#  Percolation leads to finite-size effects on the transition temperature and
center of mass energy required for the quark-gluon plasma formation

J. C. Texca García  Facultad de Ciencias Físico Matemáticas, Benemérita
Universidad Autónoma de Puebla, Apartado Postal 165, 72000 Puebla, Pue.,
México  D. Rosales Herrera  Facultad de Ciencias Físico Matemáticas,
Benemérita Universidad Autónoma de Puebla, Apartado Postal 165, 72000 Puebla,
Pue., México  J. E. Ramírez<EMAIL_ADDRESS>Centro de
Agroecología, Instituto de Ciencias, Benemérita Universidad Autónoma de
Puebla, Apartado Postal 165, 72000 Puebla, Pue., México  Consejo de Ciencia y
Tecnología del Estado de Puebla, Privada B poniente de la 16 de Septiembre
4511, 72534, Puebla, Pue., México.  A. Fernández Téllez  Facultad de Ciencias
Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Apartado Postal
165, 72000 Puebla, Pue., México  C. Pajares<EMAIL_ADDRESS>Departamento de Física de Partículas and Instituto Galego de Física de Altas
Enerxías, Universidad de Santiago de Compostela, E-15782 Santiago de
Compostela, España

###### Abstract

We investigate the finite-size effects on the transition temperature
associated with the quark-gluon plasma (QGP) formation. From a percolation
perspective, the onset of the QGP in high-energy collisions occurs when the
spanning cluster of color strings emerges. The principal result presented here
is the finite-size effects on the transition temperature expressed as a power
law in terms of the nucleon number. We found that the transition temperature
is higher for small systems than for large ones. It means that minimal
triggering conditions events in pp collisions require about twenty times
higher energies than AuAu-PbPb collisions. We also estimate the center of mass
energy required for the QGP formation as a function of the nucleon number. Our
results are consistent with the minimal center of mass energies at which the
QGP has been observed.

††preprint: APS/123-QED

In high energy physics, multiparticle production is usually described in terms
of color strings stretched between the projectile and target, which decay into
new strings through color neutral $q\bar{q}$ pairs production and subsequently
hadronize to produce the observed hadrons Braun  _et al._ (2015); Bautista
_et al._ (2019). Color strings may be viewed as small circular areas
distributed in the transverse plane (of the collision) filled with a color
field created by colliding partons that interact when they overlap. With the
increasing energy and size of the colliding system, the number of strings
grows. They start to overlap, forming clusters in the transverse plane, in a
similar way to the phenomena studied by 2-dimensional percolation theory
Armesto  _et al._ (1996). At one critical density, a giant cluster of color
strings appears where the quarks are no more confined. This interpretation
marks the percolation phase transition and the onset of the quark-gluon plasma
(QGP) Chernodub (2005). A first experimental evidence of the QGP was observed
in AuAu collisions at RHIC that later was confirmed in PbPb collisions at LHC,
through the study of all the harmonics of the azimuthal distributions, showing
the existence of a collective motion of quarks and gluons ALICE Collaboration
(2010); ATLAS Collaboration (2012); CMS Collaboration (2012). Furthermore, the
data analysis for pp and pA collisions at LHC CMS Collaboration (2013) and
d-Au and 3He-Au collisions at RHIC PHENIX Collaboration (2019) conclude that
in these experiments, most properties observed in heavy-ion collisions are
also present.

The general result due to the SU(3) random summation of color charges of
overlapped strings is a reduction of the multiplicity and an increase of the
string tension, hence an increase of the mean transverse momentum Pajares
(2005). For a cluster formed by $n$ strings, it is found that its multiplicity
$\mu_{n}$ and average transverse momentum square $\langle
p_{T}^{2}\rangle_{n}$ are given by Braun and Pajares (2000)

