1. Introduction

More than four decades ago, the possibility of distributing high energy over a large
volume to restore broken symmetries of the physical vacuum creating abnormal states
of nuclear matter was raised ^{[1]}.
Very early, it was pointed out that the asymptotic freedom property of QCD implies
the existence of a high-density matter formed by deconfined quarks and gluons ^{[2]}, and the exponential increase of
the hadron Hagedorn spectrum was connected with the existence of a different phase
^{[3]}. The thermalized phase of
quarks and gluons was called Quark Gluon Plasma (QGP) ^{[4]}, and the evaluations of the required high density
showed that it could be reached in relativistic heavy ion collisions ^{[5}^{,}^{6]} and several signatures of QGP were proposed.
Quarkonium suppression ^{[7]}, the
excess of photons and jet quenching ^{[8}^{,}^{9]} were some of them. At this time, it was pointed out the
relevance of percolation in the study of the phase transitions of hadronic matter
^{[10}^{,}^{11]}.

From the experimental side, there were large facilities to study the properties of
large density matter starting by the AGS and ISR, experiments later followed by SPS,
RHIC, and LHC. At SPS already several signatures hinted the onset of QGP formation
^{[12]}. The RHIC data show a
collective elliptic flow which pointed out a very low shear viscosity over entropy
density ratio ^{[13}^{-}^{17]}. The above mentioned ratio enhanced the
attention to the AdS/CFT correspondence due to the result ^{[18]}. The LHC experiments ^{[19}^{-}^{21]} have extended the study of the elliptic flow to
all the harmonics ^{[22}^{,}^{23]} confirming the obtained strong interacting quark and
gluon matter and showing that the collective behavior and the ridge structure
previously observed at RHIC in Au-Au and Cu-Cu collisions ^{[24}^{,}^{25]}; also occurs in pPb ^{[26}^{-}^{28]} and p+p collisions at high multiplicity ^{[29]}. The collective behavior of p+p
and pPb interactions are a challenge to the hydrodynamics descriptions, and they
raise the question whether the main experimental data can be explained by final
state interactions or on the contrary, the initial state configuration should
describe them.

On the other hand, the data on quarkonium confirm the validity of combined picture of
a subsequent melting of the different resonances, together the recombination of
heavy quarks and antiquarks at high energy ^{[30}^{-}^{32]}. The departure of linear dependence on the
multiplicity of the ^{[33}^{,}^{34]}, indicating multiparton interactions or
multiplicity saturation ^{[35]}.
Detailed studies on the jet quenching for identified particles have been done ^{[36]} showing features related to the
low of coherence of the gluons edited in the jet due to the high-density medium.
Finally, let us mention that at RHIC has been observed recently that the fluid
produced by heavy ions is the most vortical system ever observed ^{[37]}.

On the theoretical side, in addition to the hydrodynamics studies, the Color Glass
Condensate (CGC) approach ^{[38}^{-}^{42]} gives a good description of most of the experimental data
and is derived directly from QCD. In QCD, the gluon density ^{[43]}. The low ^{[44}^{,}^{45]}:

This dense system, called CGC, has a very high occupation number

In high energy physics experiments, the colliding objects move at velocities close to
the speed of light. Due to the Lorentz contraction, the collision of two nuclei can
be seen as a that of two sheet of colored glass where the color field in each point
of the sheets is randomly directed. Taking these field as initial conditions, one
finds that between the sheets, longitudinal color electrical and magnetic fields are
formed. The number of these color flux tubes between the two colliding nuclei is
forming the called Glasma ^{[46]},
which has been extensively compared with the experimental data.

