2016 018 26 January

Anisotropic flow at 5.02 TeV

ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration

We report the first results of elliptic ($v_{2}$), triangular ($v_{3}$) and
quadrangular flow ($v_{4}$) of charged particles in Pb–Pb collisions at a
center-of-mass energy per nucleon pair of $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV
with the ALICE detector at the CERN Large Hadron Collider. The measurements
are performed in the central pseudorapidity region $|\eta|<$ 0.8 and for the
transverse momentum range 0.2 $<p_{\rm T}<$ 5 GeV/$c$. The anisotropic flow is
measured using two-particle correlations with a pseudorapidity gap greater
than one unit and with the multi-particle cumulant method. Compared to results
from Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 2.76 TeV, the anisotropic
flow coefficients $v_{2}$, $v_{3}$ and $v_{4}$ are found to increase by
(3.0$\pm$0.6)%, (4.3$\pm$1.4)% and (10.2$\pm$3.8)%, respectively, in the
centrality range 0–50%. This increase can be attributed mostly to an increase
of the average transverse momentum between the two energies. The measurements
are found to be compatible with hydrodynamic model calculations. This
comparison provides a unique opportunity to test the validity of the
hydrodynamic picture and the power to further discriminate between various
possibilities for the temperature dependence of shear viscosity to entropy
density ratio of the produced matter in heavy-ion collisions at the highest
energies.

The goal of studies with relativistic heavy-ion collisions is to investigate
the Quark-Gluon Plasma (QGP), a state of matter where quarks and gluons move
freely over distances large in comparison to the typical size of a hadron. The
transition from normal nuclear matter to the QGP state is expected to occur at
extreme values of energy density (0.2–0.5 GeV/fm3, according to lattice
Quantum Chromodynamics calculations [1, 2]), which are accessible in ultra-
relativistic heavy-ion collisions at the Large Hadron Collider (LHC) [3, 4].
The study of such collisions provides the unique opportunity to probe the
properties of the QGP in a region of the QCD phase diagram where a cross-over
between the deconfined phase and normal nuclear matter is expected [5, 6, 7,
8, 9].

Studies of the azimuthal anisotropy of particle production have contributed
significantly to the characterization of the system created in heavy-ion
collisions [10, 11]. Anisotropic flow, which measures the momentum anisotropy
of the final-state particles, is sensitive both to the initial geometry of the
overlap region and to the transport properties and equation of state of the
system. By using a general Fourier series decomposition of the azimuthal
distribution of produced particles:

$\frac{dN}{d\varphi}\propto
1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\varphi-\Psi_{n})]\,,$ (1)

anisotropic flow is quantified with coefficients $v_{n}$ and corresponding
symmetry planes $\Psi_{n}$ [12]. Due to the approximately ellipsoidal shape of
the overlap region in non-central heavy-ion collisions (i.e. collisions that
correspond to large impact parameter), the dominant flow coefficient is
$v_{2}$, referred to as elliptic flow. In the transition from highest RHIC to
LHC energies, elliptic flow increases by 30% [13], as predicted by
hydrodynamic models that include viscous corrections [14, 15, 16, 17, 18].
Non-vanishing values of higher anisotropic flow harmonics $v_{3}$–$v_{6}$ at
the LHC are ascribed primarily to the response of the produced QGP to
fluctuations of the initial energy density profile of the colliding nucleons
[19, 20, 21, 22]. Moreover, due to such fluctuations each flow harmonic
$v_{n}$ has a distinct symmetry plane $\Psi_{n}$ and recent measurements of
their inter-correlations provide independent constraints on the QGP properties
[23]. The combination of all of such results demonstrates that the shear
viscosity to entropy density ratio ($\eta/s$) of the QGP produced in ultra-
relativistic heavy-ion collisions at RHIC and LHC has a value close to
$1/4\pi$, a lower bound obtained in strong-coupling calculations based on the
AdS/CFT conjecture [24].

Recently, predictions from Niemi et al. on anisotropic flow coefficients for
Pb–Pb $\sqrt{s_{{}_{\rm NN}}}=5.02$ TeV collisions using the EKRT model were
reported in [25]. These predictions have a special emphasis on the
discriminating power between various parameterizations of the temperature
dependence of $\eta/s$. It was argued that in the transition from 2.76 to 5.02
TeV, the elliptic flow estimated from two–particle correlations (denoted
further in the text as $v_{2}\\{2\\}$, where the number in the curly brackets
indicates the number of particles that are used in correlation [26]) can
increase at most $\sim 5\%$ for all centrality classes. Details of the
increase depend on the parameterization of $\eta/s(T)$. On the other hand,
higher flow harmonic observables, like $v_{3}\\{2\\}$ and $v_{4}\\{2\\}$, are
predicted to increase more rapidly, $10-30\%$. With a different approach,
where previously measured values of flow harmonics at lower LHC energies are
taken as a baseline, Noronha-Hostler et al. [27] predict a larger increase for
both elliptic and triangular flow in peripheral compared to central collisions
in transition from 2.76 to 5.02 TeV. They conclude that the anisotropic flow
already reaches saturation and its maximum value in central collisions at 2.76
TeV.

A necessary condition for the development of anisotropic flow is the initial
anisotropy in the interaction region of the two colliding ions. These
coordinate space anisotropies are described in terms of eccentricities, that
are not directly accessible experimentally. Nonetheless, the theoretical
modeling of such eccentricities is actively being studied. For instance,
hydrodynamic calculations based on a MC-Glauber model and MC-KLN initial
conditions do not agree on the details of the saturation of elliptic flow at
LHC energies [28]. However, with these two initial state models, it was shown
that the final spatial eccentricity decreases monotonically as the collision
energy increases [28], and is expected to become negative only at the very
large collision energies available at the LHC (see Fig. 9 in [28]).

In addition to the initial conditions, various other stages of evolution of
the system in a heavy-ion collision may contribute to the development of
anisotropic flow. At lower energies, the state of the system will resemble
primarily a hadronic gas, and hadron rescattering is the dominant contribution
to the anisotropic flow. At higher energies, anisotropic flow mostly develops
in the thermalized color-deconfined QGP phase. However, even at these higher
energies, the contribution from the hadronic phase can be significant. The
relative amount of time the system spends in different phases varies with
collision energy [28, 29]. Radial flow, a measure for the average velocity of
the system’s collective radial expansion, also increases as a function of
collision energy, which translates into more particles being transferred to a
higher transverse momentum ($p_{\rm T}$) region, thus leading to an increase
in average anisotropic flow values. On the other hand, the opposite dependence
of differential $v_{2}(p_{\rm T})$ is expected for light (increase at low
$p_{\rm T}$) and heavy particles (decrease at low $p_{\rm T}$) as a function
of collision energy, which might yield to the saturation of elliptic flow
signal [28]. Finally, the relative importance of various stages in the system
evolution as a function of collision energy can also vary for each flow
coefficient [29].