$\displaystyle\mu_{n}$ $\displaystyle=(nS_{n}/S_{1})^{1/2}\mu_{1},$ (1)
$\displaystyle\langle p_{T}^{2}\rangle_{n}$
$\displaystyle=(nS_{1}/S_{n})^{1/2}\langle p_{T}^{2}\rangle_{1},$ (2)

where $S_{1}=\pi r_{0}^{2}$ is the transverse area of the strings, and $S_{n}$
the area covered by the cluster, $\mu_{1}$ and $\langle p_{T}^{2}\rangle_{1}$
are the multiplicity and average transverse momentum squared for a single
string, respectively. Here $r_{0}$ is the radius of the string and takes
values between 0.2 fm and 0.3 fm. Taking into account fluctuations of the
number of strings at a fixed density, the average of $nS_{1}/S_{n}$ is

$\left\langle nS_{1}/S_{n}\right\rangle=\eta/\phi(\eta)=F(\eta)^{-2},$ (3)

with $\phi(\eta)$ being the area covered by strings and $\eta=NS_{1}/S$ the
filling factor, where $N$ is the total number of strings distributed on the
transverse plane with area $S$ Braun  _et al._ (2015); Bautista  _et al._
(2019). In particular, for the case of uniformly distributed disks, this
picture corresponds to the classical two dimensional continuum percolation of
disks, and $\phi(\eta)=1-e^{-\eta}$ in the thermodynamic limit Braun  _et al._
(2015); Kertész and Vicsek (1982). Nevertheless, $\phi(\eta)$ can
significantly deviate from $1-e^{-\eta}$ if the model considers a low number
of strings Ramírez  _et al._ (2017), fluctuations in the shape of the
transverse surface to the collisions Ramírez and Pajares (2019), interactions
between strings Ramírez  _et al._ (2021), or non-uniform string density
profiles Rodrigues  _et al._ (1999). Equation (3) defines the function

$F(\eta)=[\phi(\eta)/\eta]^{1/2},$ (4)

called the color suppression factor, which reflects the effects of the
interaction between strings on the observables of the system Braun and Pajares
(2000). Combining Eqs. (1), (2), and (4), we get Pajares (2011)

$\displaystyle\mu$ $\displaystyle=NF(\eta)\mu_{1},$ (5) $\displaystyle\langle
p_{T}^{2}\rangle$ $\displaystyle=\langle p_{T}^{2}\rangle_{1}/F(\eta).$ (6)

Note that for diluted systems, $\eta\to 0$ and $F(\eta)\to 1$, hence
$\mu=N\mu_{1}$ and $\langle p_{T}^{2}\rangle=\langle p_{T}^{2}\rangle_{1}$.
However, since $F(\eta)$ is a decreasing function, a colliding system produces
less charged particles with higher momentum as $\eta$ increases, which can be
understood as one effect of the string clustering process. Additionally,
equation (6) encodes the dependence on the centrality, energy and sizes of the
collision of the average of transverse momentum of produced particles. This
determines the slope of the low $p_{T}$ distribution and thus the radial flow.
A detailed study of the comparison with data of the $p_{T}$ distribution and
fluctuations from a color string percolation approach can be seen in Refs.
Dias de Deus  _et al._ (2005); Braun  _et al._ (2013).

On the other hand, it is well-known that the Schwinger mechanism
$dN/dp_{T}^{2}\sim e^{-\pi p_{T}^{2}/x^{2}}$ dictates the transverse momentum
distribution of the produced particles Schwinger (1951); Casher  _et al._
(1979); Andersson  _et al._ (1980), with $x$ being the string tension (color
field), which is expected to fluctuate. Assuming a Gaussian distribution for
the chromoelectric field Bialas (1999) with variance $\langle x^{2}\rangle$,
the Schwinger distribution becomes $dN/dp_{T}^{2}\sim e^{-\beta p_{T}}$, where
$\beta=(2\pi/\langle x^{2}\rangle)^{1/2}$ Dias de Deus and Pajares (2006).
Note the similarity between the latter and the Boltzmann distribution. In this
way, $\beta$ can be understood as a measure of the inverse of the temperature
of the system 111The temperature in Eq. (7) is equivalent to the one of the
blast wave model, which is split into two contributions. One is the freeze-out
temperature that increases with multiplicity. The remaining contribution comes
from a mass-dependent term representing a radial flow.. To determine $\langle
x^{2}\rangle$, it is necessary to compute $\langle p_{T}^{2}\rangle$ from the
Schwinger mechanism. Then averaging over the tension fluctuations, we get
$\langle p_{T}^{2}\rangle=\langle x^{2}\rangle/\pi$. Using Eq. (6), we can
define a dimensionless temperature for the color string percolation model as