Another approach to the initial state is the percolation of strings ^{[47}^{-}^{51]} which is not so popular as the CGC because
cannot be derived directly from QCD although it is inspired in it, and most of its
results, are a direct consequence of properties of QCD. In this approach, the
multi-particle production is described in terms of older strings stretched between
the partons of the projectile and target. These strings decay into ^{[52]} and also considering
bilocal correlations ^{[53]}. It
corresponds to the correlation length of the QCD vacuum. With increasing energy
and/or size and centrality of the colliding objects, the number of strings grows and
the strings start to overlap forming clusters similarly than the continuum
percolation theory ^{[54]}. At a
given critical density, a macroscopical cluster appears crossing the collision
surface, which marks the percolation phase transition. Therefore, the nature of this
transition is geometrical.

In string percolation, the basic ingredients are the strings, and it is necessary to know its number, rapidity extension, fragmentation and number distribution. All that requires a model and therefore, string percolation is model dependent. However, most of the QCD inspired models give similar results for most of the observables in such a way that the predictions are, by a large measure, independent of the model used.

The string percolation and the Glasma are related to each other ^{[55]}: in the limit of high density, there is a
correspondence between the physical quantities of both approaches. The number of
color flux tubes in Glasma picture,

The observed densities of our world have large differences which expand over many
orders of magnitude, from 10^{-6} nucleons/cm^{3} in average in the
Universe to 10^{38} nucleons/cm^{3} inside a nucleus and
10^{39} nucleons/cm^{3} in a neutron star. The study of the
high-density limit, *i.e*., the study of de-confinement of quarks and
gluons can be regarded as the place where high energy collision of two bodies probes
the short distances and meets the thermodynamics (many body) of this short distance
limit ^{[56]}. The lattices studies
have shown that at low chemical potential

In finite

Note that

Below the critical temperature

which measures the constituent quark masses obtained from a Lagrangian with massless quarks. At high temperature this mass melts, therefore:

Here,*i.e*., a sharp transition but without discontinuity. The
quoted value is 155±9 MeV ^{[56}^{-}^{58]}.

The energy densities resulting from lattice QCD are shown in Fig. 2 (up), indicating that even for

where

Moreover, the trace of the energy momentum tensor:

is

2. Percolation model

Let us distribute small discs of area

The critical density for the onset of continuum percolation is determined by numerical and Monte Carlo simulations, which in the 2-dimensional case gives:

In the thermodynamical limit, ^{[59}^{,}^{60]}:

It also gives the total fraction of the plane covered by discs in ^{[50]}. The number 1.13 is obtained
in case of discs uniformly distributed ^{[51}^{,}^{61}^{,}^{62]}. However, in cases when the discs are not uniformly
distributed, this number changes. For instance, in the cases of circular surfaces
with Gaussian or Wood-Saxon profiles, the number is 1.5 and the fraction of the area
covered by strings is closer to the function:

where *a* and *b* depend on
the profile function, *b* that controls the ratio between the width
of the border of the profile ^{[63]}. In the collisions of two
hadrons or two nuclei, the surface where the discs are distributed is rather an
ellipse or a circle, what gives rise to smaller values of the critical density ^{[64]}. For small systems where the
number of discs is not large (far from the thermodynamical limit) the critical
density is smaller than above values, being 0.8 for high eccentricities ^{[64]}.

In SU(3) Gauge theory, spatial clusters can be identified as those where the local
Polyakov loops ^{[65]}. Below

In high energy collisions, we expect that color strings were formed between the
projectile and target partons. These color fields must have a small transverse size
due to confinement. In this way, the strings, in the transverse plane, are small
discs in the surface of the collisions. As the number of strings grows with energy
and centrality degree of the collision, the strings start to overlap forming
clusters which eventually percolate. The phenomenological consequences in relation
to SPS, RHIC, and LHC, p+p, pA and AA data are the main subject of this brief
review. A more extended version can be found in Ref. ^{[66]}

