lattice, hence in eachcolumn a majority of red particles is present. This yields in a rather high value of the orderparameter. We now focus on the most interesting part of the parameter space, namely, the regionwith small background noise.In Figure 5, we have reported the results of our simulations at r = 0, namely, when nobackground noise is present so that the sole mechanism present in the dynamics is antici-pation: particles move in their prescribed forward direction unless an opposing particle isspotted inside the horizon region. In the picture, we show scatter plots in the plane h – ρ forboth the average current and the order parameter φ .The left panel gives evidence that at any value of the lateral move probability, the averagecurrent increases with the density if this is sufficiently small, i.e., smaller that about 0 .
3. Onthe other hand, when such a value is reached, the dynamics freezes in blocked configurations,and hence, the currents suddenly drop to zero. It is quite remarkable that the current, whendifferent from zero, does not depend very much on the lateral move probability h . On the cm-lane.tex – 18 febbraio 2020 d e n s i t y d e n s i t y d e n s i t y d e n s i t y Figure 5: Scatter plot of the current (first and third panel) and order parameter φ (secondand fourth panel) in the plane h – ρ for zero background noise, h ∈ [0 . , . ρ ∈ [0 . , . H = 5 (first two panels), and H = 20 (third and fourth panel).other hand, as we have already noted in the above Section 3.1, for intermediate values ofthe density the anticipation mechanism helps transport, in the sense that the freezing of thedynamics occurs at larger value of the density if h is large.To emphasize this point aspect in a better way, we have plotted in Figure 6 the finalsimulation configuration of the system at density ρ = 0 . h . The pictures shows that if h is small the dynamics is eventually trappedin a blocked configuration whereas as h is increased no freezing is observed, at least on thetime scale we considered, and the current results to be different from zero.We finally remark that, for r = 0, if the dynamics is not trapped then the order isperfect, namely, φ = 1 is reached. In other words, in the absence of the background noise,the anticipation mechanism guarantees a perfect lane formation, provided the dynamics isnot frozen in blocked configurations . In our opinion, this is a very valuable result, sinceit states that lanes forming in counterfows can be explained just as a consequence of theanticipation mechanism.Data referring to the case H = 20 and reported in Figures 5 and 6 can be discussedsimilarly; the only difference is that the effect of the anticipation mechanism is slightlystronger. This fact can be observed both in Figure 5 and 6.We test if the anticipation mechanism is robust with respect to the background noise,that is to say, if its ability to form lanes is still valid for r different from zero.In Figures 7 and 8 we report the scatter plot of the average current and the orderparameter on the r – ρ plane for different values of the lateral move probability. In particular,the left panel refers to the case in which the anticipation mechanism is not present in thedynamics.We pinpoint here a behavior which is very similar to the one discussed in the case r = 0.In particular, we notice that, for a fixed value of the density ρ , increasing the lateral move cm-lane.tex – 18 febbraio 2020 igure 6: From the left to the right it is depicted the final configuration of the simulationsfor the cases r = 0, ρ = 0 . H = 5 and h = 0 .
05 (first graph), H = 5, and h = 0 . H = 5, and h = 0 .
50 (third graph), H = 20 and h = 0 .
05 (fourth graph), H = 20 and h = 0 .
45 (fifth graph), and H = 20 and h = 0 .
50 (sixth graph).
0 0.0045 0.009background noise 0.1 0.2 0.3 0.4 0.5 d e n s i t y d e n s i t y d e n s i t y Figure 7: Scatter plot of the average current in the plane r – ρ for r ∈ [0 , . ρ ∈ [0 . , . H = 0 (left), H = 5 and h = 0 .
11 (center), H = 5 and h = 0 .
71 (right).probability avoids the freezing of the dynamics at larger values of the density ρ . Note also,that the current does not depend very much on the value of the disorder parameter in theconsidered range. This is quite obvious since in these pictures we are focussing on a verytiny slice of the part of the graphs shown in Figures 2 and 4, very close to the vertical axis.Finally, we remark that Figure 8 shows that the anticipation mechanism is able to explainlane formation also in the presence of a weak background noise. Such an order is eventuallydestroyed if the parameter r becomes too large as underligned by the data reported inFigure 4.
4. Discussion cm-lane.tex – 18 febbraio 2020
0 0.0045 0.009background noise 0.1 0.2 0.3 0.4 0.5 d e n s i t y d e n s i t y d e n s i t y Figure 8: Scatter plot of the order parameter φ in the plane r – ρ for the same case consideredin Figure 7.As closing note, in the same line of thinking as in Ref. [24], we argue that the keytowards an even deeper understanding of a collective behavior like lane formation lies inidentifying the principles of the behavioral algorithms followed by each individual, and also,in answering the question: How does information flow among the pedestrians? Addressingthis question requires the embedding in our model of fine environmental psychology infor-mation as well as aspects of the psychology of groups. We have not touched these aspectsat all in this contribution. This can be seen as further work. On the hand, for the presentedbi–directional pedestrian flow scenario, given two population sizes walking within the strip Λ,we are convinced that the simple combination of just 3 parameters is sufficient to predict theformation of lanes. These parameters are the horizon depth H , the lateral move probability h , and the background noise r . This level of complexity is much lower than what usuallythe social force model is offering. Furthermore, our three parameters have a clear physicalmeaning. The harder to identify is eventually the background noise level r , which on topof everything is also prone to different modeling interpretations and incorporates very muchthe specifics of the local conditions (geometry of the building, local traffic, etc.).Two main results stand out:(A) In the absence of the background noise, the anticipation mechanism guarantees a per-fect lane formation, provided the dynamics is not frozen in blocked configurations;(B) The current does not depend very much on the lateral move probability h .An interesting question is to which extent (A) and (B) hold if pedestrians would bemoving at different speeds? This question, connecting pedestrian flow to traffic flow matters,could be addressed rather naturally using a continuous time version of the present model cm-lane.tex – 18 febbraio 2020 n which pedestrian moving at different speeds would be modelled by particles moving withdifferent rates. Acknowledgements
We thank Prof. Rutger van Santen (Eindhoven, NL) for very fruitful discussions on closelyrelated matters. ENMC thanks the ´ENS de Paris for the very kind hospitality in the periodin which part of this work has been done.
A. Strict periodic boundary conditions
As we have already mentioned in the above discussion, we considered ”not strictly im-posed vertical periodic boundary conditions”. Our choice is motivated by the fact that theupper and the lower boundaries are thought as two open doors for the pedestrian motion.In this appendix, we show some minimal results obtained in the case when vertical periodicboudary conditions are strictly imposed as it is usually the case in the statistical mechanicsof lattice models. By “strictly imposed” we simply mean that the updating rule is covariantin the lattice and the rows 0 and L + 1 are respectively identified with the rows L and 1.We have repeated the simulations shown in the first two panels of Figure 5 and in thethird panel of Figures 7 and 8. Our results, now plotted in Figure 9, do not show newfeatures with respect to what has been discussed above. d e n s i t y d e n s i t y
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