# Area covered by disks in small-bounded continuum percolating systems: An
application to the string percolation model

J. E. Ramírez<EMAIL_ADDRESS>Departamento de Física de Partículas,
Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, España
Instituto Galego de Física de Altas Enerxías, Universidad de Santiago de
Compostela, E-15782 Santiago de Compostela, España  C. Pajares
<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad de
Santiago de Compostela, E-15782 Santiago de Compostela, España  Instituto
Galego de Física de Altas Enerxías, Universidad de Santiago de Compostela,
E-15782 Santiago de Compostela, España

###### Abstract

In string percolation model, the study of colliding systems at high energies
is based on a continuum percolation theory in two dimensions where the number
of strings distributed in the surface of interest is strongly determined by
the size and the energy of the colliding particles. It is also expected that
the surface where the disks are lying be finite, defining a system without
periodic boundary conditions. In this work, we report modifications to the
fraction of the area covered by disks in continuum percolating systems due to
a finite number of disks and bounded by different geometries: circle, ellipse,
triangle, square and pentagon, which correspond to the first Fourier modes of
the shape fluctuation of the initial state after the particle collision. We
find that the deviation of the fraction of area covered by disks from its
corresponding value in the thermodynamic limit satisfies a universal behavior,
where the free parameters depend on the density profile, number of disks and
the shape of the boundary. Consequently, it is also found that the color
suppression factor of the string percolation model is modified by a damping
function related to the small-bounded effects. Corrections to the temperature
and the speed of sound defined in string systems are also shown for small and
elliptically bounded systems.

##  1 Introduction

The string percolation model is an alternative description of the collective
behavior observed in nucleus-nucleus collisions due to the formation of quarks
and gluon matter which subsequently expands as a liquid with very small shear
viscosity over entropy ratio [1, 2, 3].

In string percolation multiparticle production is described in terms of color
strings stretched between the partons of the projectile and target. This
strings decays into $q-\bar{q}$ and $qq-\bar{q}\bar{q}$ pairs and subsequently
hadronize producing the observed hadrons. Due to confinement, the color of
these strings is confined to a small area, $a_{0}$ in the transverse plane.
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,
very similar to clusters of disks in two-dimensional continuum percolation
theory. At a critical density, a macroscopic cluster appears crossing the
transverse collision surface. The clusters of strings behave similarly to a
single string with a higher color field corresponding to add the individual
field of each string, implying a higher tension and thus a large mean
transverse momentum [4, 5, 6, 7, 8, 9, 10]. In the applications of string
percolation to describe RHIC and LHC data on AA collisions, plays an important
role the fraction of area covered by disks, which in the thermodynamic limit
is given by [11, 12]

$\phi_{\text{TL}}(\rho)=1-\exp(-\rho),$ (1)

where $\rho$ is the filling factor, defined by $\rho=Na_{0}/S$, where $N$ is
the number of disk, $a_{0}$ the area of one disks and $S$ the transverse area
where the disks are distributed. The number of disks (strings) formed in a
collision depends on the profile of the projectile and target, energy and
impact parameter (centrality degree) of the collision. In heavy-ion collisions
as Au-Au collisions at RHIC energies or Pb-Pb collisions at the LHC energies,
$N$ is very large, close to two thousands in Pb-Pb collisions and it is
expected to be close to the thermodynamical limit in a percolation approach.
However, this is not longer true in te case of pA or pp collisions, where $N$
is only a few tens for a minimum bias collisions [13, 14].

The experimental data on the azimuthal distributions of the momenta of the
produced particles and the corresponding Fourier coefficients, $v_{n}$, show a
collective behavior in pp and pA collisions very similar to the previously
observed in Au-Au and Pb-Pb collisions [15, 16, 17, 18, 19, 17, 20, 21, 22,
23, 24]. The studies of the collective behavior and the different harmonics in
string percolation usually assume to be close to the thermodynamic limit [13,
14, 25, 26], but it is well-known that the properties of percolating systems
(percolation threshold, cluster density) may be modified by topology, system
size and periodic boundary conditions [27, 28, 29, 30]. In this paper, we show
the modifications to the fraction of area covered by disks, due to the small
number of disks for systems free of periodic boundary conditions where the
surface $S$ takes the form of circle, ellipse, square, pentagon,$\dots$
corresponding to the different space eccentricities $e_{1}$, $e_{2}$, $e_{3}$,
$e_{4}$, $e_{5}$, $\dots$ The modification leads to corrections to the color
suppression factor $F(\rho)$ which is relevant in string percolation for
describing transverse momentum and multiplicity distributions.

The underlying physical grounds of the collective behavior produced in the
collisions of small systems like pp are under debate, in particular whether it
is originated by initial state effects or on the contrary by final state
interactions suitable to a hydrodynamical description. In this context it is
important to know whether the collective behavior occurs also for rather low
multiplicity in pp collisions. In both cases, it is crucial to know the
profile of the proton and its fluctuations to describe the experimental data.
In this line, it has been emphasized the proton shape fluctuations in the
explanation of the incoherent diffractive vector meson production [31, 32],
the importance of the core-corona proton profile to explain the collective
behavior seen in pp collisions [33] and the rather large edge and the
existence of correlated hot spots inside of the proton to explain the energy
evolution of the scattering amplitude of elastic pp collisions [34].

In this paper, we also study the effects of different profiles in pp
collisions in the framework of string percolation and the differences between
the evaluated observables in the thermodynamical limit and in the case of
rather small degrees of freedom (string) in pp collisions.

The paper is organized as follows. In Section 2, we describe the model to
generate bounded string percolating systems for a small number of disks. We
also present the algorithms to fill the system with three different density
profiles and the computation of the fraction of area covered by disks. In
Section 3, we report the modification of the fraction of area covered by disks
as a function of the number of disks, boundary shape and density profile. We
also diskuss the limit of high densities for small-bounded systems. Section 4
shows the modifications to the color suppression factor as a second damping
factor to the color field of the strings due to small-bounded string systems
and the analysis of the critical behavior of the speed of sound around the
critical temperature in small-bounded string percolating systems. Finally,
Section 5 contains the conclusions of this work.

##  2 Model

In this section, we present the algorithms to generate string systems bounded
by circles, ellipses, triangles, squares, and pentagons, considering different
density profiles. It is also presented the corresponding algorithm to
determine the fraction of area covered by disks in such systems.

###  2.1 Construction of geometric boundaries

The continuum percolation systems are defined as a collection of randomly
distributed and fully penetrable objects (disks, ellipses, rotated squares,
sticks, for example see Refs. [35, 36, 37]) on a flat surface $S$ with
periodic boundary conditions. Nevertheless, the requirements to define string
percolating systems (string number and the overlapping area) are addressed by
the parameters in high energy collisions: Size of colliding particles, energy
at the center of mass, impact parameter, etc. In particular, the string
percolating system is formed only by disks lying in the overlapping area,
being necessary a description of two-dimensional continuum percolation without
periodic boundary conditions.