The data used in this Letter were recorded with the ALICE detector [30, 31] in
November 2015 in Run 2 at the LHC with Pb–Pb collisions at $\sqrt{s_{{}_{\rm
NN}}}=$ 5.02 TeV. Minimum bias Pb–Pb events were triggered by the coincidence
of signals from the V0 detector. The V0 detector is composed of two arrays of
scintillator counters, V0-A and V0-C, which cover the pseudorapidity ranges
2.8 $<\eta<$ 5.1 and -3.7 $<\eta<$ -1.7, respectively [30]. Centrality
quantifies the fraction of geometrical cross-section of the colliding nuclei.
It is determined using the sum of the amplitudes of the V0-A and C signals,
which provides a resolution better than 0.5$\%$ up to 20$\%$ central Pb–Pb
collisions, and better than 2% for peripheral collisions [32]. The offline
event selection employs the information from two Zero Degree Calorimeters
(ZDC) [30] positioned 112.5 m from the interaction point on either side. Beam
background events are removed using timing information from the V0 and the
ZDC, respectively. To ensure a uniform acceptance and reconstruction
efficiency in the pseudorapidity region $|\eta|<$ 0.8, only events with a
reconstructed vertex within 10 cm from the center of the detector along the
beam direction were used. A sample of 140 k PbPb collisions events passed the
selection criteria. Only one low luminosity run (with trigger rate of 27 Hz)
was used, being least affected by pile-up and distortions from space charge in
the main tracking detector, the Time Projection Chamber (TPC).

Figure 1: (color online) (a) Anisotropic flow $v_{n}$ integrated over the
$p_{\rm T}$ range 0.2 $<p_{\rm T}<$ 5.0 GeV/$c$, as a function of event
centrality, for the two–particle (with $|\Delta\eta|>1$) and multi-particle
cumulant methods. Measurements for Pb–Pb collisions at $\sqrt{s_{{}_{\rm
NN}}}=$ 5.02 (2.76) TeV are shown by solid (open) markers [20]. The ratios of
$v_{2}\\{2,|\Delta\eta|>1\\}$ (red), $v_{2}\\{4\\}$ (gray) and
$v_{3}\\{2,|\Delta\eta|>1\\}$ (blue), $v_{4}\\{2,|\Delta\eta|>1\\}$ (green)
between Pb–Pb collisions at 5.02 TeV and 2.76 TeV, are presented in Fig. 1 (b)
and (c). Various hydrodynamic calculations are also presented [27, 25]. The
statistical and systematical uncertainties are summed in quadrature (the
systematic uncertainty is smaller than the statistical uncertainty, which is
typically within 5%). Data points are shifted for visibility.

Charged tracks are reconstructed using the ALICE Inner Tracking System (ITS)
and TPC [30]. This combination ensures a high detection efficiency, optimum
momentum resolution, and a minimum contribution from photon conversions and
secondary charged particles produced either from the detector material or from
weak decays. In order to reduce the contamination from secondary particles,
only tracks with a distance of closest approach (DCA) to the interaction point
of less than 3 cm, both in the longitudinal and transverse directions, are
accepted. The tracking efficiency is calculated from a Monte Carlo simulation
that uses HIJING [33] to simulate particle production. GEANT3 [34] is then
used for transporting simulated particles, followed by a full calculation of
the detector response (including production of secondary particles) and track
reconstruction performed with the ALICE reconstruction framework. The tracking
efficiency is $\sim 70\%$ at $p_{\rm T}\sim 0.2$ GeV/$c$ and increases to an
approximately constant value of $\sim 80\%$ for $p_{\rm T}>1$ GeV/$c$. The
$p_{\rm T}$ resolution is better than 5% for the region presented in this
Letter. The systematic uncertainty related to the non-uniform reconstruction
efficiency was found to be at the level of 1%. The flow coefficients from
tracks that are reconstructed from TPC space points alone were compared to
coefficients extracted from particles that used both TPC clusters and ITS
hits, which were found to agree within $\sim$2%. This difference was taken
into account in the estimation of the systematic uncertainty. Altering the
selection criteria for the tracks reconstructed with the TPC resulted in a
variation of the results of 0.5%, at most. Other selection criteria that have
been scrutinized are the centrality determination, e.g. using the Silicon
Pixel Detector (SPD), which contributed by less than 1%, the polarity of the
magnetic field of the ALICE detector and the position of the reconstructed
primary vertex, whose contributions were found to be negligible. The
systematic uncertainties evaluated for each of the sources mentioned above
were added in quadrature to obtain the total systematic uncertainty of the
measurements.

Figure 2: (color online) $v_{n}(p_{\rm T})$ using two–particle cumulant method
with $|\Delta\eta|>$ 1 for (a) 0–5% and (b) 30–40% centrality classes; (c)
$v_{2}(p_{\rm T})$ using four-particle cumulant method for the centrality
10–20%, 20–30% and 30–40%. Measurements for Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}=$ 2.76 TeV are also presented as shading. (d) The
ratio of $v_{2}\\{4\\}$ in 20–30% from two collision energies is also shown
here. The statistical and systematical uncertainties are summed in quadrature
(the systematic uncertainty is smaller than the statistical uncertainty, which
is typically within 5%).

.

In this Letter, we report the anisotropic flow measurements obtained from two-
and multi-particle cumulants, using the approach proposed in [35, 36, 37].
These two measurements have different sensitivities to flow fluctuations and
non-flow effects. Non-flow effects are azimuthal correlations not associated
with the symmetry planes and usually arise from resonance decays and jets.
Their contributions are expected to be suppressed when using a large
pseudorapidity gap between particle pairs. Thus, in this study, we require a
pseudorapidity gap of $|\Delta\eta|>$ 1\. This observable is denoted as
$v_{n}\\{2,|\Delta\eta|>1\\}$. On the other hand, non-flow contributions to
multi-particle cumulants $v_{n}\\{4\\}$, $v_{n}\\{6\\}$, $v_{n}\\{8\\}$ are
found to be negligible in events with large multiplicities characteristic for
heavy-ion collisions [38].

