Alexandre Nicolas
email: alexandre.nicolas@polytechnique.edu

Marcelo Kuperman

Santiago Iba Ñez

Sebasti Án Bouzat

Mechanical response of dense pedestrian crowds to the crossing of intruders

The increasing number of mass events involving large crowds calls for a better understanding of the dynamics of dense crowds. Inquiring into the possibility of a mechanical description of these dynamics, we experimentally study the crossing of dense static crowds by a cylindrical intruder, a mechanical test which is classical for granular matter. The analysis of our experiments reveals robust features in the crowds' response, comprising both similarities and discrepancies with the response of granular media. Common features include the presence of a depleted region behind the intruder and the short-range character of the perturbation. On the other hand, unlike grains, pedestrians anticipate the intruder's passage by moving much before contact and their displacements are mostly lateral, hence not aligned with the forces exerted by the intruder. Similar conclusions are reached when the intruder is not a cylinder, but a single crossing pedestrian. Thus, our work shows that pedestrian interactions even at high densities (3 to 6 ped/m 2 ) do not reduce to mechanical ones. More generally, the avoidance strategies evidenced by our findings question the incautious use of force models for dense crowds.

Introduction

Magritte's surrealistic painting Golconda depicts a suburban landscape with scores of passers-by stiffly standing in mid-air, as if they were 'raining' over the city. On close inspection the individuals are dissimilar, but overall they look nearly identical, as though being part of a large group rendered their differences as faint as those between rain droplets. This analogy between pedestrians and beads would considerably facilitate the prediction of pedestrian crowd flows, if one were to take it at face value.

Predictions for pedestrian flows are of great avail for the architectural design of large facilities and for the management of throngs, whether it be in underground stations [START_REF] Hoogendoorn | Applying microscopic pedestrian flow simulation to railway station design evaluation in lisbon, portugal[END_REF] , at mass religious events 2 or sports gatherings [START_REF] Bain | Dynamic response and hydrodynamics of queuing crowds[END_REF] , or in front of a shop just before its opening on Black Fridays in the United States. The fear of stampedes in such assemblies has turned much of the research effort on dense crowds towards extreme situations, such as past crowd disasters [START_REF] Helbing | Dynamics of crowd disasters: An empirical study[END_REF][START_REF] Krausz | Automatic video analysis of a crowd disaster[END_REF] or scenarios of evacuation through a door [START_REF] Hoogendoorn | Pedestrian behavior at bottlenecks[END_REF][START_REF] Faure | Crowd motion from the granular standpoint[END_REF][START_REF] Garcimartín | Flow of pedestrians through narrow doors with different competitiveness[END_REF][START_REF] Nicolas | Pedestrian flows through a narrow doorway: Effect of individual behaviours on the global flow and microscopic dynamics[END_REF] , but at high densities complex collective phenomena are also expected in ordinary situations and require proper modelling. Nowadays, pedestrian simulation software is routinely entrusted with the task of predicting the dynamics of crowds. Nevertheless, their underlying models still lack extensive validation, especially at very high densities. Admittedly, the outputs of simulations made with commercial software have been compared to some high-density real-life situations [START_REF] Berrou | Calibration and validation of the legion simulation model using empirical data[END_REF][START_REF] Dridi | Pedestrian flow simulation validation and verification techniques[END_REF][START_REF] Isenhour | Verification of a pedestrian simulation tool using the NIST recommended test cases[END_REF] , but such comparisons are generally restricted to macroscopic observables -for example evacuation times -and visual impressions. In this vein, it is symptomatic that Fruin's Levels of Service [START_REF] Fruin | Pedestrian planning and design[END_REF] , widely used in safety and design handbooks, amalgamate all densities above 2.15 ped/m 2 into Level F, without further distinction. Accordingly, dense pedestrian crowds in generic settings deserve more attention than they have received so far [START_REF] Haghani | Crowd behaviour and motion: Empirical methods[END_REF] .

As a matter of fact, beyond Magritte's painting, analogies between the dynamics of these crowds and the mechanics of natural systems, in particular granular media, have long been hinted at [START_REF] Helbing | Similarities between granular and traffic flow[END_REF] , and have now been partly confirmed [START_REF] Helbing | Dynamics of crowd disasters: An empirical study[END_REF][START_REF] Ma | New insights into turbulent pedestrian movement pattern in crowd-quakes[END_REF][START_REF] Zuriguel | Clogging transition of many-particle systems flowing through bottlenecks[END_REF][START_REF] Pastor | Experimental proof of faster-is-slower in systems of frictional particles flowing through constrictions[END_REF] . The comparisons concern the passage of a bottleneck [START_REF] Zuriguel | Clogging transition of many-particle systems flowing through bottlenecks[END_REF][START_REF] Pastor | Experimental proof of faster-is-slower in systems of frictional particles flowing through constrictions[END_REF] and the onset of turbulence [START_REF] Helbing | Dynamics of crowd disasters: An empirical study[END_REF][START_REF] Ma | New insights into turbulent pedestrian movement pattern in crowd-quakes[END_REF] . Inspiration could thus be drawn from these scientifically riper fields, where progress was made by identifying the elementary microscopic processes by which the material deforms (motion of dislocations in crystals, shear transformations in amorphous solids) as well as the response halo generated by these elementary perturbations. Along these lines, since crowds are often traversed by intruders (pedestrians, bicycles, vehicles, etc.), it would be valuable to know the crowd's response to such crossings. It so happens that intruding a cylinder into a medium is actually a classical mechanical test. In particular, the response to such a perturbation strongly discriminates between granular media and viscous fluids, with an exponential radial decay of the perturbation in the former case [START_REF] Seguin | Experimental velocity fields and forces for a cylinder penetrating into a granular medium[END_REF][START_REF] Kolb | Flow fields around an intruder immersed in a 2d dense granular layer[END_REF] and a very long-ranged impact in the latter case [START_REF] Kaplun | Low reynolds number flow past a circular cylinder[END_REF] .

In this paper, we inspect the response of a dense static crowd induced by the crossing of an intruder. One may think that steric constraints will bring the crowd's response close to that of a granular medium, but an exciting question is how the pedestrians' decision-making processes will couple with the strong mechanical constraints. More generally, we aim to (i) collect controlled experimental data on dense crowds that can be used to test and calibrate models, (ii) explore the similarities of the crowd's response to the intrusion of a cylinder with that of granular media, and (iii) discuss the case in which the intruder is a single pedestrian, a situation often encountered in daily-life. In so doing, we lay the first stone for the empirical development of the (continuum) mechanics of dense crowds.

Presentation of the experiments

In order to study the response of a static crowd to the passage of an intruder, two series of experiments were performed in gymnasiums in Orsay (France) and Bariloche (Argentina) in June 2017 and September 2017, respectively. They involved between 35 and 40 voluntary participants (men and women), aged 20-60, mostly students.

During the experiments, the participants stood in a delimited area, which allowed us to control the global density ρ (between 2 and 6 ped/m 2 ) This static crowd was successively crossed from end to end [along the y-axis, see Fig. 1(a)], or (ii) a randomly chosen participant, asked to make his/her way to the other side. Meanwhile, people in the standing crowd either faced the incoming intruder (France and Argentina), or were prescribed random orientations (France), or turned their backs to the crowd (Argentina). Furthermore, we gave distinct instructions in France and in Argentina: In France, the participants were asked to behave casually, as if they were standing on an underground platform , whereas in Argentina they were to make way for the intruder only upon coming in contact with it.

The experiments were recorded from above, at an altitude of about 4 meters, with high-resolution 60 Hz action cameras (Tomtom Bandit in France, GoPro Hero 3 in Argentina). After correction of the optical distortion of the cameras, home-made software allowed us to detect and track the coloured hats worn by the participants, as shown in Fig. 1. The extracted data were then post-processed manually. As detailed in Ref. [START_REF] Appert-Rolland | Experimental study of collective pedestrian dynamics[END_REF] , we found an absolute experimental uncertainty of the order of 10 to 15 cm in most cases. The uncertainty cannot be reduced much below, unless one corrects for the different participants' heights or one positions the camera extremely high [START_REF] Boltes | Influences of extraction techniques on the quality of measured quantities of pedestrian characteristics[END_REF] . Note, however, that the error concerning the displacement of a given participant will be much lower than that on absolute positions. More extensive manual post-processing was required for the experiments conducted in Argentina, because red headscarves with a white mark were used there instead of hats of various colours.