A general description of two-dimensional continuum percolation is given
through the filling factor. In a system with fixed both the number of disks
and the shape of the boundary, one way to get variations in the filling factor
is changing the surface of the overlapping area. In this way the dimensions of
$S$ can be written as a function of $\rho$, $N$, and the geometry information.
For a boundary shape defined by a $n$-sides regular polygon, the length of the
sides is given by

$\rho=\frac{4N\pi r_{0}^{2}\tan(\pi/n)}{nl^{2}},$ (2)

where $r_{0}$ is the radio of the small disk and $l$ is the length of the
sides of the regular polygon. Note that in the above expression, for a given
number of disks $N$ and density one can determine the dimension of the regular
polygon of $n$ sides which confines the system by the following relation

$l=\sqrt{\frac{4N\pi\tan(\pi/n)}{n\rho}}r_{0}.$ (3)

In the limit $n\to\infty$, we found that the expression in Eq. (3) takes the
form $l\to 0$, which is in agreement with the boundary circle limit, i. e., in
this case an $n$-sides regular polygon looks like a circle. Moreover, the
semi-axes for an elliptic boundary are given by

$\displaystyle A^{2}=\frac{r_{0}^{2}N}{\rho\sqrt{1-\varepsilon^{2}}},$
$\displaystyle B^{2}=\frac{r_{0}^{2}N\sqrt{1-\varepsilon^{2}}}{\rho},$ (4)

where $A$ and $B$ are the major and the minor semi-axes, respectively, and
$\varepsilon$ is the eccentricity (this case has been recently diskussed in
Ref. [38]). The particular case $\varepsilon=0$ corresponds to a circular
boundary, which is the classical continuum percolation model bounded by a
circle, already studied by several authors [39, 10, 40, 41].

Once determined the dimensions of the confinement surface, we take $N$ random
points distributed according to a particular density profile, representing the
center of the disks and those should be at least a distance $r_{0}$ from any
border, so the disks must be embedded entirely in the overlapping area.

###  2.2 Density profiles

Usually, in the studies of continuum percolation the considered disks are
uniformly distributed. However, the nuclear profiles function considered in
heavy-ion collisions are more realistic. The most important feature of these
density profiles is that they are denser in the center of the overlapping area
and more dilute as we go away from the central region [42, 43]. One way to
simulate the density profile is generating random points according to a
Gaussian distribution function, defined as follow:

$f(x,y)=\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp\left[-\frac{1}{2}\left(\frac{(x-x_{0})^{2}}{\sigma_{x}^{2}}+\frac{(y-y_{0})^{2}}{\sigma_{y}^{2}}\right)\right],$
(5)

where the vector $(x_{0},y_{0})$ is the center of the geometric boundary shape
and we have considered non-correlated distribution over the axes $x$ and $y$.
Note that there is not a unique way to define the values of the standard
deviation. For regular polygons we choice $\sigma_{x}=\sigma_{y}=\sigma$,
taking the following values

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle R_{I},$ (6a)
$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle R_{I}/2^{1/2},$ (6b)

where $R_{I}$ is the radius of the inscribed circle in the bounded region.
However, for the elliptical shape, we use different dispersion along the semi-
axes:

$\displaystyle\sigma_{x}=(A-r_{0}),$ $\displaystyle\sigma_{y}=(B-r_{0}),$ (7a)
$\displaystyle\sigma_{x}=(A-r_{0})/2^{1/2},$
$\displaystyle\sigma_{y}=(B-r_{0})/2^{1/2}.$ (7b)

We denote by 1S and SQ the profiles defined by the standard deviation
(6a)-(7a) and (6b)-(7b) respectively. The case of disks uniformly distributed
is denoted by U. The cases U and SQ in the thermodynamical limit have been
used by different groups to study high energy colliding systems. Figures 1 and
2 show examples of the three different cases.

Figure 1: Samples of string percolating system at $\rho=0.7$ for small number
of disks ($N=13$), different density profile (columns) and boundary shape
condition (rows). Figure 2: Samples of high populated disk percolating system
($N=500$) at $\rho=1.1$ for different density profile (columns) and boundary
shape condition (rows).

###  2.3 Measurement of the area covered by disks

Figure 3: Scheme of the Monte Carlo integration by cell to measure the
fraction of area covered by disks. We count only points lying in disks on the
cell or any of the nearest and next-to-nearest cells (black crosses).

In order to determine the fraction of area covered by disks in string systems,
we generate percolating systems according to the boundary and density profiles
described in the Sec. 2.1 and Sec. 2.2. Below we draw an imaginary square
lattice with spacing $2r_{0}$ centered in the geometric center of the
boundary, which allows us to map the string system into a matrix. The disk
centers lying in the $i$, $j$ cell are stored in the $i$, $j$ entry. We then
determine the fraction of the area covered by disks in each cell by Monte
Carlo integration. To do this, we generate random pairs $(x,y)$ uniformly
distributed in the cell and counting the number points whose distance to any
disks be less than $r_{0}$ in the cell or the nearest and the next-to-nearest-
neighbor cells, as is depicted in Fig. 3, where the black crosses are the
points lying in a disk (and red crosses are not). This process is repeated for
all cell in the lattice. Finally, the total fraction of area covered by disks
is determined as $4r_{0}^{2}\mathcal{N}_{\text{in}}/\mathcal{N}S$, where
$\mathcal{N}_{\text{in}}$ is the total number of points laying in a disk of
the total $\mathcal{N}$ generated points.

##  3 Results

In this section, we present the simulation result about the fraction of the
area covered by disks for small-bounded string systems for the density
profiles described in Sec. 2.

The fraction of area covered by disks is determined by averaging the area
covered by disks over $2\times 10^{5}$ realizations at $N$, $\rho$, and
boundary shape fixed. Additionally, the Monte Carlo integration was performed
generating $10^{4}$ points per cell. For each value of $N$=13, 55, 96 and 500,
and each geometry, we measured the fraction of area covered by disks as a
function of the filling factor, starting in $\rho=0.2$ until $\rho=2.8$, with
increments of $\Delta\rho=0.2$. This procedure is repeated for all three
density profiles. In Fig. 4, we show the explicit dependence of the fraction
of area covered by disks on the number of disks, the boundary shape, and the
density profile for string systems. Note that the fraction of area covered by
disks in the uniform density profile recovers its behavior at the
thermodynamic limit as $N$ increases and it becomes independent on the
boundary shape, as is shown in Fig. 4 a). This is expected because in the
thermodynamic limit there is no difference between large circles and ellipses
or regular polygons.