Figure 1 (a) presents the centrality dependence of $v_{2}$, $v_{3}$ and
$v_{4}$ from two- and multi-particle cumulants, integrated over the $p_{\rm
T}$ range 0.2 $<p_{\rm T}<$ 5.0 GeV/$c$, for 2.76 and 5.02 TeV Pb–Pb
collisions. To elucidate the energy evolution of $v_{2}$, $v_{3}$ and $v_{4}$,
the ratios of anisotropic flow measured at 5.02 TeV to 2.76 TeV are presented
in Fig 1. (b) and (c). Assuming that non-flow effects are suppressed by the
pseudorapidity gap, the remaining differences between two- and multi-particle
cumulants of $v_{2}$ can be related to the strength of elliptic flow
fluctuations, which are expected to give a positive and a negative
contribution to the two- and multi-particle cumulant estimates, respectively
[11]. Moreover, the multi-particle cumulants $v_{2}\\{4\\}$, $v_{2}\\{6\\}$
and $v_{2}\\{8\\}$ are all observed to agree within 1%, which indicates that
non-flow effects are largely suppressed. It is seen that
$v_{2}\\{2,|\Delta\eta|>1\\}$ increases from central to peripheral collisions,
and reaches a maximum value of 0.104 $\pm$ 0.001 (stat.) $\pm$ 0.002 (syst.)
in the 40–50% centrality class. For the higher harmonics, i.e. $v_{3}$ and
$v_{4}$, the values are smaller and the centrality dependence is much weaker.

Furthermore, the predictions of anisotropic flow coefficients $v_{n}$ from the
previously mentioned hydrodynamic model [27] are compared to the measurements
in Fig. 1. (a). These predictions combine the changes in initial spatial
anisotropy and the hydrodynamic response (treated as systematic uncertainty
and shown by the width of the bands). The predictions are compatible with the
measured anisotropic flow $v_{n}$ coefficients. At the same time, a different
hydrodynamic calculation [25], which employs both constant $\eta/s=$ 0.20 and
temperature dependent $\eta/s$, can also describe the increase in anisotropic
flow measurements of $v_{2}$ (shown in Fig 1. (b)), $v_{3}$ and $v_{4}$ (see
Fig 1. (c)). In particular, among the different scenarios proposed in [25],
the measurements seem to favor a constant $\eta/s$ going from
$\sqrt{s_{{}_{\rm NN}}}=$ 2.76 to 5.02 TeV Pb–Pb collisions.

The increase of $v_{2}$ and $v_{3}$ from the two energies is rather moderate,
while for $v_{4}$ it is more pronounced. In addition, none of the ratios of
flow harmonics exhibit a significant centrality dependence in the centrality
range 0–50%, and thus the results of a fit with a constant value over these
ratios are reported. An increase of (3.0$\pm$0.6)%, (4.3$\pm$1.4)% and
(10.2$\pm$3.8)%, is obtained for elliptic, triangular and quadrangular flow,
respectively, over the centrality range 0–50% in Pb–Pb collisions when going
from 2.76 TeV to 5.02 TeV. This increase of anisotropic flow is compatible
with theoretical predictions described in [25, 27]. Overall, these
measurements support a low value of $\eta/s$ for the system created in Pb–Pb
collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV and seem to indicate that it
does not increase significantly with respect to Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}=2.76$ TeV.

The anisotropic flow coefficients $v_{2}\\{2,|\Delta\eta|>1\\}$,
$v_{3}\\{2,|\Delta\eta|>1\\}$ and $v_{4}\\{2,|\Delta\eta|>1\\}$ as a function
of transverse momentum ($p_{\rm T}$) are presented in Fig. 2 for the 0–5% and
30–40% centrality classes. For the 0–5% centrality class, at $p_{\rm T}>$ 2
GeV/$c$ $v_{3}\\{2\\}$ is observed to become larger than $v_{2}\\{2\\}$, while
$v_{4}\\{2\\}$ is compatible with $v_{2}\\{2\\}$, within uncertainties. For
the 30–40% centrality class, we see that $v_{2}\\{2\\}$ is higher than
$v_{3}\\{2\\}$ and $v_{4}\\{2\\}$ for the entire $p_{\rm T}$ range measured,
with no crossing of the different order flow coefficients observed. Figure 2
(c) presents the $p_{\rm T}$-differential $v_{2}\\{4\\}$ for the 10–20%,
20–30% and 30–40% centrality classes. The $v_{2}\\{4\\}$ decreases from mid-
central to central collisions over the $p_{\rm T}$ range measured. The
comparison with the corresponding measurements from Pb–Pb collisions at
$\sqrt{s_{{}_{\rm NN}}}=2.76$ TeV exhibits comparable values, as illustrated
by the ratio of $v_{2}\\{4\\}$ for the two energies in Fig. 2 (d). This
indicates that the increase observed in the $p_{\rm T}$ integrated flow
results seen in Fig. 1 can be attributed to an increase of mean transverse
momentum $\langle p_{\rm T}\rangle$. The measurements of $p_{\rm
T}$-differential flow are more sensitive to initial conditions and $\eta/s$,
and are expected to provide important information to constrain further details
of the theoretical calculations, e.g. determination of radial flow and freeze-
out conditions.

Figure 3: Integrated elliptic flow ($v_{2}\\{4\\}$) for the 20–30% most
central Pb–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV compared with $v_{2}$
measurements at lower energies with similar centralities (see [13] for
references to all data points).

Figure 3 presents the comparison of the fully $p_{\rm T}$ integrated $v_{2}$
measured in the 20–30% centrality in Pb–Pb collisions at the LHC with results
at lower energies. This integrated value in the full $p_{\rm T}$ range is
determined using two methods. The first uses fits to the efficiency-corrected
charged-particle spectra and the $p_{\rm T}$-differential $v_{2}\\{4\\}$
presented in Fig. 2, extrapolated to $p_{\rm T}=0$. The error on the
integrated $v_{2}$ is estimated both from the uncertainty on the $p_{\rm
T}$-differential measurements and from different parameterizations that
provide a good fit of the data. The second calculates $v_{2}\\{4\\}$ using
tracklets formed from SPD hits in the ITS, which have an acceptance of $p_{\rm
T}\gtrsim$ 50 MeV/c. As each method uses different ALICE sub-detectors, they
can provide independent measurements of $v_{2}$ coefficients. For this
centrality range, they agree within 1% for both energies. The values presented
in the figure are weighted averages of these two measurements, using the
inverse of the variance of each of them as weights. A continuous increase of
anisotropic flow for this centrality has been observed from SPS/RHIC to LHC
energies. For these fully $p_{\rm T}$ integrated coefficients, an increase of
$4.9\pm 1.9\%$ is observed going from $\sqrt{s_{{}_{\rm NN}}}=2.76$ to 5.02
TeV, which is close to values of the previously-mentioned hydrodynamic
calculations [25, 27].