$T^{*}(\eta)=\langle p_{T}^{2}\rangle_{1}^{-1/2}T(\eta)=[2F(\eta)]^{-1/2}.$
(7)

This local temperature becomes the temperature of the thermal distribution as
clusters grow and cover most of the collision surface Dias de Deus and Pajares
(2006). So the quark-gluon plasma formation can be associated with the
emergence of the spanning cluster Satz (2001); Gattringer (2010); Endrődi  _et
al._ (2014) and then the transition temperature can be defined as the
evaluation of the temperature (7) at the percolation threshold ($\eta_{c}$),
i.e., $T_{c}=T(\eta_{c})$. The value of $\langle p_{T}^{2}\rangle_{1}$ can be
determined by comparing $T_{c}$ in Eq. (7) with the estimation made from other
models, for example, lattice QCD Karsch (2007). However, the dimensionless
temperature $T_{c}^{*}$ may be helpful in the analysis of the behavior of
observables through the order parameter
$\epsilon=(T-T_{c})/T_{c}=(T^{*}-T_{c}^{*})/T_{c}^{*}$ which is independent of
$\langle p_{T}^{2}\rangle_{1}$ Ramírez and Pajares (2019).

This paper aims to analyze the finite-size effects of the transition
dimensionless temperature as a function of the system size. To do this for
finite systems, we estimate the percolation threshold and the area covered by
disks by computer simulation, then we compute the transition temperature and
the minimal center of mass energy required for the formation of the QGP on pp
and AA collisions. The main result presented here is that such finite-size
effects are expressed as a power-law in terms of the nucleon number. This lets
us estimate $T_{c}^{*}$ for different projectiles in the collisions, from pp
to PbPb.

In our simulations, we adopt the microcanonical ensemble scheme developed by
Mertens and Moore in Ref. Mertens and Moore (2012) for continuum percolation
systems, so that strings are added one by one in a square surface of side $L$.
We then measure the value of the observable $O$ after adding exactly $n$
strings, which is denoted by $O_{n}$. Thus, the mean value of the observable
$O$ at any value of $\eta$ is estimated by convoluting the set of values
$\\{O_{n}\\}$ and the Poisson distribution with average $\alpha=\eta L^{2}/\pi
r_{0}^{2}$ Mertens and Moore (2012), i.e.,

$O(\eta)=e^{-\alpha}\sum_{n=0}^{\infty}O_{n}\frac{\alpha^{n}}{n!}.$ (8)

We compute Poisson weights $w_{n}\propto\alpha^{n}/n!$ by using the recursive
formula provided by Mertens and Moore in Ref. Mertens and Moore (2012). By
plugging the $w_{n}$ distribution instead of Poisson weights in Eq. (8), it is
necessary to normalize the sum by dividing by $\sum w_{n}$.

We apply this algorithm to estimate the percolation threshold, area covered by
disks, color reduction function, and temperature for color string percolation
systems at finite transverse plane.