Figure 4: Fraction of the area covered by disks in small continuum percolating
systems for $N=$13 (purple), 96 (green) and 500 (cyan), and bounded by circles
(circles), ellipses (crosses and stars), triangles (triangles), squares
(squares) and pentagons (pentagons). The systems where filled according to
three different density profiles: a) U, b) 1S and c) SQ. Dashed lines and dot-
dash line are the fraction of area covered by disks in the thermodynamic limit
for the U and SQ density profiles, respectively. Figure 5: Deviation of the
fraction of area covered by disks in small-bounded system from its value in
the thermodynamic limit for the uniform density profile for systems with
$N=$13 (purple), 96 (green) and 500 (cyan), and bounded by circles (circles),
ellipses (crosses and stars), triangles (triangles), squares (squares) and
pentagons (pentagons), and considering different density profiles: a) U, b) 1S
and c) SQ.

In order to analyze the data, it is convenient to define the deviation of the
fraction of the area covered by disks from the thermodynamic limit

$\Delta\phi(\rho)=\phi_{\text{TL}}(\rho)-\phi(\rho),$ (8)

where $\phi_{\text{TL}}$ is the thermodynamical limit value given by Eq. (1)
and $\phi$ is the evaluated value by simulation. Note that $\Delta\phi$ takes
a sigmoid form and from $\phi$, it inherits all dependencies on the number of
disks, boundary shape, and density profile, as we shown in Fig 5. The data of
$\Delta\phi$ is well fitted by

$\Delta\phi(\rho)=a\left[1+\tanh\left(\frac{\rho-\rho^{*}}{\Delta
L}\right)\right],$ (9)

where $a$ corresponds to the value where $\Delta\phi$ reaches a plateau
behavior, $\Delta L$ is the width of the sigmoid transition, and $\rho^{*}$ is
the filling factor value where $\Delta\phi(\rho^{*})=a/2$. In this way, we
found the modified fraction of area covered by disks in small-bounded systems
can be written as

$\phi(\rho)=1-\exp(-\rho)-a\left[1+\tanh\left(\frac{\rho-\rho^{*}}{\Delta
L}\right)\right],$ (10)

where the fit parameters $a$, $\Delta L$ and $\rho^{*}$ take particular values
for each number of disks, boundary shape and density profile, which are
summarized in Table 1. It is necessary to remark that in the range of analyzed
values of the filling factor we do not found a substantial difference in the
fraction of area covered by disks between circles and ellipses independently
on the number of disks and density profile. This last fact it is clearly
appreciate in Fig. 4 and Fig. 5. As is expected, $\Delta\phi$ vanishes as $N$
increases, which naturally corresponds to the fraction of area covered by
disks in the thermodynamic limit for the uniform density profile. It is worth
to mention the fraction of area covered by disks in the Gaussian density
profile significatively deviates from its value in the thermodynamic limit,
however, for very populated systems ($N\sim 5000$), that functional structure
is (dash-dot line in Fig. 4 c)), given by

$\phi^{\text{Gauss}}_{\text{TL}}(\rho)=\frac{1}{1+a^{\text{Gauss}}\exp(-(\rho-\rho_{c})/b^{\text{Gauss}})},$
(11)

where $\rho_{c}\sim 1.5$ is the percolation threshold for the Gaussian density
profile [42]. The parameters $a^{\text{Gauss}}=1.5$ and $b^{\text{Gauss}}$
(0.35 for pp and pA collisions, and 0.75 for AA collisions) depend on the
profile function [44]. In particular $b^{\text{Gauss}}$ controls the ratio
between the width of the border of the profile and the total area [4].

Table 1: Fit parameters of $\Delta\phi$ (Eq. (9)).