In summary, we have presented the first anisotropic flow measurements of
charged particles in Pb–Pb collisions at $\sqrt{s_{{}_{\rm NN}}}=$ 5.02 TeV at
the LHC. An average increase of (3.0$\pm$0.6)%, (4.3$\pm$1.4)% and
(10.2$\pm$3.8)%, is observed for the transverse momentum integrated elliptic,
triangular and quadrangular flow, respectively, over the centrality range
0–50% going from 2.76 TeV to 5.02 TeV. The transverse momentum dependence of
anisotropic flow has also been investigated, and does not change appreciably
between the two LHC energies. Therefore, the increase in integrated flow
coefficients can be attributed mostly to an increase in average transverse
momentum. The measurements are found to be compatible with predictions from
hydrodynamic models [27, 25]. Further comparisons of $p_{\rm T}$-differential
flow measurements and theoretical calculations, which are not available at the
time of writing, will provide extra constraints on the initial conditions and
the transport properties of the QGP.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians
for their invaluable contributions to the construction of the experiment and
the CERN accelerator teams for the outstanding performance of the LHC complex.
The ALICE Collaboration gratefully acknowledges the resources and support
provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG)
collaboration. The ALICE Collaboration acknowledges the following funding
agencies for their support in building and running the ALICE detector: State
Committee of Science, World Federation of Scientists (WFS) and Swiss Fonds
Kidagan, Armenia; Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Financiadora de Estudos e Projetos (FINEP), Fundação de
Amparo à Pesquisa do Estado de São Paulo (FAPESP); National Natural Science
Foundation of China (NSFC), the Chinese Ministry of Education (CMOE) and the
Ministry of Science and Technology of China (MSTC); Ministry of Education and
Youth of the Czech Republic; Danish Natural Science Research Council, the
Carlsberg Foundation and the Danish National Research Foundation; The European
Research Council under the European Community’s Seventh Framework Programme;
Helsinki Institute of Physics and the Academy of Finland; French CNRS-IN2P3,
the ‘Region Pays de Loire’, ‘Region Alsace’, ‘Region Auvergne’ and CEA,
France; German Bundesministerium fur Bildung, Wissenschaft, Forschung und
Technologie (BMBF) and the Helmholtz Association; General Secretariat for
Research and Technology, Ministry of Development, Greece; National Research,
Development and Innovation Office (NKFIH), Hungary; Department of Atomic
Energy and Department of Science and Technology of the Government of India;
Istituto Nazionale di Fisica Nucleare (INFN) and Centro Fermi - Museo Storico
della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Italy; Japan Society
for the Promotion of Science (JSPS) KAKENHI and MEXT, Japan; Joint Institute
for Nuclear Research, Dubna; National Research Foundation of Korea (NRF);
Consejo Nacional de Cienca y Tecnologia (CONACYT), Direccion General de
Asuntos del Personal Academico(DGAPA), México, Amerique Latine Formation
academique - European Commission (ALFA-EC) and the EPLANET Program (European
Particle Physics Latin American Network); Stichting voor Fundamenteel
Onderzoek der Materie (FOM) and the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO), Netherlands; Research Council of Norway
(NFR); National Science Centre, Poland; Ministry of National
Education/Institute for Atomic Physics and National Council of Scientific
Research in Higher Education (CNCSI-UEFISCDI), Romania; Ministry of Education
and Science of Russian Federation, Russian Academy of Sciences, Russian
Federal Agency of Atomic Energy, Russian Federal Agency for Science and
Innovations and The Russian Foundation for Basic Research; Ministry of
Education of Slovakia; Department of Science and Technology, South Africa;
Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas
(CIEMAT), E-Infrastructure shared between Europe and Latin America (EELA),
Ministerio de Economía y Competitividad (MINECO) of Spain, Xunta de Galicia
(Consellería de Educación), Centro de Aplicaciones Tecnológicas y Desarrollo
Nuclear (CEADEN), Cubaenergía, Cuba, and IAEA (International Atomic Energy
Agency); Swedish Research Council (VR) and Knut $\&$ Alice Wallenberg
Foundation (KAW); Ukraine Ministry of Education and Science; United Kingdom
Science and Technology Facilities Council (STFC); The United States Department
of Energy, the United States National Science Foundation, the State of Texas,
and the State of Ohio; Ministry of Science, Education and Sports of Croatia
and Unity through Knowledge Fund, Croatia; Council of Scientific and
Industrial Research (CSIR), New Delhi, India; Pontificia Universidad Católica
del Perú.