3. String percolation

3.1 String models

The basic ingredient of the string percolation are the strings. Despite
differences, most of them coincide in basic postulates as the number of strings
and its dependence on energy and centrality, which is taken from the
Glauber-Gribov Model.We will concentrate in models with color exchange between
projectile and target as the Dual Parton Model (DPM) ^{[67}^{-}^{69]}, Quark Gluon String Model (QGSM) ^{[70]}, Venus and EPOS ^{[71]}. They are based on the ^{[67]}. In DPM or QGS, the
multiplicity distribution *k*
strings

where

where

The momentum distribution used for the valence quarks, valence diquarks, sea
quarks and antiquarks are *k*
partons in the proton is:

where ^{[72]}, the cross section is calculated as follows:

where:

and *g* is the coupling of the pomeron to the proton, *C* is a parameter describing the inelastic
diffractive states. Summing over *k*, we obtain the total cross
section.

The rise of *k* pomerons:

where *N* is the mean multiplicity production when cutting one
pomeron, therefore, the multiplicity distribution is:

where *s* of
the short strings contributions is due to the increase of the invariant mass of
the short strings, formed between quarks and antiquarks of the sea, and to the
*s*-dependence of the weights. DPM can be generalized to hA
and AA collisions in the following way ^{[73]}, consider a collision with

where ^{[74,75]}.
The inclusive spectra, as in the p+p case, are given by a convolution of
momentum distribution and fragmentation functions. In the case of A=B, we have
approximately;

where we have introduced the possibility of having *k* multiple
scattering in the individual nucleon-nucleon collisions, which was neglected in
Eq.(21). Notice that there is
not any reason to assume that the term proportional to *Nk*
strings which for heavy nuclei collisions and high energy is very large number,
even larger than 1500. Due to that, we expect interactions between them and they
will not fragment in an independent way.

In the case of pA collisions, the Eqs. (21) and (22) transform into:

where

3.2 String fusion and percolation

As we have said before, at large energy we expect that the strings overlap in the
transverse plane. The transverse space occupied by a cluster of overlapping
strings splits into a number of areas with different number of strings overlap,
including areas where no overlapping takes place. In each area color field
coming from the overlapping strings add together. As a result, the cluster is
split in domains with different color strength. One may assume that emission of

where *m* the transverse momentum and the mass of the
emitted parton. The tension, according to the Schwinger mechanism, is
proportional to the field, and thus to the color charge of the ends of the
string ^{[76}^{-}^{80]}, which we denote by

and in the overlapping area each string will have color:

The total color in the overlap area will be a vector sum of the two overlapping
colors ^{[50}^{,}^{51]}. Thus

Notice that due to the vector nature, the color in the overlap is less than the
sum of the two overlapping colors. This effect has important consequences
concerning the saturation of multiplicities and the rise of the mean transverse
momentum with multiplicity, which we will study in the next section. Thus,
assuming independent emission from the three regions of Fig. 6, we obtain for the multiplicity weighted by the
multiplicity for a single string (

and for the mean transverse momentum squares (we divide the total transverse momentum squared by the multiplicity):

where we have used the property *N* of overlapping
strings, we have:

where the sum runs over all individual overlaps of *n* overlapping strings

The total area can be easily computed in the thermodynamic limit. One finds that
the distribution of overlap strings over the total surface *S* in
the variable is Poissonian with mean *n* the
percolation context. Therefore, the fraction of the total area covered by
strings will be

Finally, we can write for the mean values:

In the rest of this review, these last equations will be used extensively.

3.3 Quenching of the low

The ^{[81}^{,}^{82]}. In our case, the situation is different and is
more similar to a charge particle moving in an external electromagnetic field.
The corresponding force causes a loss of energy, which is given by ^{[83]}:

where *E* is the external electric field. This equation leads to
the quenching formula:

where

which will be used to compute the harmonic of the ^{[84]}.