|  |  | $a$ |  |  | $\rho^{*}$ |  |  | $\Delta L$ |
---|---|---|---|---|---|---|---|---|---|---
$N$ | Geom. | U | 1S | SQ | U | 1S | SQ | U | 1S | SQ
13 | e2 | 0.108 $\pm$0.002 | 0.117 $\pm$0.002 | 0.129 $\pm$0.002 | 1.20 $\pm$0.02 | 1.12 $\pm$0.02 | 1.07 $\pm$0.02 | 0.70 $\pm$0.04 | 0.68 $\pm$0.03 | 0.68 $\pm$0.04
| e3 | 0.158 $\pm$0.003 | 0.182 $\pm$0.004 | 0.202 $\pm$0.004 | 1.25 $\pm$0.03 | 1.13 $\pm$0.03 | 1.03 $\pm$0.03 | 0.75 $\pm$0.05 | 0.75 $\pm$0.05 | 0.73 $\pm$0.05
| e4 | 0.133 $\pm$0.002 | 0.147 $\pm$0.003 | 0.161 $\pm$0.003 | 1.23 $\pm$0.03 | 1.14 $\pm$0.03 | 1.07 $\pm$0.02 | 0.73 $\pm$0.04 | 0.72 $\pm$0.05 | 0.70 $\pm$0.04
| e5 | 0.124 $\pm$0.002 | 0.131 $\pm$0.002 | 0.139 $\pm$0.003 | 1.22 $\pm$0.02 | 1.17 $\pm$0.02 | 1.12 $\pm$0.03 | 0.71 $\pm$0.03 | 0.71 $\pm$0.04 | 0.71 $\pm$0.05
55 | e2 | 0.079 $\pm$0.002 | 0.091 $\pm$0.002 | 0.104 $\pm$0.002 | 1.42 $\pm$0.03 | 1.37 $\pm$0.03 | 1.30 $\pm$0.03 | 0.87 $\pm$0.05 | 0.87 $\pm$0.05 | 0.85 $\pm$0.05
| e3 | 0.103 $\pm$0.003 | 0.135 $\pm$0.003 | 0.165 $\pm$0.004 | 1.43 $\pm$0.04 | 1.30 $\pm$0.03 | 1.18 $\pm$0.03 | 0.86 $\pm$0.05 | 0.84 $\pm$0.05 | 0.83 $\pm$0.06
| e4 | 0.092 $\pm$0.003 | 0.108 $\pm$0.002 | 0.127 $\pm$0.003 | 1.45 $\pm$0.04 | 1.36 $\pm$0.03 | 1.25 $\pm$0.03 | 0.88 $\pm$0.06 | 0.84 $\pm$0.05 | 0.83 $\pm$0.05
| e5 | 0.086 $\pm$0.002 | 0.096 $\pm$0.002 | 0.106 $\pm$0.003 | 1.44 $\pm$0.04 | 1.40 $\pm$0.03 | 1.34 $\pm$0.04 | 0.86 $\pm$0.05 | 0.84 $\pm$0.05 | 0.84 $\pm$0.05
96 | e2 | 0.066 $\pm$0.002 | 0.075 $\pm$0.002 | 0.088 $\pm$0.002 | 1.50 $\pm$0.04 | 1.42 $\pm$0.03 | 1.33 $\pm$0.03 | 0.94 $\pm$0.05 | 0.90 $\pm$0.05 | 0.89 $\pm$0.05
| e3 | 0.084 $\pm$0.002 | 0.113 $\pm$0.003 | 0.145 $\pm$0.003 | 1.48 $\pm$0.04 | 1.33 $\pm$0.04 | 1.19 $\pm$0.03 | 0.88 $\pm$0.06 | 0.87 $\pm$0.06 | 0.84 $\pm$0.06
| e4 | 0.073 $\pm$0.002 | 0.091 $\pm$0.002 | 0.109 $\pm$0.003 | 1.48 $\pm$0.04 | 1.41 $\pm$0.04 | 1.28 $\pm$0.04 | 0.88 $\pm$0.06 | 0.88 $\pm$0.06 | 0.86 $\pm$0.06
| e5 | 0.070 $\pm$0.002 | 0.079 $\pm$0.002 | 0.088 $\pm$0.002 | 1.49 $\pm$0.04 | 1.45 $\pm$0.04 | 1.38 $\pm$0.04 | 0.88 $\pm$0.06 | 0.89 $\pm$0.06 | 0.87 $\pm$0.05
500 | e2 | 0.034 $\pm$0.001 | 0.040 $\pm$0.001 | 0.050 $\pm$0.001 | 1.66 $\pm$0.05 | 1.49 $\pm$0.04 | 1.32 $\pm$0.04 | 1.14 $\pm$0.06 | 1.03 $\pm$0.05 | 0.99 $\pm$0.07
| e3 | 0.040 $\pm$0.001 | 0.064 $\pm$0.002 | 0.096 $\pm$0.002 | 1.55 $\pm$0.05 | 1.32 $\pm$0.03 | 1.14 $\pm$0.03 | 0.92 $\pm$0.06 | 0.88 $\pm$0.05 | 0.83 $\pm$0.06
| e4 | 0.036 $\pm$0.001 | 0.046 $\pm$0.001 | 0.063 $\pm$0.001 | 1.59 $\pm$0.05 | 1.40 $\pm$0.04 | 1.22 $\pm$0.03 | 0.95 $\pm$0.07 | 0.89 $\pm$0.05 | 0.86 $\pm$0.06
| e5 | 0.034 $\pm$0.001 | 0.040 $\pm$0.001 | 0.047 $\pm$0.001 | 1.57 $\pm$0.04 | 1.53 $\pm$0.05 | 1.39 $\pm$0.04 | 0.93 $\pm$0.06 | 0.94 $\pm$0.06 | 0.92 $\pm$0.06

Moreover, for a percolating system with $N$ fixed, there is a maximum value of
filling factor that the system can reach. This occurs when the boundary is
small enough to concentrate all disks center on the same point, which
coincides with the geometrical center of the boundary. At this point, the
filling factor is

$\rho_{\text{max}}=\left\\{\begin{array}[]{cc}N\sqrt{1-\varepsilon^{2}}&\text{for
ellipses,}\\\ \frac{\pi N}{n\tan(\pi/n)}&\text{for
polygons.}\end{array}\right.$ (12)

When the percolating system is filled at $\rho_{\text{max}}$, both the minor
axis of an ellipse or the apothem of a polygon takes the value $r_{0}$. Then,
in the limit of high densities, the fraction of area covered by disks reach an
asymptotic value, which corresponds to the packing fraction given by
$\rho_{\text{max}}/N\leq 1$. Figure 6 shows the packing fraction as a function
of $\varepsilon$ and $\pi/n$ for ellipses and polygons, respectively. Note
that in the limit $\varepsilon\to 0$ and $n\to\infty$, the fraction of area
covered by disks goes to one, which is in full agreement with the circular
boundary.

Figure 6: Fraction of the area covered by disks of small-bounded continuum
percolating systems at $\rho_{\text{max}}$ for ellipses (red line) as a
function of the eccentricity and polygons (blue dotted line) as a function of
$\pi/n$.

##  4 Application to string percolation model

In the string percolation model, the number of string and therefore the
filling factor of its percolating picture depends strongly on the nature of
the particles in the collision as well as the energy in the center of mass.
For example, in pp collisions, a rise in the string number is expected as the
energy of the center of mass increase. Then, the disks start to overlap
forming clusters. According to the summation rules of the color field, the
color of the cluster is different from those single strings. In the following
subsection, we estimate the modification of the color field of a cluster of
overlapped disks.

Figure 7: Scheme of the area covered by two overlapped strings.

###  4.1 Color field of overlapped strings and the color suppression factor

We start this diskussion assuming that the emission of $q\bar{q}$ pairs from
the cluster of strings proceeds independently and it is governed by the
strength of the color field of the cluster. Evidently, this cluster has not
only different color field than the individual strings that shape it but also
different transverse area. First, it is necessary to analyze the case of an
isolated string. In this case, a single string with transverse area $S_{1}$
emits partons with transverse momentum distribution

$\frac{d\sigma}{dyd^{2}p}=C\exp\left(-\frac{m_{T}^{2}(p_{T})}{t_{1}}\right),$
(13)

where $t_{1}$ is the tension of the string and $m_{T}^{2}=m^{2}+p_{T}^{2}$,
being $p_{T}$ and $m$ the transverse momentum and mass of the emitted parton.
The tension, according to the Schwinger mechanism, is proportional to the
field, and thus to the color charge [45, 46], which we denote by $Q_{0}$. The
mean square transverse momentum is $\langle p_{T}^{2}\rangle_{1}=t_{1}$ and
proportional to $Q_{0}$. Then the density of color charge is $q=Q_{0}/S_{1}$.
We denote the mean multiplicity of produced particles by a single string per
unit of rapidity as $\mu_{1}$ which is also proportional to the color charge.
Now, we compute the case of two overlapped strings. In this scenary, it is
expected that the total production of charged particles comes from the three
independent regions $S^{(1)}$, $S^{(2)}$ (overlapping region) and $S^{(3)}$
depicted in Fig. 7. Note that the regions are not independent and can be
related as follows $S^{(3)}=S^{(1)}=S_{1}-S^{(2)}$. Then, the color in each of
the non overlapping areas will be $Q_{1}=Q_{0}S^{(1)}/S_{1}$. On the other
hand, the total color in the overlap area $Q_{2}$ will be a vector sum of the
two overlapping colors $qS^{(2)}$. In this summation the total color squared
should be conserved [47]. Thus
$Q_{2}^{2}=(\mathbf{Q}_{ov}+\mathbf{Q}_{ov}^{\prime})^{2}$, where
$\mathbf{Q}_{ov}$ and $\mathbf{Q}_{ov}^{\prime}$ are the two vector colors in
the overlap area. Since the colors in the two strings may generally be
oriented in arbitrary directions respective and independent to one another,
the average $\langle\mathbf{Q}_{ov}\mathbf{Q}_{ov}^{\prime}\rangle$ is zero,
which leads to $Q_{2}=\sqrt{2}qS^{(2)}=\sqrt{2}Q_{0}S^{(2)}/S_{1}$ [12].
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. Thus, assuming independent emission
from the three regions of Fig. 7, we obtain for the multiplicity weighted by
the multiplicity for a single string ($\mu_{1}$)