## References

  * [1] S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg, C. Ratti, and K. K. Szabo, “The QCD equation of state with dynamical quarks,” JHEP 11 (2010) 077, arXiv:1007.2580 [hep-lat].
  * [2] HotQCD Collaboration, A. Bazavov et al., “Equation of state in ( 2+1 )-flavor QCD,” Phys. Rev. D90 (2014) 094503, arXiv:1407.6387 [hep-lat].
  * [3] ALICE Collaboration, K. Aamodt et al., “Centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV,” Phys. Rev. Lett. 106 (2011) 032301, arXiv:1012.1657 [nucl-ex].
  * [4] ALICE Collaboration, J. Adam et al., “Centrality dependence of the charged-particle multiplicity density at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV,” arXiv:1512.06104 [nucl-ex].
  * [5] G. Odyniec, “RHIC Beam Energy Scan Program: Phase I and II,” PoS CPOD2013 (2013) 043.
  * [6] STAR Collaboration, L. Adamczyk et al., “Inclusive charged hadron elliptic flow in Au + Au collisions at $\sqrt{s_{NN}}$ = 7.7 - 39 GeV,” Phys. Rev. C86 (2012) 054908, arXiv:1206.5528 [nucl-ex].
  * [7] PHENIX Collaboration, S. S. Adler et al., “Transverse-energy distributions at midrapidity in p+p , d+Au , and Au+Au collisions at $\sqrt{s_{NN}}=62.4-200$ GeV and implications for particle-production models,” Phys. Rev. C89 no. 4, (2014) 044905, arXiv:1312.6676 [nucl-ex].
  * [8] R. V. Gavai and S. Gupta, “The Critical end point of QCD,” Phys. Rev. D71 (2005) 114014, arXiv:hep-lat/0412035 [hep-lat].
  * [9] P. Braun-Munzinger, V. Koch, T. Schaefer, and J. Stachel, “Properties of hot and dense matter from relativistic heavy ion collisions,” arXiv:1510.00442 [nucl-th].
  * [10] J.-Y. Ollitrault, “Anisotropy as a signature of transverse collective flow,” Phys. Rev. D46 (1992) 229–245.
  * [11] S. A. Voloshin, A. M. Poskanzer, and R. Snellings, “Collective phenomena in non-central nuclear collisions,” arXiv:0809.2949 [nucl-ex].
  * [12] S. Voloshin and Y. Zhang, “Flow study in relativistic nuclear collisions by Fourier expansion of Azimuthal particle distributions,” Z. Phys. C70 (1996) 665–672, arXiv:hep-ph/9407282 [hep-ph].
  * [13] ALICE Collaboration, K. Aamodt et al., “Elliptic flow of charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV,” Phys. Rev. Lett. 105 (2010) 252302, arXiv:1011.3914 [nucl-ex].
  * [14] STAR Collaboration, K. H. Ackermann et al., “Elliptic flow in Au + Au collisions at $\sqrt{s_{NN}}=130$ 130 GeV,” Phys. Rev. Lett. 86 (2001) 402–407, arXiv:nucl-ex/0009011 [nucl-ex].
  * [15] M. Luzum and P. Romatschke, “Viscous Hydrodynamic Predictions for Nuclear Collisions at the LHC,” Phys. Rev. Lett. 103 (2009) 262302, arXiv:0901.4588 [nucl-th].
  * [16] T. Hirano, P. Huovinen, and Y. Nara, “Elliptic flow in U+U collisions at $\sqrt{s_{NN}}=200$ GeV and in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV: Prediction from a hybrid approach,” Phys. Rev. C83 (2011) 021902, arXiv:1010.6222 [nucl-th].
  * [17] T. Hirano, U. W. Heinz, D. Kharzeev, R. Lacey, and Y. Nara, “Hadronic dissipative effects on elliptic flow in ultrarelativistic heavy-ion collisions,” Phys. Lett. B636 (2006) 299–304, arXiv:nucl-th/0511046 [nucl-th].
  * [18] H.-J. Drescher, A. Dumitru, and J.-Y. Ollitrault, “The Centrality dependence of elliptic flow at LHC,” in Workshop on Heavy Ion Collisions at the LHC: Last Call for Predictions Geneva, Switzerland, May 14-June 8, 2007.  2007\.  arXiv:0706.1707 [nucl-th].  https://inspirehep.net/record/752929/files/arXiv:0706.1707.pdf.
  * [19] B. Alver and G. Roland, “Collision geometry fluctuations and triangular flow in heavy-ion collisions,” Phys. Rev. C81 (2010) 054905, arXiv:1003.0194 [nucl-th]. [Erratum: Phys. Rev.C82,039903(2010)].
  * [20] ALICE Collaboration, K. Aamodt et al., “Higher harmonic anisotropic flow measurements of charged particles in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV,” Phys. Rev. Lett. 107 (2011) 032301, arXiv:1105.3865 [nucl-ex].
  * [21] ATLAS Collaboration, G. Aad et al., “Measurement of the azimuthal anisotropy for charged particle production in $\sqrt{s_{NN}}=2.76$ TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C86 (2012) 014907, arXiv:1203.3087 [hep-ex].
  * [22] CMS Collaboration, S. Chatrchyan et al., “Measurement of higher-order harmonic azimuthal anisotropy in PbPb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV,” Phys. Rev. C89 no. 4, (2014) 044906, arXiv:1310.8651 [nucl-ex].
  * [23] ATLAS Collaboration, G. Aad et al., “Measurement of event-plane correlations in $\sqrt{s_{NN}}=2.76$ TeV lead-lead collisions with the ATLAS detector,” Phys. Rev. C90 no. 2, (2014) 024905, arXiv:1403.0489 [hep-ex].
  * [24] P. Kovtun, D. T. Son, and A. O. Starinets, “Viscosity in strongly interacting quantum field theories from black hole physics,” Phys. Rev. Lett. 94 (2005) 111601, arXiv:hep-th/0405231 [hep-th].
  * [25] H. Niemi, K. J. Eskola, R. Paatelainen, and K. Tuominen, “Predictions for 5.023 TeV Pb+Pb collisions at the LHC,” arXiv:1511.04296 [hep-ph].
  * [26] N. Borghini, P. M. Dinh, and J.-Y. Ollitrault, “A New method for measuring azimuthal distributions in nucleus-nucleus collisions,” Phys. Rev. C63 (2001) 054906, arXiv:nucl-th/0007063 [nucl-th].
  * [27] J. Noronha-Hostler, M. Luzum, and J.-Y. Ollitrault, “Hydrodynamic predictions for 5.02 TeV Pb-Pb collisions,” arXiv:1511.06289 [nucl-th].
  * [28] C. Shen and U. Heinz, “Collision Energy Dependence of Viscous Hydrodynamic Flow in Relativistic Heavy-Ion Collisions,” Phys. Rev. C85 (2012) 054902, arXiv:1202.6620 [nucl-th]. [Erratum: Phys. Rev.C86,049903(2012)].
  * [29] J. Auvinen and H. Petersen, “Evolution of elliptic and triangular flow as a function of beam energy in a hybrid model,” J. Phys. Conf. Ser. 503 (2014) 012025, arXiv:1310.7751 [nucl-th].
  * [30] ALICE Collaboration, K. Aamodt et al., “The ALICE experiment at the CERN LHC,” JINST 3 (2008) S08002.
  * [31] ALICE Collaboration, P. Cortese et al., “ALICE: Physics performance report, volume II,” J. Phys. G32 (2006) 1295–2040.
  * [32] ALICE Collaboration, B. Abelev et al., “Centrality determination of Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV with ALICE,” Phys. Rev. C88 no. 4, (2013) 044909, arXiv:1301.4361 [nucl-ex].
  * [33] X.-N. Wang and M. Gyulassy, “HIJING: A Monte Carlo model for multiple jet production in p p, p A and A A collisions,” Phys. Rev. D44 (1991) 3501–3516.
  * [34] R. Brun, F. Carminati, and S. Giani, “GEANT Detector Description and Simulation Tool,” 1994.  https://cds.cern.ch/record/118715?ln=en.
  * [35] Y. Zhou, X. Zhu, P. Li, and H. Song, “Investigation of possible hadronic flow in $\sqrt{s_{NN}}=5.02$ TeV $p-Pb$ collisions,” Phys. Rev. C91 (2015) 064908, arXiv:1503.06986 [nucl-th].
  * [36] A. Bilandzic, R. Snellings, and S. Voloshin, “Flow analysis with cumulants: Direct calculations,” Phys. Rev. C83 (2011) 044913, arXiv:1010.0233 [nucl-ex].
  * [37] A. Bilandzic, C. H. Christensen, K. Gulbrandsen, A. Hansen, and Y. Zhou, “Generic framework for anisotropic flow analyses with multiparticle azimuthal correlations,” Phys. Rev. C89 no. 6, (2014) 064904, arXiv:1312.3572 [nucl-ex].
  * [38] ALICE Collaboration, B. B. Abelev et al., “Multiparticle azimuthal correlations in p -Pb and Pb-Pb collisions at the CERN Large Hadron Collider,” Phys. Rev. C90 no. 5, (2014) 054901, arXiv:1406.2474 [nucl-ex].