3.4 Multiplicity distributions

The multiplicity distributions in the DPM of QGSM in p+p and AA collisions are
given by Eqs. (12)-(13) and Eqs. (21)-(22), respectively. However, as the
energy or centrality of the collision increases one expects interaction among
strings. As discussed before, due to the randomness of the color field in color
space non-abelian field the resulting color field in a cluster of
*n* overlapping strings is only

where *A*, whereas the number of strings as ^{[85]}:

where:

Here, the parameter

Taking into account the interaction of strings, we can write a closed formula for
the multiplicity distribution in AA in terms of the multiplicity distribution of
p+p, namely ^{[85]}:

where:

and

Moreover, the dependence of the multiplicity on *N* in a p+p collision is known. At low energies it is 2,
growing as:

Notice that here a single parameter ^{[85}^{-}^{87]} and central Cu-Cu ^{[88]} and for Au-Au and Pb-Pb ^{[89]}. The results of the dependence of the the
multiplicity per participant nucleon on the number of participants is shown in
Fig. 8 together with the experimental
data Cu-Cu, Au-Au and Pb-Pb at different energies.

The evolution outside the central rapidity region has been studied extensively,
extending Eq.(43) to all
rapidities ^{[90}^{-}^{95]}. The limiting fragmentation property is not
satisfied exactly. In Fig. 9, we show the
results together with the experimental data for p+p collisions at all rapidities
at different energies ^{[94}^{,}^{95]} and in Fig.
10 the results for Cu-Cu, Au-Au ^{[96]} and Pb-Pb ^{[97]} together the experimental data. In Fig. 11, we compare the results ^{[95}^{-}^{98]} for d-Au collisions together
the experimental data. A good description of all experimental data is
obtained.

The behavior of

3.5 Multiplicity and transverse momentum distributions

Let us start considering a set of overlapping strings, which depend both on the
number of strings and the overlapping area, which combine to give an average
multiplicity *N*. We may characterize the different overlaps just
by the average multiplicity that combines both the number of collisions and the
area. With a lot of overlapping strings *N* will change
practically continuously. We can introduce a probability *W(N)*
to have overlaps with size *N* in a collision and write the total
multiplicity distribution as:

where *N*, which we take Poissonian with the average multiplicity
*N*:

The normalization conditions

For the weight function we assume the gamma distribution ^{[99}^{-}^{101]}:

where *N* corresponding to a higher color field in the cluster.
This transformation, similar to the block transformation of Wilson type, can be
seen as a transformation of the cluster size probability of the type:

Transformations of this type were studied long time ago by Jona Lasinio in
connection with the renormalization group in probabilistic theory ^{[102]}, showing that the only
probability distribution function ^{[103}^{-}^{107]}. We will come back to this point, studying
the underlying events when one high *k* of Eq.(52) is energy independent. This
property is a consequence of the invariance of the gamma functions under the
transformation of type Eq.(51)
^{[107]}. Let us discuss now the
transverse momentum distribution (TDM) *x*;

Actually, *x* denotes the inverse of the color field in the
cluster, which depends not only on the size but also on the degree of
overlapping strings inside the cluster. Assuming that *x* varies
continuously, one can write the total TMD, similarly to the multiplicity
distribution case as:

We must realize the normalization condition:

which gives the relation:

Comparing the latter with Eq.(50b), we can make the identification *k* n *r* given by:

with

and:

The mean value and the dispersion of the distributions and are:

The distribution is a negative binomial distribution. Eqs.(58) and (59) are superposition of clusters
and

and:

We observe that:

Eqs.(63) and (64) give the distributions for any
projectile, target, energy and degree of centrality and are universal functions
which depend of only two parameters,

At ^{[103]} for central Au-Au collisions.

Let us now discuss the interplay of low and high

The normalization on the number of collisions in the latter, essentially
eliminates

Here

which is independent of

and

At low density in the region where decreases with the string density ^{[108]}. The results for
the TMD for ^{[109]}. In Fig.
13, we show the ratios kaons/pion, and proton/pion as a function of

The experimental data on p+p in the range ^{[110]}. In this case, the
values of ^{[66]}.