Results

Crowd's response to a standardized perturbation

This section addresses the following questions: What are the main features of the crowd's response to its crossing by a cylindrical intruder? How similar is it to what has been reported for two-dimensional granular media [START_REF] Kolb | Flow fields around an intruder immersed in a 2d dense granular layer[END_REF][START_REF] Seguin | Dense granular flow around a penetrating object: Experiment and hydrodynamic model[END_REF] ?

Density field around the intruder

First, we consider the density fields around the intruder, averaged over time and over similar realisations in the intruder's frame and plotted in Fig. 2 for crowds facing the intruder. The local densities were computed with a novel method (detailed in the Methods section), which is based on a Voronoi tessellation of the plane, but suitably modified to account for edge effects.

In all situations, a depleted region behind the cylinder clearly stands out in the density maps. It reflects the empty space created in the intruder's wake, which is then filled by pedestrians in a matter of seconds, similarly to the cavity formed downstream from the intruder in granular media [START_REF] Kolb | Flow fields around an intruder immersed in a 2d dense granular layer[END_REF][START_REF] Kolb | Rigid intruder inside a two-dimensional dense granular flow: Drag force and cavity formation[END_REF] . However, it is remarkable that, in contrast with granular matter (below jamming), compaction is observed mostly along the transverse direction: high-density 'wings' adorn both sides of the cylinder. We emphasise that these features are witnessed at all densities, with participants facing the intruder [Fig. Upstream from the cylinder, instead of the compression reported in granular matter below jamming [START_REF] Kolb | Flow fields around an intruder immersed in a 2d dense granular layer[END_REF] , we observe a redistribution of the density, with higher values on the sides, and either lower or average density along the central axis [Figs. 3]. This can be ascribed to the pedestrians' dodging out of the intruder's trajectory with anticipation, as one clearly sees in the videos. Consistently with this idea, the effect is more marked when pedestrians face the intruder than for random orientation, which also transpires from the density profiles along the longitudinal axis x = 0, presented in Fig. When pedestrians are told not to anticipate (in Argentina), the density field becomes more reminiscent of (but probably not as marked as) the granular case, with some compaction upstream. It should be noted that in spite of the instructions, some pedestrians facing the intruder [Fig.

Velocity fields around the intruder

To shed light on the emergence of the density inhomogeneities, we take a closer look at the pedestrians' displacements. To this end, we measure the velocity fields v(x, y) = δ u(x, y)/δt around the intruder. Here δ u(x, y) is the displacement between t and t + δt of a pedestrian standing at position (x, y) (relative to the intruder) at time t. Continuous fields are obtained via the smoothing procedure described in Methods.

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Let us first consider a crowd facing the intruder (France) 4(a) presents the mean velocity field measured for ρ ≈ 3.5 ped/m 2 . Interestingly, the velocities are almost exclusively directed transversally (along x), i.e., perpendicularly to the trajectory of the intruder: Ahead of it, the pedestrians step aside (towards larger |x|) to avoid the anticipated contacts, while behind it they reoccupy the depleted region (small |x|), again via lateral moves.

Very similar flow patterns are observed when the density is varied, although the displacement amplitude decreases with the density, and seems to extend farther transversally in the densest crowd [Supplemental Figure S1(a,b,c)]. The observed overall similarity is noteworthy, because it spans across the transition from a regime dominated by 'social' interactions at low density to a tightly packed one with prominent physical contacts.

When participants are randomly oriented [Supplemental Figure S1(d,e,f)], the mean directions of motion remain identical, i.e. transverse outwards (transverse inwards) in front of (behind) the cylinder. However, the perturbed region ahead of it shrinks significantly, reflecting a lesser degree of anticipation, as the participants do not all face the intruder. This shrinkage is even more marked when the participants were asked not to anticipate the contact [Fig. 4(c)]. In this case, the velocity field, still localised in the direct vicinity of the cylinder, is no longer transverse, but mostly radial (at least, in the half-plane y > 0). Besides, behind the cylinder, the velocities seem to acquire a small backwards longitudinal component, as though the participants, now that they can see the cylinder, wanted to stand away from it. In fact, this flow pattern is strongly reminiscent of that observed around an intruder in a granular medium, shown in Supplementary Fig. S2; the displacements are then probably due

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LastPage LastPage to mechanical pushes by the intruder, without any preliminary avoidance manoeuvres. The wake of the intruder (y < 0) is still reoccupied via lateral moves, but the latter are less unidirectional than in the other situations.

Spatial extent of the perturbation along the transverse direction

We have noticed that the regions of highest density extend on both lateral sides of the intruder, while the velocity perturbation seems to decay very quickly along the transverse direction x. To quantify how much each pedestrian is affected by the intruder's passage on the whole, we now measure the amplitude ∆u ( j) of the total displacement of pedestrian j defined by ∆u

( j) x = max t∈[t i ,t f ] x ( j) (t) -min t∈[t i ,t f ]
x ( j) (t) and ∆u

( j) y = max t∈[t i ,t f ] y ( j) (t) -min t∈[t i ,t f ] y ( j) (t),
(1) eq:delta where times t i and t f mark the start and the end of the intruder's passage, respectively. 5, after smoothing (see Methods), we plot the variation of ∆u with respect to the transverse distance between the initial pedestrian's position (x(t i ), y(t i )) and the intruder's trajectory x cyl (t), y cyl (t) , i.e., x(t i )x cyl (t c ) where t c is such that y cyl (t c ) = y(t i ).

Consistently with our foregoing observations, we notice that the longitudinal displacement amplitude ∆u y is much smaller than the transverse one ∆u x , especially in not too dense crowds. As expected, the peak amplitude ∆u x slightly exceeds the cylinder's radius. Besides, the amplitude ∆u x exhibits a fast decay in the transverse direction x and drops to half of the peak value over a typical length scale x c D, i.e., typically one radius beyond the outer boundary of the cylinder (note that experimentally the crowd was larger than 5 x c in width). This length scale does not vary much with the density and the experimental conditions, insofar as x c always ranges between 0.8 D and 1.5 D (perhaps excepting the experiments in which participants turned their backs to the intruder, for which x c could not be determined precisely). figure6 So far we have unveiled the main features of the crowd's mechanical response to a standardized perturbation, created by an intruder significantly larger than a pedestrian. It is not obvious a priori whether our findings apply to the case where the intruder itself is a single pedestrian. Therefore, we performed a second set of experiments, in which single pedestrians crossed the static crowd, one by one.

Perturbation induced by the crossing of a single pedestrian

To begin with, let us underscore the disparity between these new perturbations and those investigated above: Unlike the cylinder, crossing pedestrians are not of circular shape and have a much smaller cross section. In addition, while traversing they rotate their chest so as to 'squeeze' through inter-pedestrian gaps in the crowd, which enhances the difference in shape and size. Finally, instead of walking straight, they followed winding trajectories, presumably favouring regions of lower density and

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LastPage LastPage larger spacings between pedestrians (some examples of trajectories are shown in Supplemental Fig. S3). This is consistent with the observation made in numerical simulations of a similar setting [START_REF] Metivet | How to push one's way through a dense crowd[END_REF] that making use of the empty spaces at the cost of a longer trajectory is favorable when the surrounding crowd is very dense.

These differences between the cylindrical and pedestrian intruders naturally affect the crowd's response. Because of their smaller cross sections, the latter hardly alter the density of fairly sparse crowds. For much denser crowds, however, some pattern emerges in the perturbed density field, shown in Fig. 6(a), with a clear dip in density in the wake of the intruder. On the other hand, we find no evidence of the 'wing'-like structure observed with the cylinder, but considerably more data would be needed to make a definite statement in this respect. Technically, it should nonetheless be mentioned that the winding intruder's trajectory may add a 'motion blur' along the x-coordinate to the density maps.