##  Appendix A The ALICE Collaboration

J. Adamorg39&D. Adamováorg84&M.M. Aggarwalorg88&G. Aglieri Rinellaorg35&M.
Agnelloorg110&N. Agrawalorg47&Z. Ahammedorg132&S. Ahmadorg19&S.U. Ahnorg68&S.
Aiolaorg136&A. Akindinovorg58&S.N. Alamorg132&D.S.D. Albuquerqueorg121&D.
Aleksandrovorg80&B. Alessandroorg110&D. Alexandreorg101&R. Alfaro
Molinaorg64&A. Aliciorg104,org12&A. Alkinorg3&J.R.M. Almarazorg119&J.
Almeorg37&T. Altorg42&S. Altinpinarorg18&I. Altsybeevorg131&C. Alves Garcia
Pradoorg120&C. Andreiorg78&A. Andronicorg97&V. Anguelovorg94&T.
Antičićorg98&F. Antinoriorg107&P. Antonioliorg104&L. Aphecetcheorg113&H.
Appelshäuserorg53&S. Arcelliorg27&R. Arnaldiorg110&O.W. Arnoldorg36,org93&I.C.
Arseneorg22&M. Arslandokorg53&B. Audurierorg113&A. Augustinusorg35&R.
Averbeckorg97&M.D. Azmiorg19&A. Badalàorg106&Y.W. Baekorg67&S.
Bagnascoorg110&R. Bailhacheorg53&R. Balaorg91&S. Balasubramanianorg136&A.
Baldisseriorg15&R.C. Baralorg61&A.M. Barbanoorg26&R. Barberaorg28&F.
Barileorg32&G.G. Barnaföldiorg135&L.S. Barnbyorg101&V. Barretorg70&P.
Bartaliniorg7&K. Barthorg35&J. Bartkeorg117&E. Bartschorg53&M. Basileorg27&N.
Bastidorg70&S. Basuorg132&B. Bathenorg54&G. Batigneorg113&A. Batista
Camejoorg70&B. Batyunyaorg66&P.C. Batzingorg22&I.G. Beardenorg81&H.
Beckorg53&C. Beddaorg110&N.K. Beheraorg50,org48&I. Belikovorg55&F.
Belliniorg27&H. Bello Martinezorg2&R. Bellwiedorg122&R. Belmontorg134&E.
Belmont-Morenoorg64&V. Belyaevorg75&P. Benacekorg84&G. Bencediorg135&S.
Beoleorg26&I. Berceanuorg78&A. Bercuciorg78&Y. Berdnikovorg86&D.
Berenyiorg135&R.A. Bertensorg57&D. Berzanoorg35&L. Betevorg35&A.
Bhasinorg91&I.R. Bhatorg91&A.K. Bhatiorg88&B. Bhattacharjeeorg44&J.
Bhomorg128,org117&L. Bianchiorg122&N. Bianchiorg72&C. Bianchinorg134&J.
Bielčíkorg39&J. Bielčíkováorg84&A. Bilandzicorg81,org36,org93&G. Biroorg135&R.
Biswasorg4&S. Biswasorg4,org79&S. Bjelogrlicorg57&J.T. Blairorg118&D.
Blauorg80&C. Blumeorg53&F. Bockorg74,org94&A. Bogdanovorg75&H. Bøggildorg81&L.
Boldizsárorg135&M. Bombaraorg40&J. Bookorg53&H. Borelorg15&A. Borissovorg96&M.
Borriorg83,org124&F. Bossúorg65&E. Bottaorg26&C. Bourjauorg81&P. Braun-
Munzingerorg97&M. Bregantorg120&T. Breitnerorg52&T.A. Brokerorg53&T.A.
Browningorg95&M. Brozorg39&E.J. Bruckenorg45&E. Brunaorg110&G.E. Brunoorg32&D.
Budnikovorg99&H. Bueschingorg53&S. Bufalinoorg35,org26&P. Buncicorg35&O.
Buschorg94,org128&Z. Butheleziorg65&J.B. Buttorg16&J.T. Buxtonorg20&J.
Cabalaorg115&D. Caffarriorg35&X. Caiorg7&H. Cainesorg136&L. Calero
Diazorg72&A. Calivaorg57&E. Calvo Villarorg102&P. Cameriniorg25&F.
Carenaorg35&W. Carenaorg35&F. Carnesecchiorg27&J. Castillo
Castellanosorg15&A.J. Castroorg125&E.A.R. Casulaorg24&C. Ceballos
Sanchezorg9&J. Cepilaorg39&P. Cerelloorg110&J. Cerkalaorg115&B. Changorg123&S.
Chapelandorg35&M. Chartierorg124&J.L. Charvetorg15&S. Chattopadhyayorg132&S.
Chattopadhyayorg100&A. Chauvinorg93,org36&V. Chelnokovorg3&M. Cherneyorg87&C.
Cheshkovorg130&B. Cheynisorg130&V. Chibante Barrosoorg35&D.D.
Chinellatoorg121&S. Choorg50&P. Chochulaorg35&K. Choiorg96&M.
Chojnackiorg81&S. Choudhuryorg132&P. Christakoglouorg82&C.H.
Christensenorg81&P. Christiansenorg33&T. Chujoorg128&S.U. Chungorg96&C.
Cicaloorg105&L. Cifarelliorg12,org27&F. Cindoloorg104&J. Cleymansorg90&F.
Colamariaorg32&D. Colellaorg59,org35&A. Colluorg74,org24&M. Colocciorg27&G.
Conesa Balbastreorg71&Z. Conesa del Valleorg51&M.E.
Connorsidp1779728,org136&J.G. Contrerasorg39&T.M. Cormierorg85&Y. Corrales
Moralesorg110&I. Cortés Maldonadoorg2&P. Corteseorg31&M.R. Cosentinoorg120&F.
Costaorg35&P. Crochetorg70&R. Cruz Albinoorg11&E. Cuautleorg63&L.
Cunqueiroorg54,org35&T. Dahmsorg93,org36&A. Daineseorg107&M.C. Danischorg94&A.
Danuorg62&D. Dasorg100&I. Dasorg100&S. Dasorg4&A. Dashorg79&S. Dashorg47&S.
Deorg120&A. De Caroorg12,org30&G. de Cataldoorg103&C. de Contiorg120&J. de
Cuvelandorg42&A. De Falcoorg24&D. De Gruttolaorg12,org30&N. De Marcoorg110&S.
De Pasqualeorg30&A. Deistingorg97,org94&A. Delofforg77&E. Dénesorg1350&C.