$\mu/\mu_{1}=2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1}),$ (14)

and for the mean square transverse momentum (we divide the total transverse
momentum by the multiplicity)

$\displaystyle\frac{\langle p_{T}^{2}\rangle}{\langle p_{T}^{2}\rangle_{1}}$
$\displaystyle=$
$\displaystyle\frac{2(S^{(1)}/S_{1})+\sqrt{2}\sqrt{2}(S^{(2)}/S_{1})}{2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1})}$
(15) $\displaystyle=$
$\displaystyle\frac{2}{2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1})},$

where we have used the property $S^{(1)}+S^{(2)}=S_{1}$. Generalizing to any
number $N$ of overlapping strings, we have

$\displaystyle\frac{\mu}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1}),$ (16)
$\displaystyle\frac{\langle p_{T}^{2}\rangle}{\langle p_{T}^{2}\rangle_{1}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{i}(S^{(i)}/S_{1})}{\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1})}=\frac{N}{\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1})},$
(17)

where the sum runs over all individual overlaps of $n_{i}$ strings having
areas $S^{(i)}$. We have used the identity $\sum_{i}S^{(i)}=NS_{1}$. These
equations are not easy to apply because we have to identify all individuals
overlaps of any number of strings with their areas. However one can avoid
these difficulties realizing that one can combine all terms with a given
number of overlapping strings $n_{i}=n$ into a single term, which sums all
such overlaps into a total area of exactly $n$ overlapping strings
$S_{n}^{Tot}$. Then, one can write

$\displaystyle\frac{\mu}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\sqrt{n}(S_{n}^{Tot}/S_{1}),$ (18)
$\displaystyle\frac{\langle p_{T}^{2}\rangle}{\langle p_{T}^{2}\rangle_{1}}$
$\displaystyle=$
$\displaystyle\frac{N}{\sum_{n=1}^{N}\sqrt{n}(S_{n}^{Tot}/S_{1})}.$ (19)

The total area can be determined in the thermodynamic limit. One finds that
the distribution of overlap strings over the total surface $S$ in the variable
$n$ is Poissonian with mean $\rho=NS_{1}/S$, which corresponds to the filling
factor in the percolation context. Therefore, the fraction of the total area
covered by strings will be $1-\exp(-\rho)$, which matches with the
corresponding fraction of area covered by disks in two-dimensional continuum
percolation in Eq. (1). Note that the multiplicity in Eq. (17) is damped by a
factor

$F_{\text{TL}}(\rho)=\sqrt{\frac{\phi_{\text{TL}}(\rho)}{\rho}},$ (20)

which can be interpreted as a factor that suppress the total color charge of
overlapped strings. In the following, we name $F(\rho)$ as the color
suppression factor. Finally, we can write for the mean values of the
multiplicity and the square transverse momentum as [4, 5, 12]

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle
NF_{\text{TL}}(\rho)\mu_{1},$ (21) $\displaystyle\langle p_{T}^{2}\rangle$
$\displaystyle=$ $\displaystyle\langle
p_{T}^{2}\rangle_{1}/F_{\text{TL}}(\rho).$ (22)

Note that both the multiplicity and the transverse momentum distributions
depend explicitly on the color suppression factor, and therefore in the
fraction of area covered by disks. Then any modification from any source in
the latter leads corrections in the phenomenological description of the
experimental results in high energy physics.

###  4.2 Modifications to the color suppression factor

Figure 8: Comparison between the color suppression factor computed from the
simulation data (figures) and the computed from the fit function (solid lines)
on Eq. (9) for small-bounded string systems for $N=$13, and bounded by circles
(circles), ellipses (crosses and stars), triangles (triangles), squares
(squares) and pentagons (pentagons), and for the a) U, b) 1S and c) SQ density
profiles. Dashed lines and dash-dot line are the corresponding color
suppression factor in the thermodynamic limit for the uniform and SQ density
profiles. Figure 9: Damping factor due to small-bounded effects in string
systems for $N=13$ all shapes of the boundary e2 (purple line), e3 (green
line), e4 (blue line), and e5 (red line), for the density profiles: a) U, b)
1S, and c) SQ.

As we diskussed in Sec. 3, the fraction of area covered by disks deviates from
$\phi_{\text{TL}}$ due to the size and the boundary shape of the percolating
system. Assuming the same arguments to deduce the color suppression factor, we
propose change $\phi\to\phi_{\text{TL}}$ in Eq. (20), thus

$F(\rho)=\sqrt{\frac{\phi(\rho)}{\rho}},$ (23)

is the color suppression factor for small-bounded systems. The latter can be
expressed in terms of $\Delta\phi$, $\phi_{\text{TL}}$ and $F_{\text{TL}}$ as
follows

$F(\rho)=F_{\text{TL}}(\rho)\left(1-\frac{\Delta\phi(\rho)}{\phi_{\text{TL}}(\rho)}\right)^{1/2},$
(24)

where the second factor in the right-hand side may be interpreted as a damping
function of the color field due to small-bounded effects, which becomes
relevant for small percolating systems. For example, in the case of $N=$13,
where the maximum deviation of the area covered is observed, the data of the
color suppression factor is well fitted by the expression in (24) for
$0.4<\rho<3$, as we shown in Fig. 8. It is remarkable since the most
interesting phenomena in high energy physics are observed in systems with
densities in the range $0.5<\rho<2$ for the pp case. We also show in Fig. 9
the explicit dependence of this damping function on the filling factor for all
shapes of the boundary and all density profiles for string system with number
of disks $N=13$. The value $N=$13 is close to the values found for minimum
bias in pp collisions at $\sqrt{s}=$7TeV. The deviation of the color
suppression factor from its value in the thermodynamic limit is given by

$\frac{F_{\text{TL}}(\rho)-F(\rho)}{F_{\text{TL}}(\rho)}=1-\left(1-\frac{\Delta\phi(\rho)}{\phi_{\text{TL}}(\rho)}\right)^{1/2}.$
(25)

In both Fig. 8 and Fig. 9, we have obviated the error bars from the propagated
error since the relative error due to the fit parameter in $\Delta\phi$ are of
the order of 0.3%, which are technically unperceived.