The mean velocity fields plotted in Fig. 6(c-d) give sharper insight into the induced perturbation. Remarkably, albeit noisier, the flow patterns bear vivid resemblance with those induced by the cylinder, even at fairly low density. Indeed, anticipated lateral moves are seen ahead of the intruder, while transverse inwards displacements fill the depleted region in its wake. Furthermore, the displacements are virtually all in the transverse direction. Recall that these are on no account trivial features, since they are not present in granular media.

The displacement amplitudes ∆u [an example of which is plotted in Fig. 6(b)] reflect both the similarity in the crowd's response and the difference in the perturbation: For sparse crowds (data not shown), the pedestrian-induced perturbation is too weak and no clear profile emerges, but at higher density the profile of ∆u(x) resembles that observed with the cylindrical intruder, with a fast transverse decay. Two subtle differences may nevertheless be pointed out. Firstly, the peak amplitude is only slightly above 20 cm, consistently with the pedestrian's smaller 'cross section'. Secondly, the longitudinal displacement amplitude ∆u y is not always significantly smaller than ∆u x , possibly because of the winding trajectory of the crossing pedestrian.

All in all, the responses to the cylinder's and pedestrian's crossings share several common features. This qualitative robustness is encouraging for a prospective development of continuum crowd mechanics, based on empirical data rather than postulated rules.

Discussion

Comparison with the response of granular media Let us now discuss whether our observations on pedestrian crowds crossed by a cylindrical intruder match previous findings for granular media. On the one hand, the gradually refilled depletion zone in the intruder's wake mirrors the cavity formed behind the intruder in granular media. Besides, we observed perturbations that decay quite fast (in the transverse direction), with little sensitivity to the crowd's density. Similarly, in granular media, the perturbation decays exponentially in space [START_REF] Seguin | Experimental velocity fields and forces for a cylinder penetrating into a granular medium[END_REF] . Recall that this stands in stark contrast with the very long range effects encountered in incompressible viscous fluids [START_REF] Kaplun | Low reynolds number flow past a circular cylinder[END_REF] and also in the linear response of compressible elastic media to a point force [START_REF] Karimi | Elasticity of frictionless particles near jamming[END_REF] . This similarity points to the importance of the granularity of the crowd1 .

On the other hand, some striking differences were also found between the crowd's response and that of granular media. Ahead of the intruder, the pedestrians' avoidance strategies lead to anticipated lateral moves prior to contact. As a result, the compacted region does not take the shape of a dome as in granular matter below jamming, but rather forms a wing-like structure on both sides of the intruder. Still, the most striking difference between pedestrians and grains concerns the displacement field. In the case of pedestrians, the displacements are almost exclusively directed laterally (outwards or inwards), at odds with the loop-like ('dipolar') pattern with recirculation eddies seen in grains. It is remarkable that this feature survives when the intruder is a single moving pedestrian.

Two specificities of pedestrians contribute to the realization of such lateral displacements. The first one is the ability to anticipate, which is reflected by the fact that pedestrians start to move much before contact. This ability also allows static pedestrians to forecast that they only need to move laterally and let the intruder pass. Incidentally, anticipation is so anchored in the pedestrians' behaviour that, even when they are told not to anticipate, they still have some tendency to do so (provided they see the intruder coming). The second specificity of pedestrians is self propulsion; they can move in a direction which is not aligned with the force applied on them, unlike grains.

Only when anticipation was precluded by the given instructions and the crowd's orientation (with an intruder coming from behind) did the pedestrian displacement field turn more similar to that of grains, whose motion results from contact forces. A sensible hypothesis is that, in the absence of anticipation, the pedestrians are simply shoved by contact forces which they cannot withstand, while that they lack information to determine what moves would be optimal. where x j denotes the transverse (x) distance between the initial pedestrian's position and the intruder's trajectory, i.e., x(t i )x cyl (t c ) where t c is such that y cyl (t c ) = y(t i ).

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  fig:figure1 fig:figure1 1(b)] by (i) an intruder of cylindrical shape (diameter: 74 cm in France, 68 cm in Argentina) moved along a more or less straight line x = cst by a staff member located inside it [see Fig.

  fig:figure1 fig:figure1

  fig:figure1 fig:figure1

Figure 1 .

 1 Figure 1. Crossing of a fairly dense crowd ( ρ ≈ 3.5ped/m 2 ) by a cylindrical intruder. (a) Snapshot of an experiment (France), after automatic detection and tracking of the participants' hats. (b) Trajectories of the intruder (thick red line) and of the participants over a time window of around 5 seconds.

  figure1

  fig:figure2 fig:figure2

  fig:figure2 fig:figure2 2] as well as randomly oriented ones [Fig. fig:figure3 fig:figure3 3].Upstream from the cylinder, instead of the compression reported in granular matter below jamming[START_REF] Kolb | Flow fields around an intruder immersed in a 2d dense granular layer[END_REF] , we observe a redistribution of the density, with higher values on the sides, and either lower or average density along the central axis [Figs.

  fig:figure2 fig:figure2 2-

  fig:figure2 fig:figure22(b) for moderately dense crowds (the local densities plotted in the density profiles were smoothed over a length scale ξ = 9 cm according to the procedure described in the Methods).When pedestrians are told not to anticipate (in Argentina), the density field becomes more reminiscent of (but probably not as marked as) the granular case, with some compaction upstream. It should be noted that in spite of the instructions, some pedestrians facing the intruder [Fig.

  fig:figure3 fig:figure33(c)] still had a tendency to anticipate, visible in the videos. Only when they were turning their back to the intruder was anticipation fully suppressed. As a result, it is only in this case that a clear high-density layer coats the fore part of the intruder [see Fig.

  fig:figure3 fig:figure3 3(d)].

Figure 2 .

 2 Figure 2. Pedestrian densities (in ped/m 2 ) in the frame of the moving cylindrical intruder, in crowds facing the intruder (France). Panels (a) and (c) show the average density fields in moderately dense crowds ( ρ ≈ 2.5, average over 7 crossings, which we abbreviate into 'x7') and very dense crowds ( ρ ≈ 6, x7), respectively. Panel (b) displays the density profiles along the 'streamline' x = 0 in fairly dense crowds ( ρ ≈ 3.5); the result for participants facing the intruder (light blue, x9) is compared with its counterpart for randomly oriented participants (dark blue, x9). The error bars represent the standard deviation over all similar crossings and the dashed red lines delimit the portion of space occupied by the intruder.

  figure2

Figure 3 .

 3 Figure 3. Average density fields (in ped/m 2 ) around the cylindrical intruder for various types of dense crowds. (a) Fairly dense crowd ( ρ ≈ 3.5) with randomly oriented participants (France, x9). (b) Very dense crowd ( ρ ≈ 6) with randomly oriented participants (France, x5). (c) Dense crowd ( ρ ≈ 4.5) with non-anticipating participants facing the intruder (Argentina, x15). (d) Dense crowd ( ρ ≈ 4.5) with participants turning their backs to the intruder (Argentina, x10).

  figure3

  fig:figure4 fig:figure4

  fig:figure4 fig:figure4 4(b)], in which case the velocity field (whose overall pattern remains similar) localises closer to the intruder. The contrast sharpens with participants turning their backs to the intruder [Fig. fig:figure4 fig:figure4

Figure 4 .

 4 Figure 4. Mean velocity fields around the cylindrical intruder. (a) For a fairly dense crowd ( ρ ≈ 3.5 ped/m 2 ) with participants facing the intruder (France, x9). (b) For a dense crowd ( ρ ≈ 4.5 ped/m 2 ) with non-anticipating participants facing the intruder (Argentina, x15). (c) For a dense crowd ( ρ ≈ 4.5 ped/m 2 ) with non-anticipating participants turning their backs to the intruder (Argentina, x9). The velocities are computed on the basis of the displacements performed over a time interval δt = 0.7 s and the arrows are magnified by a factor 3.

  figure4

Figure 5 .

 5 Figure 5. Smoothed overall displacement amplitudes ∆u induced by the cylindrical intruder, for (a) fairly dense crowds ( ρ ≈ 3.5 ped/m 2 ) of participants facing the intruder and (b) dense crowds ( ρ ≈ 4.5 ped/m 2 ) of non-anticipating participants facing the intruder. The grey-to-white overlay gives a rough idea of the background noise, due to the pedestrians' residual motion in the absence of a perturbation and (to a lesser extent) to measurement errors. The dashed red lines delimit the portion occupied by the intruder.

  figure5

Figure 6 .