To compute the percolation threshold, we randomly add disk by disk over the
square $[0,L]\times[0,L]$ until the appearance of the spanning cluster. The
Union-Find algorithm is used for the clustering process. The spanning cluster
is detected when at least one string at the left border and other at the
opposite side belong to the same cluster. After the emergence of the spanning
cluster, the simulation is stopped, from which we obtain the critical number
$n_{c}$ of disks added. We compute the probability $f_{n}$ that a spanning
cluster appears after adding exactly $n$ disks as the quotient between the
frequency of $n$ and the total of trials performed. Thus
$\mathcal{F}_{n}=\sum_{k=0}^{n}f_{k}$ defines the probability of observing the
spanning cluster after adding $n$ disks in the system. Therefore, we compute
the percolation probability $P(\eta)$ at any filling factor by convoluting the
distribution $\mathcal{F}_{n}$ with the Poisson distribution as in Eq. (8).
The percolation probability has a sigmoid form whose width transition becomes
smaller as $L$ increases. Moreover, in the thermodynamic limit, $P(\eta)$ is a
Heaviside step function $H(\eta-\eta_{c})$, with $\eta_{c}$ the percolation
threshold Saberi (2015). In this way, the percolation probability for finite
$L$ is well-fitted to the function
$P_{L}(\eta)=0.5\left[1+\tanh\left((\eta-\eta_{cL})/\Delta_{L}\right)\right]$.
The subscript in $P$ and $\eta_{c}$ point out their dependence on $L$.
$\Delta_{L}$ is the width of the sigmoid transition for finite size systems
Rintoul and Torquato (1997); Saberi (2015). The estimation of $\eta_{c}$ in
the thermodynamic limit requires a finite-size effects analysis on $\eta_{cL}$
Mertens and Moore (2012); Newman and Ziff (2000, 2001).

Figure 1: a) Percolation probability for finite-size systems with $L=$ 64
(solid line), 128 (dashed line), and 256 (dash-dotted line). b) Width
transition $\Delta_{L}$ (squares) of the percolation probability as a function
of the system side $L$. The dashed line is the function $f(L)=aL^{b}$, with
$b=-0.758(7)$. c) Results of $\eta_{cL}$ as a function of $L$. Note the
suitable match between our results and the function $mL^{-2/\nu}+\eta_{c}$
(dashed line), with $\eta_{c}=$1.1279(1) being the estimation of the
percolation threshold in the thermodynamic limit.

On the other hand, we use a grid sample method to compute the area covered by
disks $\phi_{n}$ after adding exactly $n$-strings. Our selection of the grid
spacing is $r_{0}/20$. Thus, $\phi_{n}$ is approximated as
$\mathcal{N}_{n}r_{0}^{2}/400L^{2}$, where $\mathcal{N}_{n}$ counts only those
cells whose center lies on a disk after adding exactly $n$-strings. It is
expected that $\phi_{n}$ be a random variable because the center of the
strings are randomly allocated. We thus define $\bar{\phi}_{n}$ as the average
of $\phi_{n}$ over the number of trials performed. Therefore, the fraction of
covered area by disks at any filling factor value is estimated using the
convolution in Eq. (8). Since the distribution $\bar{\phi}_{n}$ is estimated
for a finite number of strings, we perform the convolution for a finite number
of terms. The sum runs from $N_{\text{min}}=\bar{n}-5\sigma$ to
$N_{\text{max}}=\bar{n}+5\sigma$, where $\bar{n}=\lfloor\alpha\rfloor$ and
$\sigma=\lfloor\alpha^{1/2}\rfloor$. Therefore, the critical area covered by
disks is computed as

$\phi_{cL}=e^{-\alpha_{cL}}\sum_{n=N_{\text{min}}}^{N_{\text{max}}}\bar{\phi}_{n}w_{n}\left/\sum_{n=N_{\text{min}}}^{N_{\text{max}}}w_{n}\right.,$
(9)

with $\alpha_{cL}=\eta_{cL}L^{2}/\pi r_{0}^{2}$. In what follows, we use the
latter scheme to calculate the value of the color suppression factor and
dimensionless temperature at the percolation threshold just by replacing
$\bar{\phi}_{n}$ by $F_{n}$ and $T_{n}^{*}$, which are computed as
$(\bar{\phi}_{n}L^{2}/n\pi r_{0}^{2})^{1/2}$ and $(2F_{n})^{-1/2}$,
respectively.

In the simulations, we set the diameter of the strings as the typical length
of the system. This allows us to represent the length side of the system as
integer multiples of the string diameter. To take into account the finite-size
effects on the percolation threshold and other observables of interest,
simulations were carried out by setting $L=$6, 8, 12, 16, 24, 32, 48, 64, 96,
128, 192, and 256. The data analysis is achieved over a compilation of 104
trials for each $L$-value.