Due to the form of $\Delta\phi$, the relative deviation starts at a minimum
value. Then, for $\rho^{*}-L/2<\rho<\rho^{*}+L/2$, it increases linearly until
reaching a maximum value, where it will hold constantly for
$\rho^{*}+L/2<\rho<3$. In the region of high densities, the percentage of the
deviation takes values around 11%-19% for string systems with disks uniformly
distributed. However, in the case of the Gaussian density profile, where the
disks are more concentrated in the geometrical center of the boundary, this
percentage becomes higher, taking values around 15%-25%, for ellipses and
triangular boundaries, respectively.

Figure 10: Damping factor due to small-bounded string systems for the color
suppression factor in the SQ density profile for a) $N=$13, b) $N=$55 y c)
$N=$96 bounded by circles and ellipses (purple line), triangles (red line),
squares (cyan line) and pentagons (green line).

On the other hand, it is possible to determine the corresponding deviation of
the fraction of area covered by disks in the Gaussian density profile from its
value in the thermodynamic limit for the same Gaussian profile in Eq. (11)
using the data for $a^{\text{Gauss}}$, $b^{\text{Gauss}}$ and $\rho_{c}$
reported in the literature for pp and pA collisions. In this case, we analyze
only the systems constituted by $N$=13, 55 and 96 strings. In Fig. 4 c) we
plot the thermodynamical limit for both uniform (dashed line) and Gaussian
(dash-dot line) density profiles together the simulation data.

Similarly as we have determined the deviation of the fraction of area covered
by disks for $\phi$ from $\phi_{\text{TL}}$, we define

$\Delta\phi^{\text{Gauss}}(\rho)=\phi^{\text{Gauss}}_{\text{TL}}(\rho)-\phi(\rho),$
(26)

where $\phi$ is the fraction of area covered by disks measured by simulation
and $\phi^{\text{Gauss}}_{\text{TL}}$ is its corresponding value in the
thermodynamic limit for the Gaussian profile in Eq. (11). As $\phi$ is well
characterized by Eq. (10) as a function of $\phi_{\text{TL}}$ and
$\Delta\phi$, and its corresponding fit parameters are also determined (see
Table 1), then $\Delta\phi^{\text{Gauss}}$ is well determined and it fits
correctly the simulation data. In this way, we re-write the color suppression
factor as follow

$F(\rho)=F^{\text{Gauss}}_{\text{TL}}(\rho)\left(1-\frac{\Delta\phi^{\text{Gauss}}(\rho)}{\phi^{\text{Gauss}}_{\text{TL}}(\rho)}\right)^{1/2},$
(27)

where $F^{\text{Gauss}}_{\text{TL}}$, as usual in the string percolation
model, is computed by replacing $\phi_{\text{TL}}^{\text{Gauss}}$ instead
$\phi_{\text{TL}}$ in Eq. (20). This last expression has the same structure
than the color suppression factor in Eq. (24), where it has been replaced the
functions $F_{\text{TL}}\to F^{\text{Gauss}}_{\text{TL}}$,
$\Delta\phi\to\Delta\phi^{\text{Gauss}}$ and
$\phi_{\text{TL}}\to\phi^{\text{Gauss}}_{\text{TL}}$ defined for the Gaussian
density profile. In the same way as in Eq. (24), the second factor in the
right-hand side can be interpreted as a second damping factor due to small-
bounded effects. Substituting the explicit form of
$\Delta\phi^{\text{Gauss}}_{\text{TL}}$, we found that the damping factor is
re-written as $(\phi/\phi^{\text{Gauss}}_{TL})^{1/2}$, which is plotted in
Fig. 10. In this way, the deviation of $F$ from $F^{\text{Gauss}}_{TL}$ is
given by

$\frac{F^{\text{Gauss}}_{\text{TL}}(\rho)-F(\rho)}{F^{\text{Gauss}}_{\text{TL}}(\rho)}=1-\left(\frac{\phi(\rho)}{\phi^{\text{Gauss}}_{\text{TL}}(\rho)}\right)^{1/2},$
(28)

which is well defined through known functions. Notice that in the case of low-
density systems ($\rho<1$), the relative deviation is large, taking values
between 100%-150%. However, for $1<\rho<2$, this deviation becomes smaller as
a crossover effect of $F^{\text{Gauss}}_{\text{TL}}$ and the simulation data.
Finally, for high density percolating systems, the deviation reaches a plateau
region, and it becomes very similar to the case of the uniform density profile
since for high-density systems there are not substantial differences between
$F_{\text{TL}}$ and $F^{\text{Gauss}}_{\text{TL}}$, as can be seen in Fig. 8
c).

###  4.3 Color suppression factor at $\rho_{\text{max}}$

As we mentioned in Sec. 3, the filling factor is physically upper bounded in
small-bounded percolating systems and the corresponding fraction of the area
covered is given by the packing fraction. Then, substituting in Eq. (20), we
found the color suppression factor at the maximum filling factor is

$F(\rho_{\text{max}})=\frac{1}{\sqrt{N}}.$ (29)

Note that the color suppression factor at $\rho_{\text{max}}$ is a function
depending only on the number of strings in the system.

###  4.4 Temperature in string percolating systems

It is well-known that the distribution of transverse momentum of the charged
particle produced in high energy physics experiments can be determined by
first principles. According to the Schwinger mechanism for the particle
production, the distribution for massless particles expressed in terms of
$p_{T}^{2}$ and the string tension $x$ is given by

$\frac{dn}{dp_{T}^{2}}\sim\exp\left(-\pi\frac{p_{T}^{2}}{x^{2}}\right).$ (30)

The superposition of clusters with different number of strings each satisfying
Eq. (30) gives rise to the transverse momentum distribution (the cluster size
distribution is a gamma function)

$\frac{dn}{dp_{T}^{2}}\sim\left(1+\frac{p_{T}^{2}}{k\langle
p_{T}^{2}\rangle}\right)^{-k}=\left(1+\frac{F_{\text{TL}}(\rho)p_{T}^{2}}{k\langle
p_{T}^{2}\rangle_{1}}\right)^{-k}$ (31)

which describes rightly the pp experimental data. At low $p_{T}$, the Eq. 31
becomes