 6 Figure 6. Crowd's response to its crossing by a pedestrian. (a) Average density field around the crossing pedestrian in a very dense crowd ( ρ ≈ 6 ped/m 2 ) of participants facing the intruder (France, x10). (b) Smoothed overall displacement amplitude ∆u in a fairly dense crowd ( ρ ≈ 3.5 ped/m 2 ) of participants facing the intruder (France, x10). (c-d) Mean flow fields (based on the displacements over a time window δt = 0.7 s) around the crossing pedestrian in crowds of participants facing the intruder at (c) fairly low density ( ρ ≈ 1.5 ped/m 2 ) and (d) fairly high density ( ρ ≈ 3.5 ped/m 2 ) (France, x5 and x8, respectively). The velocity arrows are magnified by a factor 2.5.

  fig:figure6 fig:figure6

  fig:figure6 fig:figure6

  fig:figure6 fig:figure6

Figure 7 .

 7 Figure 7. Modified Voronoi tesselation used to compute local densities. The border B of the convex hull of the group of pedestrians (blue dots) is represented as a red line and delimits the modified Voronoi cells of border points. Besides, the ridges of the initial Voronoi cells that extend out of the convex hull (dotted yellow lines) are replaced by the dashed black lines. The solid and dashed segments in black (together with the convex hull) delimit the final Voronoi cells.

  figure7

Note, however, that more generic explanations could also be put forward. For instance, in fluids, shear-thinning generically promotes localisation near the intruder

.

Acknowledgements

We thank all participants for taking part in the experiments. A.N. and C.A.-R. acknowledge help from Tom Marzin, Étienne Pinsard, Aymeric Duigou-Majumdar, and Ioannis Touloupas for the organisation of the experiments A.N. and C.A.-R. acknowledge funding from PALM (ANR-10-LABX-0039-PALM) through the project PERCEFOULE. M.K., S.I., and S.B. acknowledge help from A. Kolton and from the staff of Grupo de Física Estadística e interdisciplinaria (CAB)

Comments and perspectives

The foregoing differences in the response of pedestrians as compared to grains (whose origin we ascribed to anticipation and self-propulsion) cast doubt on the the incautious use of force models, even at densities where physical forces are commonly believed to dominate. In particular, the observed prominence of transverse displacements cannot be reproduced by models based on radial interpedestrian forces. Our findings would rather lend credence to descriptions based on (contact avoidance) strategies.

More generally, the collection of experimental results on high density crowds should foster the development of data-driven modelling efforts. These efforts are encouraged by our finding of robust characteristics in the crowd's response, which emerge in spite of the randomness stemming from the pedestrians' free will and their diverse body sizes, and resist variations in the crowd's density. These characteristics did not require extensive averaging (less than ten realisations). Nonetheless, it would be desirable to extend our work to larger crowds and to vary the intruder's size. Besides, the response to the crossing of a single pedestrian deserves further study. Indeed, in this case, unlike that of the cylinder, the static pedestrians may expect the intruder to also take his share of the efforts associated with the crossing, by adapting his trajectory. How this possibly impacts the crowd's response is still unclear.

Methods

Experiments

The experiments were approved by the local ethics committees (C3E and Grupo de Higiene y Seguridad) prior to their realisation and they were conducted and analysed in accordance with the relevant guidelines and regulations. Informed consent was obtained from all participants.

Voronoi-based local densities with reduced edge effects

The Voronoi tessellation is a convenient tool to delimit the area 'belonging' to each pedestrian; local densities ρ are then defined as the inverse area A -1 of the local Voronoi cell. But this definition fails for points (pedestrians) at the edge of the system, namely, (i) the points P B that form part of the border B of the convex hull of the system and (ii) the points P V whose Voronoi cells spread beyond the convex hull. These points are not surrounded in all directions and their personal space is thus bounded only in a portion of the spatial directions, which we call the populated sector. More precisely, the populated sector of a point P B is the angular sector limited by the two adjacent points in B, while the populated sector of a point P V is delimited by the intersections between their Voronoi cell and B, as shown in Fig. 7. For all edge points, the local density is the density seen in the populated sector (which forms an angle α), i.e.,

where A is the area of the Voronoi cell that lies in the populated sector. Mean values for the local densities are obtained by averaging areas (i.e., 1/ρ , and not ρ ) over time and over realisations.

Smoothing of the displacement and velocity fields

To get smoother (continuous) displacement fields out of our finite data set, we perform a convolution of the displacements u i of all pedestrians i as follows

where φ is a smooth function (almost everywhere) with a compact support. Here, we have chosen (ξ = 0.25 m),

The continuous velocity field is obtained from the displacement field over a time interval δt as v(r,t) = u(r,t) δt

Displacement amplitudes

The displacement amplitude ∆u ( j) of a pedestrian j is defined by Eq. ( eq:deltau eq:deltau 1). In order to get a smooth transverse profile ∆u(x), we compute the following convolution with the coarse-graining function φ introduced in Eq. ( eq:phi eq:phi 3), viz.,

Additional information

The authors declare no competing interests. The raw trajectories collected in relation with the experiments described here and the scripts used for the analysis are available upon request to the authors.


Hatem Baloudi

AND Sayda Belgacem

Aref Jeribi

THE DISCRETE LAPLACIAN ACTING ON 2-FORMS AND APPLICATION

Keywords: 2010 Mathematics Subject Classification. 05C12, 05C63, 47B25, 81Q35 locally finite graph, essential self-adjointness, adjacency matrix, unboundedness, spectral graph theory. 1

In the current paper, we study the discrete Laplacian acting on 2-forms which was introduced and investigated by Chebbi (2018). We establish a new criterion of essential self-adjointness using the Nelson lemma. Moreover, we give an upper bound on the infimum of the essential spectrum. Furthermore, we establish a link between the adjacency matrix and the discrete Laplacian on 2-forms.

operators on infinite graphs have attracted a lot of interest, see [START_REF] De Verdière | Essential self-adjointness for combinatorial schrödinger operators. II metrically non complete graphs[END_REF][START_REF] Hung | A note on selfadjoint extensions of the Laplacian on weighted graphs[END_REF][START_REF] Milatovic | Essential self-adjointness of magnetic Schrödinger operators on locally finite graphs[END_REF][START_REF] Golénia | The problem of deficiency indices for discrete Schrödinger operators on locally finite graph[END_REF]. There exist other definitions of the discrete Laplacian, e.g., [START_REF] Mohar | A survey on spectra of infinite graphs[END_REF][START_REF] Golénia | Hardy inequality and assymptotic eigenvalue distribution for discrete laplacians[END_REF][START_REF] Balti | On the eigenvalues of weighted directed graphs[END_REF][START_REF] Ayadi | Spectra of Laplacians on an infinite graph[END_REF]. The one we are studying here is the discrete Laplacian acting on 2-forms and denoted by L 2,skew , where skew stands for skew-symmetric. This operator was introduced by Chebbi in [START_REF] Chebbi | The discrete Laplacian of a 2-Simplicial complex[END_REF]. The author shows the relation between the χ-completeness geometric hypothesis for the graph and essentially selfadjointness for the discrete Laplacian L 2,skew . More specifically, the author has proved that L 2,skew is essential self-adjoint, when the triangulation (we refer to Section 2 for a precise definition) is χ-complete.

The current study has two major aims. It first aims to discuss the question of essential self-adjointness for L 2,skew . It is worth noting that this operator depends on the weight R on oriented triangular faces and the weight E on oriented edges, see Section 4 for more details. In the setting of electrical networks, the weight E correspond to the conductance. We establish a hypothesis on the weights and involve essential self-adjointness by using the Nelson commutator theorem. The technique of the proof are inspired from [START_REF] Baloudi | The adjacency matrix and the discrete Laplacian acting on forms[END_REF]. Moreover, we give an upper bound on the infimum of the essential spectrum σ ess (L F 2,skew ), where L F 2,skew is the Friedreichs extension of L 2,skew . Secondly, the paper aims to identify the link between the adjacency matrix and the discrete Laplacian L 2,skew . To achieve this goal, we analyze the structure of L 2,skew . Note that this discrete Laplacian was introduced on a skew-symmetric statistic on the space of 2-forms. We can define the discrete Laplacian L 2,sym in the symmetric case by the same expression of L 2,sym . In the case of a tri-partite graph, we prove that the two operators are unitarily equivalent. Furthermore, L 2,sym is unitarily equivalent to the adjacency matrix of the triangular graph, see Section 7 for more details. We recall that the spectral theory of adjacency matrix acting on graphs is useful for the study of some gelling polymers, of some electrical networks, and in number theory, see [START_REF] Doyle | Random walks and electric networks[END_REF][START_REF] Davidoff | Elementary number theory, group theory, and Ramanujan graphs[END_REF][START_REF] Mohar | The spectrum of infinite graphs with bounded vertex degrees, Graphs, hypergraphsGraphs, hypergraphs and applications[END_REF].