Figure 1 a) illustrates the percolation probability sigmoid shape. Notice that
the width transition $\Delta_{L}$ becomes smaller as $L$ increases. This fact
leads to the well-known scaling relation $\Delta_{L}\propto L^{-1/\nu}$, with
$\nu$ the critical exponent associated to the (cluster radii) correlation
length Stauffer and Aharony (2014). This behavior is depicted in Fig. 1 b),
where $\Delta_{L}$ is well described by a linear function in a logarithmic
scale whose slope corresponds to the critical exponent. We found
$1/\nu=0.758(7)$ for the systems described above. This value agrees with the
existing result for percolation systems in two-dimensions, where $1/\nu=0.75$
Stauffer and Aharony (2014).

The percolation threshold in the thermodynamic limit is determined by
analyzing the scaling relation of $\eta_{c}-\eta_{cL}$as a function of $L$,
which is determined by two factors. The first comes from the fact that the
width transition scales as $L^{-1/\nu}$. The second factor comes from the
imposed boundary conditions, which contributes another $L^{-1/\nu}$. These two
factors combine to give

$\eta_{c}-\eta_{cL}\propto L^{-2/\nu},$ (10)

leading to stronger finite-size effects than the universal scaling relation
for the percolation threshold. Figure 1 c) shows the behavior of $\eta_{cL}$
as a function of $L^{-2/\nu}$. Notice that our results on the percolation
threshold for finite systems satisfy the scaling relation (10) even for small
values of $L$. The percolation threshold in the thermodynamic limit
($L\to\infty$) is determined by extrapolating the trendline of the data. We
found $\eta_{c}=1.1279(1)$, which is in accordance with the most precise value
reported in the literature (1.12808737(6)) Mertens and Moore (2012).

In addition, from the analysis of the finite-size effects, we found the
following power-law behaviors for the area covered by disks and critical color
suppression factor

$\displaystyle\phi_{c}-\phi_{cL}$ $\displaystyle\propto L^{-2},$ (11)
$\displaystyle F_{c}-F_{cL}$ $\displaystyle\propto L^{-1.3},$ (12)

respectively. Moreover, the average transverse momentum squared at the
percolation threshold behaves as

$\langle p_{T}^{2}\rangle_{c}-\langle p_{T}^{2}\rangle_{cL}\propto L^{-1.33}.$
(13)

Note that the multiplicity diverges as $L^{2}$ because $\mu$ is proportional
to the number of strings in the system. By replacing $N=\eta L^{2}/S_{1}$ in
Eq. 5, we can define the multiplicity density as follows $M=\mu/L^{2}=\eta
F(\eta)M_{1}$, with $M_{1}=\mu_{1}/S_{1}$ being the multiplicity density for a
single string. Thus, we found

$M_{c}-M_{cL}\propto L^{-1.66}.$ (14)

Figures 2 a), b), c) and d) show the scaling relations in Eqs. (11), (12),
(13), and (14), respectively. Error bars are computed as the usual standard
deviation, where the second moment is calculated according to the convolution
in Eq. (9).

Figure 2: Area covered by disks [a)], color suppression factor [b)],
multiplicity density [c)], and average transverse momentum squared [d)] at
$\eta_{cL}$ as a function of $L$. Dashed lines are the corresponding scaling
relation $mL^{a}+b$, with $b=$0.6757(7), 0.7742(1), 0.8731(6), and 1.2917(2)
being the estimation in the thermodynamic limit for $\phi_{c}$, $F_{c}$,
$M_{c}/M_{1}$, and $\langle p_{T}^{2}\rangle_{c}/\langle
p_{T}^{2}\rangle_{1}$, respectively. Error bars are computed as the square
root of the variance.