Even though the parametrization describes well the data up to 5-10 GeV/c, most of
the considerations concerning the string fragmentation are only valid for low
and intermediate

The differences between the baryon and meson spectrum are not only due to the
mass differences, which results

In some sense, the coalescence picture of particle production is incorporated in
a natural way. An effective way of taking into account these flavor
considerations can be seen in Ref.^{[111]}. Very often, it is used an exponential instead
of a gaussian for the decay of one string. Indeed, the tension of a cluster
fluctuates around its mean value because the chromo-electric field is not
constant. Such fluctuations lead to a Gaussian distribution of the string
tension ^{[109}^{-}^{113]}:

which give rise to the thermal distribution:

where ^{[112}^{,}^{114]}:

Now the total TMD is changed and instance of the gamma distribution in Eq (57) a Tsallis type distribution is obtained, namely:

There are several scaling properties found in TMD related to string percolation.
The experimental data for p+p collisions exhibit a universal behavior in a
suitable variable ^{[115}^{,}^{116]}. Indeed, the TMD of p+p
and ^{[117]} at
different centralities.

The experimental data on the mean

3.6 Transverse momentum fluctuations

The event by event fluctuations of thermodynamical quantities as the temperature
were proposed as a probe for the deconfined phase. Due to that, the study of the
fluctuations on the mean

and the correlation between the transverse momemtum:

where

and ^{[118]}:

In Fig. 15, we show the result for

3.7 Forward-backward correlations

The width of the KNO scaling shape is related to the fluctuations on the number
of strings or the number of clusters (independent color sources). This width is
also related to the forward-backward (*F-B*) correlations. These
correlations can be described by a linear approximation:

where *b* measures the correlation
forward-backward:

Usually, the *F* and *B* rapidity intervals are
taken separated by a central rapidity window ^{[50}^{,}^{119}^{-}^{124]}. Let us consider symmetric *F*
and *B* intervals and having *N* strings which
decay into *b* can be split into
short range (SR) and long range (LR) correlations ^{[121]}:

where

*F* or *B*
rapidities, respectively. We also take *b* becomes:

At low energy, there are not fluctuations in the number of strings,
*i.e*.,

As the energy or the centrality of the collisions increases, *b*. This behavior can be turned
as a consequence of the formation of a large cluster of overlapping strings and
consequently, a decreasing of the number of independent color sources. Notice
that if we fix the multiplicity, we eliminate many of the possible string
fluctuations and therefore, *b* will be smaller. In CGC, the main
contribution to long range correlations comes from the diagram of Fig. 17, which only contributes to short
range correlations in such a way that for a large rapidity gap between
*F* and *B* intervals we have ^{[125}^{-}^{127]}:

where *c* is a constant independent on the energy and centrality
degree. As the strong coupling constant, *b*
increases. This behavior is very similar to the one described above for string
percolation. The analysis of *F-B* correlations has been extended
not only to two rapidity separated windows, but also to different azimuthal
windows which help to separate short and long range correlations ^{[121]}. In this case, the
coefficient *b* is given by:

where now ^{[121]}. The
*F-B* correlations have been studied not only for
multiplicities in the *F-B* intervals, but also for transverse
momentum-multiplicity

where:

^{[128}^{,}^{129]} which predicted a rise of
*b* with centrality up to around 30% decreasing above this
centrality value.

3.8 Underlying event of high

The study of the underlying effect can be useful to understand the particle
production mechanism. It has been shown that selecting events of determinate
high ^{[130]}. Let us show that
this is a rather general property which is satisfied by events of a determined
class of scatterings, for instance diffractive and non diffractive or inelastic
and elastic or soft and hard scatterings. In a multiple scattering approach,
there will be events that is sufficient to have one elementary scattering of
being of this class to be the final result of this class. The non diffractive,
the inelastic and the hard events satisfy this requirement. It is said that
these events are only shadowed by themselves and in fact, the evaluation of the
cross section for these selected events only appears the cross section of the
elementary cross section of these events, not the elementary cross section of
all kind of events ^{[131]}.
Concerning the associated multiplicity distribution to these events, it is shown
that in terms of multiple scatterings, the original distribution and the new one
are related for a factor *N* which translate into a
multiplicative factor *n* in the multiplicity in such a way that
^{[103}^{-}^{107]}:

If we go on the process of the selection of high

In a similar way than the one in Sec. 3.5, the only stable distributions under
these transformation are the generalized gamma function, being the gamma
function the most simple of them (see Eq.(51). This function satisfies KNO scaling if *k*
is independent of energy. We have seen above that *k* increases
with the energy for p+p collisions in the studied range, as

3.9 Bose-Einstein Correlations

The Bose-Einstein Correlations (BEC) are very interesting in order to determinate
the extension of the source of multi-particle production, as well as to know the
degree of coherence of the emitted particles. The correlation strength is
characterized by the parameter ^{[132}^{-}^{134]}. In this interpretation ^{[135}^{-}^{137]}. As the number of collisions increases no longer
decreases, even it increases reaching values of 0.6-0.7. At SPS energies, the
values of ^{[138}^{,}^{139]}. The strings of the Lund
type fragmentation according to totally chaotic sources, ^{[140]}.
Under this assumptions one can write:

where

and the total number of pairs of identical particles produced is:

where ^{[138]}.

The three body BEC have been also studied in percolation ^{[139]}, predicting the strength of the three
particle BEC, which is in good overall agreement with data ^{[141]}.

3.10

The ALICE collaboration has found a departure from linearity on the dependence of ^{[142]}. In fact assuming that as in any hard
process, the number of produced

From Eq.(35) we can write:

thus:

At low multiplicities,

therefore:

Note that the linear behavior changes to quadratic at high multiplicities. In
Fig. 21, we show the results together
with the experimental data ^{[33]}, as well as the results for the forward rapidity
region together with the experimental data. In the forward rapidity region we
have less number of strings, and as a consequence, the departure from the linear
behavior starts at higher multiplicity. In both cases, central and forward
rapidity region, a good agreement is obtained.

Notice that only there are two assumptions, namely, the ^{[7]}, due to
the high density reached. In this case, there are not nuclear suppression
effects, then we assume that the suppression is proportional to the collision
area covered by strings. In Fig. 22 we
show the results without and with

3.11 Incoherent

The incoherent photo production of ^{[143}^{,}^{144]} and theoretically ^{[145]}. The cross section of

3.12 Strangeness enhancement

The overlapping of the strings modify the strength of the color field, and hence
the string tension of the formed cluster. Due to this, the decay of these
clusters produced naturally an enhancement of the strangeness ^{[146}^{-}^{148]}. In addition to this
effect, as the clusters have at their extremes complex flavor ^{[149]}, a simplified model of string percolation which
taken into account only the different string tension of the cluster is able to
describe qualitatively the data ^{[146]}. In Fig.
24 are shown the model results (up) and the experimental data
(bottom).

4. Azimuthal dependence of the momentum distributions

4.1 Collective flow and ridge structure

The clusters formed by the strings have an asymmetric form in the transverse
plane and acquires dimensions comparable to the nuclear overlap. This azimuthal
asymmetry is at the origin of the elliptic flow in string percolation. The
partons emitted at some point inside the cluster have to pass through the strong
color field before appearing in the surface. The energy loss by the parton is
proportional to the length, and therefore, the ^{[150]}. The results of this simulation for the
different harmonics ^{[150}^{,}^{151]} are in reasonable agreement with experimental
data on the

where

Thus, the elliptic flow can be computed as follows:

Note that the latter is an analytical close expression for all energies, centralities, projectiles, and targets.

The transverse momentum dependence of ^{[152}^{-}^{154]}, as well as the hierarchy on ^{[155]}.