As for the rest of this paper, it is structured as follows: The next section is devoted to some definitions and notations for graph. We find the definitions of two different Hilbert structures on the set of faces, in Section 3. Both definitions have their own interest. This permits to define two different types of discrete Laplacian associated to faces. The relation between these two operators is clearly presented in Section 4.3. In Section 5, we discuss the question of essential self-adjointness for the discrete Laplacian L 2,skew . We establish a new criterion of essential self-adjointness using the Nelson lemma. In Section 6, we give an upper bound on the infimum of the essential spectrum. The obtained findings from the previous sections are presented in Section 7 for the purpose of investigating the questions of boundedness and essential self-adjointness for the adjacency matrix.

Acknowledgment. The authors thank Colette Anné, Nabila Torki-Hamza, Sylvain Golénia and Nassim Athmouni for useful discussions and comments on the text. They would like to thank also the anonymous referee for their numerous relevant remarks and useful suggestions.

Generalities about graphs

We start with some definitions to fix notations for graphs and refer to [START_REF] Chung | Spectral graph theory[END_REF][START_REF] De Verdière | Spectres de graphes[END_REF][START_REF] Mohar | A survey on spectra of infinite graphs[END_REF] for surveys on the matter. Let V be a countable set. We equip V with the discrete topology. Let E : V × V -→ [0, +∞) and assume that E is symmetric (i.e., E(x, y) = E(y, x), for all x, y ∈ V). Let m : V -→ (0, +∞). We say that G = (V, m, E) is a weighted graph with vertices V, weight of vertices m and weight of edges E. In the setting of electrical networks, the weights correspond to the conductances. We say that x, y are neighbors if E(x, y) = 0 and we denote it by x ∼ y. A graph G is simple if it has no loops (i.e., E(x, x) = 0), m=1 and E has values in {0, 1}. The set of neighbors of x ∈ V is denoted by

N G (x) := {y ∈ V : E(x, y) = 0}.
A graph is locally finite if ♯N G (x) is finite for all x ∈ V. The weighted degree of vertices is given by

d V (x) := 1 m(x) y∼x E(x, y).
When G is simple, d V (x) = ♯N G (x). A graph G is connected, if for all x, y ∈ V, there exists an xy-path, i.e., there is a sequence (x 1 , ..., x N +1 ) ∈ V N +1 such that x 1 = x, x N +1 = y and E(x n , x n+1 ) > 0 for all n ∈ {1, ..., N }. If no vertices appear more than once in (x 1 , ..., x N ), the path (x 1 , ..., x N +1 ) is called a simple path. The path is called a cycle or closed when the origin and the end are identical, i.e., x 1 = x N +1 . An n-cycle is a cycle with n vertices.

In the sequel, we shall always consider graphs G, which are locally finite, connected and have no loop. The set of cyclic permutations of (x, y, z) ∈ V 3 is denoted by (x, y, z) := {(x, y, z), (y, z, x), (z, x, y)}.

Let T r the set of all simple 3-cycles

F = T r/ ∼ = where ̟ 1 ∼ = ̟ 2 if and only if ̟ 1 is a cyclic permutation of ̟ 2 .
The elements of F are called triangular faces. Atriangulation is a couple (G, F) where G is a graph and F is the set of all triangular faces.

In the sequel, we represent the triangular faces by their vertices. For a triangular face ̟ = (x, y, z), we have ̟ = (x, y, z) = (y, z, x) = (z, x, y).

Let (G = (V, E, m), F) be a triangulation and let R :

V × V × V -→ [0, +∞) such that (x, y, z) ∈ F ⇐⇒ R(x, y, z) > 0.
We assume that R is symmetric, i.e. R(x, y, z) = R(σ(x, y, z))

For all (x, y, z) ∈ V × V × V and for any permutation σ(x, y, z) of (x, y, z). We say that T = (V, m, E, R) is a weighted triangulation with weight of triangular faces R. We say that T is simple if G := (V, m, E) is simple and the weights of the faces equal 1. Choosing an orientation of triangulation consists of defining a partition of F:

F = F + ⊔ F -, (x 1 , x 2 , x 3 ) ∈ F + ⇐⇒ (x 3 , x 2 , x 1 ) ∈ F -. For (x 1 , x 2 , x 3 ) ∈ F, we denote -(x 1 , x 2 , x 3 ) = (x 3 , x 2 , x 1 ).
The set of neighbors of the edge (x, y) is given by

F (x,y) := N G (x) ∩ N G (y).
The weighted degree of edges is given by:

d E (x, y) := 1 E(x, y) z∈F (x,y) R(x, y, z). When T is simple, d E (x, y) = ♯F (x,y) .
3. The symmetric and skew-symmetric spaces 3.1. Hilbert structures on the set of edges. Let T = (V, m, E, R) be a weighted triangulation. Let E := {(x, y) ∈ V × V : E(x, y) > 0}. The set of 1-cochains (or 1-forms) is given by:

C skew (E) := f : E → C, f (x, y) = -f (y, x) for all x, y ∈ V ,
where skew stands for skew-symmetric. This corresponds to fermionic statistics. The set of functions with finite support is denoted by C c skew (E). Concerning bosonic statistics, we define:

C sym (E) := f : E → C, f (x, y) = f (y, x) for all x, y ∈ V .
The set of functions with finite support is denoted by C c sym (E). We turn to the Hilbert structures.

ℓ 2 skew (E) :=    f ∈ C skew (E) such that f 2 := 1 2 x,y∈V E(x, y)|f (x, y)| 2 < ∞    and ℓ 2 sym (E) :=    f ∈ C sym (E) such that f 2 := 1 2 x,y∈V E(x, y)|f (x, y)| 2 < ∞    .
The associated scalar product is given by

f, g := 1 2 (x,y)∈E E(x, y)f (x, y)g(x, y),
when f and g are both in ℓ 2 skew (E) or in ℓ 2 sym (E).

3.2. Hilbert structures on the set of faces. Let T = (V, m, E, R) be a weighted triangulation. The set of 2-cochains or 2-forms is given by

C skew (R) = f : F -→ C : f (x, y, z) = -f (z, y, x) .
The set of functions with finite support is denoted by C c skew (R). Concerning the case symmetric, we define

C sym (R) = f : F -→ C : f (x, y, z) = f (z, y, x) .
The set of functions with finite support is denoted by C c sym (R). Let us define the Hilbert spaces ℓ 2 skew (R) and ℓ 2 sym (R) as the sets of cochains with finite norm, we have

ℓ 2 skew (R) :=    f ∈ C skew (R) : f 2 = 1 2 (x,y,z)∈F R(x, y, z) | f (x, y, z) | 2 < ∞    and ℓ 2 sym (R) :=    f ∈ C sym (R) : f 2 = 1 2 (x,y,z)∈F R(x, y, z) | f (x, y, z) | 2 < ∞    .
The associated scalar product is given by

f, g := 1 2 (x,y,z)∈F R(x, y, z)f (x, y, z)g(x, y, z) := 1 6 (x,y)∈E z∈F (x,y) R(x, y, z)f (x, y, z)g(x, y, z)
when f and g are both in ℓ 2 skew (R) or in ℓ 2 sym (R).