Deplanoorg82&P. Dhankherorg47&D. Di Bariorg32&A. Di Mauroorg35&P. Di
Nezzaorg72&M.A. Diaz Corcheroorg10&T. Dietelorg90&P. Dillensegerorg53&R.
Diviàorg35&Ø. Djuvslandorg18&A. Dobrinorg82,org62&D. Domenicis
Gimenezorg120&B. Dönigusorg53&O. Dordicorg22&T. Drozhzhovaorg53&A.K.
Dubeyorg132&A. Dublaorg57&L. Ducrouxorg130&P. Dupieuxorg70&R.J.
Ehlersorg136&D. Eliaorg103&E. Endressorg102&H. Engelorg52&E. Eppleorg136&B.
Erazmusorg113&I. Erdemirorg53&F. Erhardtorg129&B. Espagnonorg51&M.
Estienneorg113&S. Esumiorg128&J. Eumorg96&D. Evansorg101&S. Evdokimovorg111&G.
Eyyubovaorg39&L. Fabbiettiorg93,org36&D. Fabrisorg107&J. Faivreorg71&A.
Fantoniorg72&M. Faselorg74&L. Feldkamporg54&A. Felicielloorg110&G.
Feofilovorg131&J. Ferenceiorg84&A. Fernández Téllezorg2&E.G. Ferreiroorg17&A.
Ferrettiorg26&A. Festantiorg29&V.J.G. Feuillardorg15,org70&J.
Figielorg117&M.A.S. Figueredoorg124,org120&S. Filchaginorg99&D.
Finogeevorg56&F.M. Fiondaorg24&E.M. Fioreorg32&M.G. Fleckorg94&M.
Florisorg35&S. Foertschorg65&P. Fokaorg97&S. Fokinorg80&E. Fragiacomoorg109&A.
Francesconorg35,org29&U. Frankenfeldorg97&G.G. Fronzeorg26&U. Fuchsorg35&C.
Furgetorg71&A. Fursorg56&M. Fusco Girardorg30&J.J. Gaardhøjeorg81&M.
Gagliardiorg26&A.M. Gagoorg102&M. Gallioorg26&D.R. Gangadharanorg74&P.
Ganotiorg89&C. Gaoorg7&C. Garabatosorg97&E. Garcia-Solisorg13&C.
Gargiuloorg35&P. Gasikorg93,org36&E.F. Gaugerorg118&M. Germainorg113&A.
Gheataorg35&M. Gheataorg35,org62&P. Ghoshorg132&S.K. Ghoshorg4&P.
Gianottiorg72&P. Giubellinoorg110,org35&P. Giubilatoorg29&E. Gladysz-
Dziadusorg117&P. Glässelorg94&D.M. Goméz Coralorg64&A. Gomez Ramirezorg52&A.S.
Gonzalezorg35&V. Gonzalezorg10&P. González-Zamoraorg10&S. Gorbunovorg42&L.
Görlichorg117&S. Gotovacorg116&V. Grabskiorg64&O.A. Grachovorg136&L.K.
Graczykowskiorg133&K.L. Grahamorg101&A. Grelliorg57&A. Grigorasorg35&C.
Grigorasorg35&V. Grigorievorg75&A. Grigoryanorg1&S. Grigoryanorg66&B.
Grinyovorg3&N. Grionorg109&J.M. Gronefeldorg97&J.F. Grosse-Oetringhausorg35&R.
Grossoorg97&F. Guberorg56&R. Guernaneorg71&B. Guerzoniorg27&K.
Gulbrandsenorg81&T. Gunjiorg127&A. Guptaorg91&R. Guptaorg91&R. Haakeorg54&Ø.
Haalandorg18&C. Hadjidakisorg51&M. Haiducorg62&H. Hamagakiorg127&G.
Hamarorg135&J.C. Hamonorg55&J.W. Harrisorg136&A. Hartonorg13&D.
Hatzifotiadouorg104&S. Hayashiorg127&S.T. Heckelorg53&E. Hellbärorg53&H.
Helstruporg37&A. Herghelegiuorg78&G. Herrera Corralorg11&B.A. Hessorg34&K.F.
Hetlandorg37&H. Hillemannsorg35&B. Hippolyteorg55&D. Horakorg39&R.
Hosokawaorg128&P. Hristovorg35&T.J. Humanicorg20&N. Hussainorg44&T.
Hussainorg19&D. Hutterorg42&D.S. Hwangorg21&R. Ilkaevorg99&M. Inabaorg128&E.
Incaniorg24&M. Ippolitovorg75,org80&M. Irfanorg19&M. Ivanovorg97&V.
Ivanovorg86&V. Izucheevorg111&N. Jacazioorg27&P.M. Jacobsorg74&M.B.
Jadhavorg47&S. Jadlovskaorg115&J. Jadlovskyorg115,org59&C. Jahnkeorg120&M.J.
Jakubowskaorg133&H.J. Jangorg68&M.A. Janikorg133&P.H.S.Y. Jayarathnaorg122&C.
Jenaorg29&S. Jenaorg122&R.T. Jimenez Bustamanteorg97&P.G. Jonesorg101&A.
Juskoorg101&P. Kalinakorg59&A. Kalweitorg35&J. Kaminorg53&J.H. Kangorg137&V.
Kaplinorg75&S. Karorg132&A. Karasu Uysalorg69&O. Karavichevorg56&T.
Karavichevaorg56&L. Karayanorg97,org94&E. Karpechevorg56&U. Kebschullorg52&R.
Keidelorg138&D.L.D. Keijdenerorg57&M. Keilorg35&M. Mohisin
Khanidp3139616,org19&P. Khanorg100&S.A. Khanorg132&A. Khanzadeevorg86&Y.
Kharlovorg111&B. Kilengorg37&D.W. Kimorg43&D.J. Kimorg123&D. Kimorg137&H.
Kimorg137&J.S. Kimorg43&M. Kimorg137&S. Kimorg21&T. Kimorg137&S.
Kirschorg42&I. Kiselorg42&S. Kiselevorg58&A. Kisielorg133&G. Kissorg135&J.L.
Klayorg6&C. Kleinorg53&J. Kleinorg35&C. Klein-Bösingorg54&S. Klewinorg94&A.
Klugeorg35&M.L. Knichelorg94&A.G. Knospeorg118,org122&C. Kobdajorg114&M.
Kofaragoorg35&T. Kolleggerorg97&A. Kolojvariorg131&V. Kondratievorg131&N.
Kondratyevaorg75&E. Kondratyukorg111&A. Konevskikhorg56&M. Kopcikorg115&P.
Kostarakisorg89&M. Kourorg91&C. Kouzinopoulosorg35&O. Kovalenkoorg77&V.