$\frac{dn}{dp_{T}^{2}}\sim\exp\left(-\frac{F_{\text{TL}}(\rho)p_{T}^{2}}{\langle
p_{T}^{2}\rangle_{1}}\right).$ (32)

On the other hand, inside each cluster, the chromoelectric field is not
constant and then the tension fluctuates around its mean value. This
fluctuation is related to the stochastic picture of the QCD vacuum. Since the
average value of the color field strength must vanish, it can not be constant
but change randomly from point to point. Such fluctuations lead to a Gaussian
distribution of the string tension [48]. Then, as

$\displaystyle\frac{dn}{dp_{T}^{2}}$ $\displaystyle\sim$
$\displaystyle\sqrt{\frac{2}{\langle
x^{2}\rangle}}\int_{0}^{\infty}dx\exp\left(-\frac{x^{2}}{2\langle
x^{2}\rangle}\right)\exp\left(-\pi\frac{p_{T}^{2}}{x^{2}}\right)$ (33)
$\displaystyle\sim$ $\displaystyle\exp\left(-p_{T}\sqrt{\frac{2\pi}{\langle
x^{2}\rangle}}\right),$

we obtain a thermal distribution. Using Eq. (33), the distribution in Eq. (32)
becomes

$\frac{dn}{dp_{T}^{2}}\sim\exp\left(-\frac{p_{T}}{T_{\text{TL}}(\rho)}\right),$
(34)

with

$T_{\text{TL}}(\rho)=\sqrt{\frac{\langle
p_{T}^{2}\rangle_{1}}{2F_{\text{TL}}(\rho)}}.$ (35)

Therefore in the string percolation framework, the fluctuations of the string
tension give rise to the thermal distribution of the transverse momentum with
a temperature which can be interpreted as the temperature of the initial state
of the system [49].

If we take as the phase transition temperature the experimentally determined
chemical freeze-out temperature $T_{c}=167.7\pm 2.6$ MeV for the percolation
threshold $\rho_{c}=1.2$, $F_{\text{TL}}(\rho_{c})=0.66$ and using Eq. (35) we
obtain $\sqrt{\langle p_{T}^{2}\rangle_{1}}=207.2\pm 3.3$MeV, which is close
to the value 200MeV used in the phenomenological applications [49].

Taking into account that the color suppression factor is damped for small-
bounded systems, this temperature is also corrected by the damping factor
$(1-\Delta\phi/\phi_{\text{TL}})^{-1/4}$. Moreover, at $\rho_{\text{max}}$,
the behavior of the temperature as a function of the number of string goes as
$T({\rho_{\text{max}}})\propto N^{1/4}$, and becomes independent of the
geometry and the density profile. Note that Eq. (35) allows us to define a
critical temperature $T_{\text{c}}$ associated with the percolation threshold
as $T_{\text{c}}=T(\rho_{\text{c}})$. In Sec. 4.5, we use this temperature
definition to analyze the small-bounded effects in the speed of sound (in the
high energy physics context) through the modifications to the color
suppression factor considering the percolation threshold for small and
elliptically bounded systems reported in Ref. [38].

###  4.5 Critical behavior of the speed of sound

Figure 11: Speed of the sound for small string systems bounded by ellipses for
$N=$13 (purple), 55 (green), 96 (yellow) and 500 (cyan), for different density
profiles: a) U and b) SQ. Solid lines correspond to the speed of sound in the
thermodynamic limit for the U (black) and Sq (blue) density profiles. Red
dashed line is the lattice QCD prediction.

The most important feature of the string percolation model is its ability to
predict several observables in the high energy physics for both theoretical
and experimental results. A completed review on the string percolation
applications can be found in Refs. [4, 5]. Particularly, coupling this
framework with the 1D Bjorken expansion, it is possible to compute the speed
of sound as [49, 4]

$c_{s}^{2}=\frac{1}{3}\left(\frac{\exp(-\rho)}{F_{\text{TL}}^{2}}-1\right)\left(0.0191\frac{1}{\rho
F_{\text{TL}}^{2}}\left(\frac{\eta}{s}\right)_{\text{TL}}^{-1}-1\right),$ (36)

where $\eta$ is the shear viscosity and $s$ is the entropy density, which in
the context of string percolation (in the thermodynamic limit) is given by

$\left(\frac{\eta}{s}\right)_{\text{TL}}=\frac{T_{\text{TL}}L}{5\phi_{\text{TL}}},$
(37)

where $L=1$fm is the longitudinal extension of the string. In the following,
we replace $F_{\text{TL}}\to F$ and $\phi_{\text{TL}}\to\phi$ in order to
include the modifications due to small-bounded effects in the speed of sound.
We have obviated the explicit dependence on $\rho$ in order to improve
notation. We also define the order parameter

$\tau=\frac{T}{T_{\text{c}}}-1,$ (38)

for $T>T_{\text{c}}$, where $T_{\text{c}}$ is the critical temperature, which
is computed as the evaluation of $T$ in the percolation threshold. As far as
we know, the percolation threshold has been reported for both U and SQ density
profiles in the thermodynamic limit (see Refs. [50, 51, 37, 42]) and small and
elliptical bounded systems considering several density profiles in Ref. [38].
In Fig. 11, we show the behavior of the speed of sound as a function on the
number of disks for the Uniform (Fig. 11 a)) and SQ (Fig. 11 b)) and
elliptically bounded string systems considering the eccentricities
$\varepsilon=$0.0, 0.5 and 0.9 together its corresponding function in the
thermodynamic limit and the lattice QCD prediction. For $N$ fixed, we observe
a smooth dependence of $c_{s}^{2}$ on the eccentricity even though the
fraction of area covered by disks does not show differences in any cases of
ellipses analyzed. This is due to the percolation threshold shows an explicit
dependence on the eccentricity for small systems [38].

Moreover, it is possible to determine the critical behavior of the speed of
sound around the critical temperature. This is done by fitting the data of
$c_{s}^{2}$ in all curves in Fig. 11 for $0<\tau<0.1$. We observe that
$c_{s}^{2}$ follows a power law scaling for $T>T_{\text{c}}$ given by

$c_{s}^{2}\sim\tau^{2\beta}.$ (39)

Figure 12 shows the values of $\beta$ as a function of the number of disks for
elliptically bounded systems for both uniform and Gaussian SQ density
profiles. Note that this exponent grows as $N$ increases. This last effect may
be understood as the capability of the string system to add more disks to the
spanning cluster and becomes denser. Particularly, we found $\beta=0.456\pm
0.003$ in the thermodynamic limit for the uniform density profile, which is
close to the value 1/2 predicted by the mean field theory for magnetization
around the critical temperature in the Ising model.