Operators

In this section, we recall the concept of exterior derivative operator associated to a faces space, we refer to [START_REF] Chebbi | The discrete Laplacian of a 2-Simplicial complex[END_REF][START_REF] Chebbi | Laplacien discret d'un 2-complexe simplicial[END_REF] for more details. This permits to define the discrete Laplacian acting on 2-forms. 4.1. Skew-symmetric case. We start with defining the operators in the skew-symmetric case. The skew-symmetric exterior operator is the operator

d 1 skew : C c skew (E) -→ C c skew (R), given by d 1 skew (f )(x, y, z) = f (x, y) + f (y, z) + f (z, x).
The skew-symmetric co-exterior derivative operator is the formal adjoint of d 1 skew , i.e. it is the operator

δ 1 skew : C c skew (R) -→ C c skew (E), given by ∀f ∈ C c skew (R), δ 1 skew (f )(x, y) = 1 E(x, y) z∈F (x,y) R(x, y, z)f (x, y, z).
Both operators are closable (see [START_REF] Chebbi | The discrete Laplacian of a 2-Simplicial complex[END_REF]Lemme 3.1]). We denote their closure by the same symbol. The skew-symmetric discrete Laplacian operator acting on 2-forms is given by

L 2,skew (f )(x, y, z) = d 1 skew δ 1 skew (f )(x, y, z) = 1 E(x, y) t∈F (x,y) R(x, y, t)f (x, y, t) + 1 E(y, z) t∈F (y,z) R(y, z, t)f (y, z, t) + 1 E( z, x) t∈F (z,x) R(z, x, t)f (z, x, t), with f ∈ C c skew (R).
4.2. Symmetric case. We turn to the symmetric case. The symmetric exterior operator is the operator

d 1 sym : C c sym (E) -→ C c sym (R), given by ∀f ∈ C c sym (E), d 1 sym (f )(x, y, z) = f (x, y) + f (y, z) + f (z, x).
The symmetric co-exterior derivative operator is the formal adjoint of d 1 sym , i.e. it is the operator δ 1 sym :

C 1 sym (R) -→ C c sym (E), given by ∀f ∈ C 1 sym (R), δ 1 sym (f )(x, y) := 1 E(x, y) z∈F (x,y) R(x, y, z)f (x, y, z).
Indeed, for f ∈ C c sym (E) and g ∈ C c sym (R) we get:

d 1 sym f, g = 1 2 (x,y,z)∈F R(x, y, z)d 1 sym f (x, y, z)g(x, y, z) = 1 6 x∼y z∈F (x,y) R(x, y, z)(f (x, y) + f (y, z) + f (z, x))g(x, y, z) = 1 2 x∼y z∈F (x,y) R(x, y, z)f (x, y)g(x, y, z) = 1 2 x∼y E(x, y)f (x, y) 1 E(x, y) z∈F (x,y) R(x, y, z)g(x, y, z) = f, δ 1 sym g .
The operators d 1 sym and δ 1 sym are closable. Indeed, since δ 1 sym :

ℓ 2 sym (R) -→ ℓ 2 sym (E)( resp. d 1 sym : ℓ 2 sym (E) -→ ℓ 2 sym (R)) is with dense domain then δ 1 sym ( resp. d 1 sym
) is closable. We denote their closure by the same symbol. The symmetric discrete Laplacian operator acting on 2-forms is the operator L 2,sym = d 1 sym δ 1 sym , given by the same expression of L 2,skew .

4.3.

Relationship between L 2,skew and L 2,sym . The two operators L 2,skew and L 2,sym have the same expression. However, they do not act on the same spaces. Namely, when T is tri-partite, we shall prove that the two operators are unitarily equivalent.

Definition 4.1. A tri-partite graph is a graph whose vertices can be partitioned into 3 disjoint sets so that there are no two vertices within the same set are adjacent. A tri-partite triangulation is a triangulation T = (G, F) such that G is tri-partite.

Theorem 4.2. Let T = (V, m, E, R) be a tri-partite weighted triangulation. Then, L 2,skew and L 2,sym are unitarily equivalent.

Proof. We consider the tri-partite decomposition

{V 1 , V 2 , V 3 }. Set V 1 × V 2 × V 3 = { (x, y, z) : (x, y, z) ∈ V 1 × V 2 × V 3 }.
Let U : ℓ 2 skew (R) -→ ℓ 2 sym (R) be the unitary map given by U (f )(x, y, z) = S(x, y, z)f (x, y, z), where S(x, y, z)

:=    1, if (x, y, z) ∈ V 1 × V 2 × V 3 , -1, if (x, y, z) ∈ V 3 × V 2 × V 1 .
Let W be the following mapping from ℓ 2 sym (R) into ℓ 2 skew (R) : W(f )(x, y, z) = S(x, y, z)f (x, y, z). Then U f, g = f, Wg , U (W(g)) = g and W(U (f )) = f for all f ∈ ℓ 2 skew (F) and g ∈ ℓ 2 sym (F). So we have

W(f ) = U -1 (f ) = U * (f ) for all f ∈ ℓ 2 sym (R). Therefore, U L 2,skew U -1 (f )(x, y, z) = L 2,sym (f )(x, y, z)
for all f ∈ C c sym (R).

A Nelson criterium

For the general theory of unbounded Hermitian operators and their extensions, we refer the reader to [START_REF] Palle | Essential self-adjointness of semibounded operators[END_REF][START_REF] Kato | Perturbation theory for linear operators[END_REF][START_REF] Schechter | Principles of functional analysis[END_REF]. Let L skew be the following mapping from C skew (R) into itself:

L skew (f )(x, y, z) = 1 E(x, y) t∈F (x,y) R(x, y, t)f (x, y, t) + 1 E(y, z) t∈F (y,z) R(y, z, t)f (y, z, t) + 1 E( z, x) t∈F (z,x) R(z, x, t)f (z, x, t).
Let L 2,max,skew be the restrictions of L skew to

D(L 2,max,skew ) := f ∈ ℓ 2 skew (R) such that L skew f ∈ ℓ 2 skew (R) . Lemma 5.1. L * 2,skew = L 2,max,skew .
Proof. Let f ∈ C c skew (R) and let g ∈ C skew (R). Let F 00 the support of f and set F 0 = (x, y, z) ∈ F : ∃u ∈ V, {(x, y, u), (y, z, u), (z, x, u)} ∩ F 00 = ∅ which is a finite set. Then, supp(L 2,skew ) ⊂ F 0 and the following relation holds: 1 2

(x,y,z)∈F 0 R(x, y, z)L 2,skew (f )(x, y, z)g(x, y, z) = 1 2 (x,y,z)∈F 0 R(x, y, z) 1 E(x, y) t∈F (x,y) R(x, y, t)f (x, y, t) + 1 E(y, z) t∈F (y,z) R(y, z, t)f (y, z, t) + 1 E( z, x) t∈F (z,x) R(z, x, t)f (z, x, t) g(x, y, z) = 1 2 (x,y,z)∈F 00 R(x, y, z)f (x, y, z) 1 E(x, y) u∈V R(x, y, u)g(x, y, u) + 1 E(y, z) u∈V R(y, z, u)g(y, z, u) + 1 E(z, x) u∈V R(z, x, u)g(z, x, u) (1) = 1 2
x,y,z∈V R(x, y, z)f (x, y, z)L 2 g(x, y, z).

Let g ∈ D(L 2,max,skew ). It follows from (1) that L 2,skew f, g = f, L 2,max,skew g for all f ∈ C c skew (R), which implies that g ∈ D(L * 2,skew ). Now let g ∈ D(L * 2,skew ). Let (x, y, z) ∈ F and let

f = 1 R(x, y, z) 1 (x,y,z) -1 (z,y,x) .
Then, f ∈ C c skew (R) and we obtain from (1):

(L * 2,skew g)(x, y, z) = f, L * 2,skew g = L 2,skew f, g = 1 2 (u,v,w)∈F R(u, v, w)L 2,skew f (u, v, w)g(u, v, w) = 1 2 (u,v,w)∈F R(u, v, w)f (u, v, w)L 2,max,skew g(u, v, w) = (L 2,max,skew g)(x, y, z).
which implies that L 2,skew g = L 2,max,skew g ∈ ℓ 2 skew (R) by the definition of the adjoint, it follows that g ∈ D(L 2,max,skew ). Hence L * 2,skew = L 2,max,skew .