Our estimation of the critical area covered by disks in the thermodynamic
limit is $\phi_{c}=$0.6757(7), which agrees with the best existing
determination (0.67634831(2)) Mertens and Moore (2012). Moreover, we estimate
$F_{c}=$0.7742(1), $M_{c}/M_{1}=0.8731(6)$, and $\langle
p_{T}^{2}\rangle_{c}/\langle p_{T}^{2}\rangle_{1}=1.2917(2)$ in the
thermodynamic limit. These values are in agreement with those computed using
the information of $\eta_{c}$ and $\phi_{c}$ reported in Ref. Mertens and
Moore (2012), whose estimations are $F_{c}=$0.77430816(4),
$M_{c}/M_{1}=$0.87348726(6), and $\langle p_{T}^{2}\rangle_{c}/\langle
p_{T}^{2}\rangle_{1}=$1.29147548(7).

We recall that the transition temperature associated with the quark-gluon
plasma formation can be computed by evaluating $T$ at the percolation
threshold. For the color string percolation model we found that $T_{cL}^{*}$
satisfy

$T_{c}^{*}-T_{cL}^{*}\propto L^{-1.32},$ (15)

where $T_{cL}^{*}$ is computed by means of the convolution in Eq. (9). Figure
3 depicts this scaling behavior for our data simulation of $T_{cL}^{*}$.
Another way to calculate $T_{cL}^{*}$ consists of computing the percolation
probability as a function of the temperature. Since $P_{L}(T)$ also exhibits a
sigmoid shape, $T_{cL}^{*}$ is estimated by using the aforementioned
procedure. Both methods give consistent results on $T_{cL}^{*}$, which
coincide up to five decimals. Additionally, we estimate $T_{c}^{*}=$0.80365(8)
in the thermodynamic limit. As expected, our estimation perfectly agrees with
the value 0.80357808(3) computed using the best determination of the
percolation threshold and the critical area covered by disks reported in Ref.
Mertens and Moore (2012). Notice the scaling relation in (15) implicates that
the transition temperature of the quark-gluon plasma formation is greater for
small systems than for large ones.

Figure 3: Transition dimensionless temperature (black squares) for the
estimated percolation threshold $\eta_{cL}$. Error bars are calculated as the
square root of the variance. Dashed line is the function
$mL^{-1.32}+T_{c}^{*}$, with $T_{c}^{*}$=0.80365(8) being the estimation of
the dimensionless temperature in the thermodynamic limit. Color boxes are
estimations and error propagation of the transition temperature of the quark-
gluon plasma formation for pp, pA and AA collisions.

To make contact with the experimental parameters of collision physics, we
enclose the transverse plane surface for central collision with a square of
side length $L=R_{A}/r_{0}$, with $R_{A}$ being the radius of the atomic
nucleus, given by $R_{A}=r_{0}^{*}A_{M}^{1/3}$, where $A_{M}$ is the nucleon
number and $r_{0}^{*}$ takes values between 1.2-1.3 fm Krane (1991).
Therefore, the finite-size effects on the transition temperature can be
rewritten as a power law on the nucleon number as

$T_{c}^{*}-T_{cL}^{*}\propto A_{M}^{-0.44}.$ (16)

This last relation is helpful because we can estimate the transition
temperature for different kinds of projectiles, as Fig. 3 illustrates. In the
computation, we use $r_{0}^{*}=1.25(5)$ fm and $r_{0}=0.25(5)$ fm. The error
propagation leads to $\sigma_{L}\approx 0.2L$ and $\sigma_{T_{cL}}\approx
0.08L^{-1.32}$, which are shown as box errors in Fig. 3. We take $A_{M}=1$ and
$A_{M}=2$ for pA-pp and dA collisions, respectively. For AA collisions, we
take for $A_{M}$ the value that corresponds to the most abundant isotope.

Figure 4: Diagram of the QGP formation as a function of the variables
$\sqrt{s}$ and $A_{M}$. The critical curve (solid line) marks the minimum
center of energy required to observe the QGP for a collision involving $A_{M}$
nucleons per projectile. The errors (grey shaded region) are computed by
assuming a normal distribution for all the parameters but keeping fixed the
ratio $r_{0}/r_{0}^{*}=0.2$. Red shaded region corresponds to the conditions
of $\sqrt{s}$ and $A_{M}$ at which the QGP may be observed. Figures are AA and
pp collision experiments at different center of mass energies.