Figure 12: Critical exponent of the power laws behavior of the speed of sound
around the critical temperature for small and elliptically bounded string
systems as a function of $N$ and the eccentricity: $\varepsilon=$0.0
(circles), 0.5 (crosses) and 0.9 (stars), for the Uniform (purple figures) and
SQ (green figures) density profiles.

On the other hand, it is possible to determine the speed of sound for high
density string systems. At $\rho_{\text{max}}$, we found

$\displaystyle\rho_{\text{max}}\sim N,$ $\displaystyle F(\rho_{max})\sim
N^{-1/2},$ $\displaystyle T(\rho_{\text{max}})\sim N^{1/4},$
$\displaystyle(\eta/s)(\rho_{\text{max}})\sim N^{1/4}.$ (40)

Therefore, the speed of sound at $\rho_{\text{max}}$ can be written as a
function that only depends on the number of disks:

$c_{s}^{2}(\rho_{\text{max}})=\frac{1}{3}\left(N\exp(-N)-1\right)(\text{const}N^{-1/4}-1).$
(41)

Taking the limit $N\to\infty$ we found that $c_{s}^{2}\to 1/3$, which is in a
full agreement with the reported in the literature [52, 53].

##  5 Conclusions

In this work, we have presented the algorithm to build a 2-dimensional
continuum percolating system free of periodic boundary conditions and
delimited by different geometries, with a low population of disks and filled
with mainly two different density profiles: Uniform and Gaussian. The second
one is relevant in the string percolation since the nuclear profile is denser
in the center of the overlapping area than the region near to the border. We
also provided the Monte Carlo integration method to measure the fraction of
area covered by disks in such percolating systems.

Simulation results indicate an explicit dependence of the fraction of area
covered by disks on the number of disks, boundary shape, and the density
profile, which is more relevant in small systems. However, the percolating
system lost this dependence as $N$ increases. This is beacause there is no
difference between very large circles (compared with the disk size) and very
large ellipses or very large regular polygons and thus the system behaves as
in the thermodynamic limit. Other limit analyzed corresponds to the situation
when all disks in the percolating system are concentric with the geometric
shape of the boundary. In this limit, the percolating system reaches its
maximum filling factor value and the fraction of area covered by disks is
exactly the packing fraction, becoming independent of the number of disks and
the density profile.

The modifications on the fraction of area covered by disks in small and
bounded systems lead corrections to the color suppression factor in the string
percolation model as a second damping function of the color due to finite size
and boundary effects. This function is well determined by Eqs. (24) and (27).
In the case of smallest system analyzed ($N=$13), this damping involves
relatives errors of the order between 11% and 25% (depending on the filling
factor and boundary shape) for the uniform density profile and high-density
systems (1.5$<\rho$). On the other hand, for low-density systems ($\rho<$0.5),
we can assure that the area covered by disks is well determined by its value
in the thermodynamic limit taking into account that it includes a relative
error of the order of 5%. However, these results change drastically for the
Gaussian density profile. Particularly, for low-density systems ($\rho<$0.75),
the damping factor takes values greater than 2.5, it means that the absolute
relative error is of the order between 150% and 250%. In spite of that, the
second damping factor becomes more similar to the corresponding to the uniform
density profile since the color suppression factor in the thermodynamic limit
for both uniform and Gaussian profiles look very similar in high-density
systems ($\rho>2.3$). Moreover, at $\rho_{\text{max}}$, the color suppression
factor just only depends on the number of strings distributed in the
overlapping surface.

Although the color suppression factor has low deviations from its value in the
thermodynamic limit even in the case of systems populated by 13 strings, it is
necessary to take into account the finite size-shape effects of all quantities
involved to compute any observable, which could give rise to significant
deviations from the determination in the thermodynamic limit. For example, the
speed of sound shows a dependence on both the strings number and on the
eccentricity around the critical temperature for elliptically bounded systems
even though the fraction of area covered by disks does not have notable
deviations as a function of the eccentricity. This is due to the dependence of
the percolation threshold on the number of disks and the eccentricity, which
carry on modifications of the critical temperature. Then, in the temperature
scale, the speed of sound is shifted as a function on the number of disks and
the boundary shape. On the other hand, we compute the speed of sound at
$\rho_{\text{max}}$, which just only depends on the number of disks as is
expected. Moreover, in the limit of high populated systems ($N\to\infty$), the
speed of sound takes the value 1/3, which is in a full agreement with the
reported by the calculations of lattice QCD.

Additionally, we determine the behavior of the speed of sound around the
critical temperature in the thermodynamic limit, finding that the
corresponding critical exponent of the power law is 0.45, which is close to
the value 1/2 for the critical exponent $\beta$ of the magnetization in the
Ising model determined by the mean field theory. This is an indication that
the speed of sound is related to the capability of the spanning cluster to add
more disks or finite clusters as the temperature increase. On the contrary,
for the Gaussian density profile, this exponent takes the value 0.2, which
means that the spanning cluster is constituted by a large proportion of disks
in the system, then, the speed of sound reaches values close to limit value
1/3 as a consequence of the saturation of the spanning cluster. These
behaviors of both the uniform and Gaussian density profiles could be verified
by analyzing the percolation susceptibility as a function of the temperature
around the percolation threshold. In the same sense, it is worth to mention
that the filling factor and the temperature for string systems define
different scales. Therefore there is not a direct way to determine the
correlation length for both parameters, and it is not possible to establish a
scaling hypothesis using the critical exponents determined for two-dimensional
percolation. To do this, it is necessary to determine the corresponding
exponent $\nu$ associated with the correlation length, considering the
parameter $\tau$ of Eq. (38) as the order parameter.

At RHIC and LHC energies the string density for minimum bias pp collisions is
in the range $0.3<\rho<0.6$ and the differences for the suppression factor
between the thermodynamic limit value and the corresponding profile already
taking into account the finite-shape effects is less than 5% (see Fig. 8), the
effects concerning global observables like multiplicity or transverse momentum
distribution is negligible. Even in the case of high multiplicity events where
the filling factor takes values larger than 1 or 1.5, the difference are of
the order of 10% which are comparable to the uncertainties of the comparison
with experimental data. However, the differences can be experimentally seen
looking at harmonics of the azimuthal distribution and some correlations
between them. In Fig. 9 we show that at high multiplicity, $\rho>1.5$, the
difference in the eccentricity $e_{3}$ are of the order of 25% in the cases of
an extended gaussian profile.

##  6 Acknowledgments

J.E.R. acknowledges financial support from CONACYT (postdoctoral fellowship
Grant no. 289198). C.P. was supported by the grant Maria de Maeztu Unit of
Excellence MDM-20-0692 and FPA project 2017-83814-P of Ministerio de Ciencia e
Inovación of Spain, FEDER and Xunta de Galicia.

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