In string percolation correlations can arise from the superposition of many
events with different number and type of string. In this way, there appears long
range correlations in rapidity. However, passing to the azimuthal dependence, if
the emission of strings is isotropic, the correlations due to their distribution
in different events will be also isotropic. Also in the central rapidity region,
the inclusive cross section is approximately independent of rapidity. This
generates a plateau in the ^{[154]}.

In Fig. 27, we show the results ^{[156]} for ^{[156]}. An
overall agreement is obtained in spite of the approximations done in the
computation.

In the case of p+p collisions, to obtain the ridge structure, we need to consider high multiplicity events (three times the minimum bias multiplicity shown in Fig. 29). This is due to the fluctuations needed to have sizable long correlations which are only obtained for these events. These fluctuations are also crucial to describe the higher harmonics of the azimuthal distributions.

We can conclude that string percolation is able to describe the ridge structure seen in p+p, pA and AA collisions. The ridge is obtained from the superposition of many events with different number and types of clusters of strings. There is not any essential difference between high multiplicity p+p for pA collisions and AA collisions. The collective flow is obtained from the configuration of the initial state as clusters of overlapping clusters and the interaction of the produced partons with the color field of the clusters. This interaction could be interpreted as final state interaction, but as far as the parton have these interactions before hadronization, it should be regarded as well as initial state interaction. In the production of heavy particles, due to their short formation time, they can be formed before than the parton get out the surface collision area. This is certainly true for central heavy ion collisions. In this case, the energy loss by the parton would be smaller, and thus the elliptic flow. As the elliptic flow for central collisions is small, the effect is difficult to be observed.

4.2 Elliptic flow scaling and energy loss

In Sec. 3.3, we discussed the quenching of low

Here the temperature,

We will take proportional to the product of the eccentricity of the overlap area
and

We expect that the elliptic flow were proportional to the strength of the quenching, so:

Using the dependence of ^{[154]} and taking ^{[157]}:

where we have choice, the scaling variable ^{[158]} and
ALICE ^{[159]} at different
centralities are shown versus the scaling function *b*=0.404, which is not
very different from 1/3. Taking into account the crude approximations done in
deriving the scaling formula Eq.(104), the result is very remarkable, confirming the quenching of
partons inside the overlap surface of the colliding objects.

5. Thermodynamics of string percolation

The thermodynamics of the string percolation can be addressed by extracting the
temperature from the transverse momentum distribution. We also can extract the
suppression factor ^{[113}^{,}^{112]}:

where ^{[160]}.

In Fig. 31, we plot the obtained energy density
over ^{[161]}:

where the mean free path is ^{[55]}, where

In Fig. 32, we show the behavior of ^{[18]}.

The arrows marks are the result of string percolation for Au-Au and Pb-Pb at RHIC and
LHC energies. Below

Moreover, the mean value of the trace of the energy momentum tensor

The minimum corresponds to the maximum of

On the other hand, it is possible to determine the speed of sound,

where

From the above equations, it is possible write

where

Another interesting thermodynamic variable which can be determined is the bulk
viscosity. Starting from ^{[162]}:

where

Note that this last expression depends on the sound speed, trace anomaly and entropy
density, which has already been computed in the string percolation context. In Fig. 36, we plot the bulk viscosity over the
entropy density as a function of the temperature, which has a maximum close to

6. Summary

The string percolation describes successfully most of the experimental data in the
soft region, namely, rapidity distributions, probability distributions of
multiplicities and transverse momentum, strength of BE correlations as a function of
multiplicities, forward-backward multiplicities as

The string percolation, although is not derived directly from QCD, has a clean
physical ground and it has the fundamental QCD feature. The non abelian character is
reflected in the coherent sum of the color fields which gives rise to an enhancement
of the mean

The collective behavior of the multiparticle production has its origin in the cluster
configuration formed in the initial state of the collisions, followed by the
interactions between the produced partons with the color fields, giving rise to
energy loss. Due to that, the elliptic flow satisfies an universal scaling law valid
for all centralities and energies. At low