Additionally, it is possible to determine the center of mass energy needed for
the formation of the QGP. To do this, we consider that the filling factor of
strings distributed on the transverse plane of pp collisions at center of mass
energy $\sqrt{s}$ is given by Bautista  _et al._ (2012)

$\eta^{pp}=\frac{\pi}{25}\left[2+4\left(\frac{r_{0}}{R_{p}}\right)^{2}\left(\frac{\sqrt{s}}{m_{p}}\right)^{2\lambda}\right],$
(17)

where $R_{p}$=0.84-0.87 fm and $m_{p}$=938 MeV/c2 are the radius and mass of
the proton, respectively. On the other hand, the filling factor for central AA
collisions is

$\eta^{AA}=\eta^{pp}A_{M}^{\alpha(\sqrt{s})},$ (18)

with

$\alpha(\sqrt{s})=\frac{1}{3}\left[1-\frac{1}{1+\ln(\sqrt{s/s_{0}}+1)}\right].$
(19)

The parameters $\lambda$ and $\sqrt{s_{0}}$ can be found by fitting the
adequate experimental data, taking the values 245(29) GeV/c2 and 0.201(3)
Bautista  _et al._ (2012), respectively. Since the onset of the QGP is
associated to the emergence of the spanning cluster, the center of mass energy
$\sqrt{s_{c}}$ for central AA collisions is estimated by solving the equation
$\eta_{c}+mA_{M}^{-2/3\nu}=\eta^{AA}$, with $m=0.316(4)$. In Fig. 4 we plot
$\sqrt{s_{c}}$ as a function of the nucleon number. We estimate
$\sqrt{s_{c}}$=184(15) GeV/c2 for AuAu collisions. Notice that our estimation
is in agreement with the data analysis that claims the QGP signatures for AuAu
($\sqrt{s}=$200 GeV/c2 at RHIC) collisions Gyulassy and McLerran (2005). For
PbPb collisions we found $\sqrt{s_{c}}$=182(15) GeV/c2. This explains why the
QGP is obtained at LHC energies for these experiments. In the case of pp
collisions we estimate $\sqrt{s_{c}}$=3.7(5) TeV/c2, which is in agreement
with the results observed in experiments at $\sqrt{s}=$5.02 TeV/c2 at LHC CMS
Collaboration (2017); ATLAS Collaboration (2017). However, the formation of
the QGP in pp collisions is expected for high multiplicity events at lower
energies CMS Collaboration (2010).

In summary, we presented the analysis of the finite-size effects on the
transition temperature of the QGP formation associated with the emergence of
the spanning cluster of color strings. Through the radius of the atomic
nucleus, these effects are expressed as a power-law in terms of the nucleon
number, which allowed us to estimate the transition temperature for different
projectiles and targets in high energy collisions systems. We found that the
transition temperature is higher for small collision systems like pp or pA
collisions than for large ones, such as AuAu or PbPb collisions. This means
that systems of color strings formed by pp or pA collisions need high filling
factor values, which require higher energy or very high multiplicity than
those at minimal triggering conditions events In particular, we estimate that
pp collisions require about twenty times bigger center mass energy than AuAu
or PbPb collisions for the QGP formation.

Our results could lead to finding scaling functions that incorporate
information on finite-size systems and the filling factor for observables in
the color string percolation framework and other models. Moreover, results
presented here can be extended to consider other centralities, fluctuations of
the initial shape of the transverse plane, non-uniform density profiles or
interactions between strings.

###### Acknowledgements.

J.E.R. acknowledges financial support from Consejo de Ciencia y Tecnología del
Estado de Puebla. C.P. thanks the grant Maria de Maeztu unit of excelence
MDM-2016 0682 of Spain, the support of Xunta de Galicia and project PID
2020-119632GB-I00. We also thank Nestor Armesto, Miguel Ángel García-Ariza and
Bogar Díaz for their valuable comments.

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