Remark 5.2. Let L sym be the mapping from C sym (R) into itself given by the same expression of L 2,sym . Then, L * 2,sym = L 2,max,sym where L 2,max,sym is the restrictions of L sym to

D(L 2,max,sym ) := f ∈ ℓ 2 sym (R) such that L sym f ∈ ℓ 2 sym (R) .
Using the Nelson commutator theorem, we prove the criterium of essential self-adjointness for L 2,skew and L 2,sym .

Theorem 5.3. Let T = (V, m, E, R) be a weighted triangulation. Set N (x, y, z) = 1 + d E (x, y) + d E (y, z) + d E (z, x). Suppose that sup x∼y, z∈F (x,y) r∈F (x,y) 1 E(x, y) R(x, y, r) | N (x, y, r) -N (x, y, z) | 2 < ∞. Then L 2,skew is essentially self-adjoint on C c skew (R) and L 2,sym is essentially self-adjoint on C c sym (R).
Proof. Let N be the operator of multiplication by N (., ., .) ant take f ∈ C c skew (F). Going over the same techniques of the proof of [3, Theorem 5.13], we obtain:

L 2,skew f 2 ≤ 2 3 x∼y,z∈F (x,y) R(x, y, z) 1 E 2 (x, y) | t∈F (x,y) R(x, y, t)f (x, y, t) | 2 + 1 E 2 (y, z) | t∈F (y,z) R(y, z, t)f (y, z, t) | 2 + 1 E 2 (z, x) | t∈F (z,x) R(z, x, t)f (z, x, t) | 2 ≤ 2 x∼y,z∈F (x,y) R(x, y, z) 1 E 2 (x, y) | t∈F (x,y) R(x, y, t)f (x, y, t) | 2 ≤ 2 x∼y,z∈F (x,y) R(x, y, z) 1 E 2 (x, y) r∈F (x,y) R(x, y, r) × t∈F (x,y) R(x, y, t) | f (x, y, t) | 2 = 2 x∼y,z∈F (x,y) R(x, y, z) 1 E(x, y) t∈F (x,y) R(x, y, t) 2 | f (x, y, z) | 2 ≤ 12 N (f ) 2 .
Moreover, we notice that N (., ., .) is symmetric and f (x, y, z) = -f (z, y, x) and let J =| f, [L 2,skew , N ]f |. We get:

J ≤ 1 12 x∼y,z∈F (x,y) R(x, y, z) | f (x, y, z) | 2 + | [L 2,skew , N ](f )(x, y, z) | 2 ≤ 1 2 N 1 2 (f ) 2 + x∼y,z∈F (x,y) R(x, y, z) | t∈F (x,y) 1 E(x, y) R(x, y, t)× N (x, y, t) -N (x, y, z) f (x, y, t) | 2 ≤ 1 2 N 1 2 (f ) 2 + x∼y,z∈F (x,y) R(x, y, z) t∈F (x,y) 1 E(x, y) R(x, y, t) × r∈F (x,y) 1 E(x, y) R(x, y, r) | N (x, y, r) -N (x, y, z) | 2 | f (x, y, r) | 2 = 1 2 N 1 2 (f ) 2 + x∼y,z∈F (x,y) R(x, y, z) t∈F (x,y) 1 E
(x, y) R(x, y, t) × r∈V 1 E(x, y) F(x, y, r) | N (x, y, r) -N (x, y, z) | 2 ≤C | f (x, y, z) | 2 ≤ ( 1 + 12C 2 ) N 1 2 (f ) .
Applying [26, Theorem X.37], the result follows. The proof of L 2,sym may be checked in the same way as the proof of L 2,skew .

Corollary 5.4. Let T = (V, m, E, R) be a simple triangulation. Assume that sup x∼y,z∈F (x,y) r∈F (x,y)

|♯F (y,z) + ♯F (z,x) -♯F (y,r) -♯F (r,x) | 2 < ∞.
Then L 2,skew is essentially self-adjoint on C c skew (R) and L 2,sym is essentially self-adjoint on C c sym (R).

Essential spectrum

Let A be a closed, densely defined linear operator on a Banach space X, and let σ(A) denote the spectrum of A. We denote by K(X) the set of compact operators on X to itself. We define the essential spectrum of the operator A by

σ ess (A) = K∈K(X) σ(A + K).
It is well known that if A is a self-adjoint operator on a Hilbert space, the essential spectrum of A is the set of limit points of the spectrum of A, i.e., all points of the spectrum except isolated eigenvalues of finite multiplicity, see [START_REF] Wolf | On the essential spectrum of partial differential boundary problems[END_REF]. Let T = (V, m, E, R) be a weighted triangulation. Note that L 2,skew is non-negative symmetric operator on C c skew (R). We consider the quadratic form

q(f, g) = f, L 2,skew g + f, g on C c skew (R) × C c skew (R). Let H 1 be the completion of C c skew (R) under the norm f q = L 2,skew f, f + f 2 .
We define the Friedrichs extension L F 2,skew of L 2,skew by: i)

A vector f is in domain D(L F 2,skew ) if and only if f ∈ H 1 and C c skew (R) ∋ g -→ f, L 2,skew g + f, g extends to a norm continuous function on ℓ 2 skew (R). (ii) For each f ∈ D(L F 2,skew
), there is a unique u f such that f, L 2,skew g + f, g = u f , g by Riesz' Theorem. The Friedrichs extension of L 2,skew , is given by L F 2,skew f = u ff . It is a self-adjoint extension of L 2,skew , e.g. see [START_REF] Reed | Methods of modern mathematical physics tome I-IV[END_REF]Theorem X.23]. Note that L F 2,skew is bounded if and only if d E (.) is bounded, e.g. see [START_REF] Chebbi | Laplacien discret d'un 2-complexe simplicial[END_REF]. Theorem 6.1. Let T = (V, m, E, R) be a weighted triangulation and let

F 0 = {K ⊂ F : K finite}. Then, inf σ(L F 2,skew ) ≤ inf (x,y,z)∈F R(x, y, z) 1 E(x, y) + 1 E(y, z) + 1 E(z, x)
.

and inf σ ess (L F 2,skew ) ≤ sup K⊂F 0 inf (x,y,z)∈K c R(x, y, z)( 1 E(x, y) + 1 E(y, z) + 1 E(z, x) .
In particular, if T is a simple triangulation then L F 2,skew is not with compact resolvent.

Proof. Let (x 0 , y 0 , z 0 ) ∈ F and let

f = 1 (x 0 ,y 0 ,z 0 ) -1 (z 0 ,y 0 ,x 0 ) R(x 0 , y 0 , z 0 )
where 1 (x 0 ,y 0 ,z 0 ) denotes the indicator function of (x 0 , y 0 , z 0 ). Then f = 1 and

f, L 2,skew f = R(x 0 , y 0 , z 0 ) 1 E(x 0 , y 0 ) + 1 E(y 0 , z 0 ) + 1 E(z 0 , x 0 )
Applying [START_REF] Keller | The essential spectrum of the Laplacian on rapidly branching tessellations[END_REF]Proposition 3], the result follows.

7. Application to the study of the adjacency matrix 7.1. Adjacency matrix. The adjacency matrix has important implications. For example, it uses the semi-boundedness in order to give meaning to the heat equation, see [START_REF] Berkolaiko | Edge connectivity and the spectral gap of combinatorial and quantum graphs[END_REF]. Let G = (V, m, E) be a weighted graph. We define the set of 0-cochains on V by

C(V) = {f : V -→ C}.
We denote by C c (V) the 0-cochains with finite support in V. We associate a Hilbert space to V :

ℓ 2 (V) = f ∈ C(V) such that f 2 = x∈V m(x)|f (x)| 2 < ∞ .
The associated scalar product is given by

f, g = x∈V m(x)f (x)g(x), for f, g ∈ ℓ 2 (V).
We define the adjacency matrix:

A G (f )(x) = 1 m(x) y∈V E(x, y)f (y), f ∈ C c (V).
It is symmetric and thus closable. We denote its closure by the same symbol. When G is simple, we have that A G is unbounded if and only if it is unbounded from above and if and only if the degree is unbounded, see [START_REF] Golénia | Unboundedness of adjacency matrices of locally finite graphs[END_REF].

((x 0 , y 0 , z 0 ), (x, y, z)) = R(x 0 , y 0 , z 0 ) R(x, y, z) E(x, y) 1 x=x 0 , y=y 0 + R(x, y, z) E(y, z) 1 y=y 0 , z=z 0 + R(x, y, z) E(z, x) 1 z=z 0 , x=x 0 if (x 0 , y 0 , z 0 ) = (x
A G (f )(x 0 , y 0 , z 0 ) := z∈V, z =z 0 R(x 0 , y 0 , z 0 )R(x 0 , y 0 , z) E(x 0 , y 0 ) f (x 0 , y 0 , z) + x∈V, x =x 0 R(x 0 , y 0 , z 0 )R(x, y 0 , z 0 ) E(y 0 , z 0 ) f (x, y 0 , z 0 ) + y∈V, y =y 0 R(x 0 , y 0 , z 0 )R(x 0 , y, z 0 ) E(z 0 , x 0 ) f (x 0 , y, z 0 ) for all f ∈ C c ( V). Proposition 7.3. Let T = (V, m, E, R) be a weighted triangulation. Then L 2,sym is unitarily equivalent to A G + Q(V )
where V (x 0 , y 0 , z 0 ) = R(x 0 , y 0 , z 0 ) R(x 0 , y 0 ) + R(x 0 , y 0 , z 0 ) E(y 0 , z 0 ) + R(x 0 , y 0 , z 0 ) E(z 0 , x 0 ) and Q(V ) be the operator of multiplication by V .

Proof. Set U : ℓ 2 sym (R) -→ ℓ 2 ( V) as the operator given by U (f )(x, y, z) = R(x, y, z)f (x, y, z).

Notice that

U -1 (f )(x, y, z) = U * (f )(x, y, z) = 1 R(x, y, z) f (x, y, z) for all f ∈ ℓ 2 ( V). Notice now that on C c ( V) U L 2,sym U -1 (f )(x 0 , y 0 , z 0 ) = t∈V R(x 0 , y 0 , z 0 )R(x 0 , y 0 , t) E(x 0 , y 0 ) f (x 0 , y 0 , t) + t∈V R(x 0 , y 0 , z 0 )R(y 0 , z 0 , t) E(y 0 , z 0 ) f (y 0 , z 0 , t) + t∈V R(x 0 , y 0 , z 0 )R(z 0 , x 0 , t) E(z 0 , x 0 ) f (z 0 , x 0 , t).
Using Remark 7.2, we obtain the result.

So, T is tri-partite. Using Theorem 4.2, Proposition 7.3 and Proposition 7.12, the result holds. Then L 2,sym does not essentially self-adjoint on C c sym (R).

Proof. Set f ∈ ℓ 2 sym (F)\{0} such that f ∈ ker(L * 2,sym + i) and such that f is constant on S n × S

M := sup n∈N * f | Sn×S 2 n+1 2 < ∞.
Then, we have

f | S n+1 ×S 2 n+2 2 ≤ M |i + 4 + s n | s n s n+2 .
From Eq. ( 3), we infer that f ∈ ℓ 2 sym (F). Using [26, Theorem X.36], we conclude that L 2,sym is not essentially self-adjoint.

Corollary 7.16. Let T = (V, m, E, R) be a simple triangular anti-tree satisfying (3). Then, A G is not essentially self-adjoint on C c ( V).

7. 2 .

 2 Triangular graph. Let T = (V, m, E, R) be a weighted triangulation. Set V = F/ ∼, where ̟ ∼ -̟. Definition 7.1. Let T = (V, m, E, R) be a weighted triangulation. Set V = F/ ∼, where ̟ ∼ -̟. The triangular graph of T is the graph G = ( V, m, E) where m = 1 and E

Remark 7 . 2 .

 72 , y, z) and 0 otherwise.A triangulation TTriangular graph G Let T = (V, m, E, R) be a triangulation. The adjacency matrix on G is given by

7. 5 .

 5 Triangular anti-tree. Let T = (V, m, E, R) be a weighted triangulation. The sphere of radius n ∈ N around a vertex v ∈ V is the setS n (v) := w ∈ V : d V (v, w) = n .We recall that C n denotes the n-cycle graph, i.e. V = Z/nZ, whereE(x, y) > 0 if and only if |x -y| = 1. Let G(S n (o)) = (S n (v), E ′ ) where E ′ = E | Sn(v)×Sn(v) .Definition 7.14. Let T = (V, m, E, R) be a weighted triangulation. We say that T is anti-tree if there exists a vertex o ∈ V such that 1) For all n ∈ N * and v ∈ S n (o), we haveN G (v)\S n (o) = S n-1 (o) ∪ S n+1 (o).

2 )

 2 For all n ∈ N * , G(S n (o)) ≃ C ♯Sn(o) . o S 0 (o) S 1 (o) S 2 (o) S 3 (o)An triangular anti-tree with spheres S 0 (o), S 1 (o), S 2 (o), S 3 (o). Theorem 7.15. Let T = (V, m, E, R) be a simple triangular anti-tree whose root in o. Set s n = ♯S n (o) and assume that (3) n -→ s 2 n s n+2 ∈ ℓ 1 (N).

  2 n+1 ∪ S 2 n × S n+1 , n ∈ N * . We denote the constant value by C n . It takes the value 0 on S 2 n . We have the following equation: (s n + 4 + i)C n + s n+2 C n+1 = 0.

	Therefore,
		f | S n+1 ×S 2 n+2	2 = f | S 2 n+1 ×S n+2	2
				=	1 2	s n+2 s n+1 |C n+1 | 2
				≤	|s n + 4 + i| 2 s n s n+2	n+1 f | Sn×S 2	2 .
	Since lim n→∞	s 2 n s n+2	= 0, we get by induction:

Corollary 7.4. Let T = (V, m, E, R) be a weighted 3-partite triangulation. If V is bounded on F, then A G ≥ sup (x,y,z)∈F V (x, y, z).

In particular, A G ≥ -3 when T is simple.

Proof. Since L 2,sym is non-negative operator, then

Proof. Combine Proposition 7.3, Theorem 5.3 and Theorem 4.2.

7.3. Geometric Hypothesis. We recall the two following definitions:

Definition 7.6. [1, Definition 8] The graph G := (V, m, E) is χ-complete if there exists an increasing sequence of finite set (V n ) n such that V = ∪ n V n and there exist related functions χ n satisfying the following three conditions:

We recall the criterion obtained in [START_REF] Chebbi | The discrete Laplacian of a 2-Simplicial complex[END_REF].

Theorem 7.8. Let T = (V, m, E, R) be a χ-complete weighted triangulation then L 2,skew is essentially self-adjoint on C c skew (R).

Corollary 7.9. Let T = (V, m, E, R) be a χ-complete weighted triangulation. If T is tri-partite then A G is essentially self-adjoint on C c ( V).

Proof. Combine Proposition 7.3, Theorem 4.2 and Theorem 7.8.

7.4. Book-like triangulation. We recall the definition of 1-dimensional decomposition given in [START_REF] Bonnefont | Essential spectrum and Weyl asymptotics for discrete Laplacians[END_REF] for the case of graphs.

Definition 7.10.

[6] A 1-dimensional decomposition of the graph G = (V, m, E) is a family of finite sets (S n ) n∈N which forms a partitions of V, that is V = ∪ n∈N S n , and such that for all x ∈ S n , y ∈ S m ,

The following definition is introduced in [START_REF] Chebbi | The discrete Laplacian of a 2-Simplicial complex[END_REF].

Definition 7.11. Let T := (V, m, E, R) be a weighted triangulation and

A book-like triangulation

We recall [7, Proposition 6.5]: Proposition 7.12. Let T be a simple book-like triangulation. Assume that

Then, L 2,skew is not essentially self-adjoint on C c skew (R). Corollary 7.13. Let T = (V, m, E, R) be a simple book-like triangulation satisfying [START_REF] Ayadi | Spectra of Laplacians on an infinite graph[END_REF]. Then, A G is not essentially self-adjoint on C c ( V).

Proof. Set

Hatem Baloudi, Department of Mathematics, Faculty of Sciences of Gafsa, University of Gafsa, 2112 Zarroug, Tunisia Sayda Belgacem Aref Jeribi, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, 3000 Sfax, Tunisia