Kovalenkoorg131&M. Kowalskiorg117&G. Koyithatta Meethaleveeduorg47&I.
Králikorg59&A. Kravčákováorg40&M. Krivdaorg59,org101&F. Krizekorg84&E.
Kryshenorg86,org35&M. Krzewickiorg42&A.M. Kuberaorg20&V. Kučeraorg84&C.
Kuhnorg55&P.G. Kuijerorg82&A. Kumarorg91&J. Kumarorg47&L. Kumarorg88&S.
Kumarorg47&P. Kurashviliorg77&A. Kurepinorg56&A.B. Kurepinorg56&A.
Kuryakinorg99&M.J. Kweonorg50&Y. Kwonorg137&S.L. La Pointeorg110&P. La
Roccaorg28&P. Ladron de Guevaraorg11&C. Lagana Fernandesorg120&I.
Lakomovorg35&R. Langoyorg41&C. Laraorg52&A. Lardeuxorg15&A. Lattucaorg26&E.
Laudiorg35&R. Leaorg25&L. Leardiniorg94&G.R. Leeorg101&S. Leeorg137&F.
Lehasorg82&R.C. Lemmonorg83&V. Lentiorg103&E. Leograndeorg57&I. León
Monzónorg119&H. León Vargasorg64&M. Leoncinoorg26&P. Lévaiorg135&S.
Liorg7,org70&X. Liorg14&J. Lienorg41&R. Lietavaorg101&S. Lindalorg22&V.
Lindenstruthorg42&C. Lippmannorg97&M.A. Lisaorg20&H.M. Ljunggrenorg33&D.F.
Lodatoorg57&P.I. Loenneorg18&V. Loginovorg75&C. Loizidesorg74&X. Lopezorg70&E.
López Torresorg9&A. Loweorg135&P. Luettigorg53&M. Lunardonorg29&G.
Luparelloorg25&T.H. Lutzorg136&A. Maevskayaorg56&M. Magerorg35&S.
Mahajanorg91&S.M. Mahmoodorg22&A. Maireorg55&R.D. Majkaorg136&M.
Malaevorg86&I. Maldonado Cervantesorg63&L. Malininaidp3840864,org66&D.
Mal’Kevichorg58&P. Malzacherorg97&A. Mamonovorg99&V. Mankoorg80&F.
Mansoorg70&V. Manzariorg103,org35&M. Marchisoneorg65,org126,org26&J.
Marešorg60&G.V. Margagliottiorg25&A. Margottiorg104&J. Marguttiorg57&A.
Marínorg97&C. Markertorg118&M. Marquardorg53&N.A. Martinorg97&J. Martin
Blancoorg113&P. Martinengoorg35&M.I. Martínezorg2&G. Martínez Garcíaorg113&M.
Martinez Pedreiraorg35&A. Masorg120&S. Masciocchiorg97&M. Maseraorg26&A.
Masoniorg105&A. Mastroserioorg32&A. Matyjaorg117&C. Mayerorg117&J.
Mazerorg125&M.A. Mazzoniorg108&D. Mcdonaldorg122&F. Meddiorg23&Y.
Melikyanorg75&A. Menchaca-Rochaorg64&E. Meninnoorg30&J. Mercado Pérezorg94&M.
Meresorg38&Y. Miakeorg128&M.M. Mieskolainenorg45&K. Mikhaylovorg66,org58&L.
Milanoorg35,org74&J. Milosevicorg22&A. Mischkeorg57&A.N. Mishraorg48&D.
Miśkowiecorg97&J. Mitraorg132&C.M. Mituorg62&N. Mohammadiorg57&B.
Mohantyorg132,org79&L. Molnarorg55,org113&L. Montaño Zetinaorg11&E.
Montesorg10&D.A. Moreira De Godoyorg54&L.A.P. Morenoorg2&S. Morettoorg29&A.
Morrealeorg113&A. Morschorg35&V. Mucciforaorg72&E. Mudnicorg116&D.
Mühlheimorg54&S. Muhuriorg132&M. Mukherjeeorg132&J.D. Mulliganorg136&M.G.
Munhozorg120&R.H. Munzerorg93,org36,org53&H. Murakamiorg127&S. Murrayorg65&L.
Musaorg35&J. Musinskyorg59&B. Naikorg47&R. Nairorg77&B.K. Nandiorg47&R.
Naniaorg104&E. Nappiorg103&M.U. Naruorg16&H. Natal da Luzorg120&C.
Nattrassorg125&S.R. Navarroorg2&K. Nayakorg79&R. Nayakorg47&T.K.
Nayakorg132&S. Nazarenkoorg99&A. Nedosekinorg58&L. Nellenorg63&F. Ngorg122&M.
Nicassioorg97&M. Niculescuorg62&J. Niedzielaorg35&B.S. Nielsenorg81&S.
Nikolaevorg80&S. Nikulinorg80&V. Nikulinorg86&F. Noferiniorg104,org12&P.
Nomokonovorg66&G. Noorenorg57&J.C.C. Norisorg2&J. Normanorg124&A.
Nyaninorg80&J. Nystrandorg18&H. Oeschlerorg94&S. Ohorg136&S.K. Ohorg67&A.
Ohlsonorg35&A. Okatanorg69&T. Okuboorg46&L. Olahorg135&J. Oleniaczorg133&A.C.
Oliveira Da Silvaorg120&M.H. Oliverorg136&J. Onderwaaterorg97&C.
Oppedisanoorg110&R. Oravaorg45&M. Oravecorg115&A. Ortiz Velasquezorg63&A.
Oskarssonorg33&J. Otwinowskiorg117&K. Oyamaorg94,org76&M. Ozdemirorg53&Y.
Pachmayerorg94&D. Paganoorg26&P. Paganoorg30&G. Paićorg63&S.K. Palorg132&J.
Panorg134&A.K. Pandeyorg47&V. Papikyanorg1&G.S. Pappalardoorg106&P.
Pareekorg48&W.J. Parkorg97&S. Parmarorg88&A. Passfeldorg54&V.
Paticchioorg103&R.N. Patraorg132&B. Paulorg100&H. Peiorg7&T. Peitzmannorg57&H.
Pereira Da Costaorg15&D. Peresunkoorg80,org75&C.E. Pérez Laraorg82&E. Perez
Lezamaorg53&V. Peskovorg53&Y. Pestovorg5&V. Petráčekorg39&V. Petrovorg111&M.
Petroviciorg78&C. Pettaorg28&S. Pianoorg109&M. Piknaorg38&P.
Pillotorg113&L.O.D.L. Pimentelorg81&O. Pinazzaorg104,org35&L.
Pinskyorg122&D.B. Piyarathnaorg122&M. Płoskońorg74&M. Planinicorg129&J.
Plutaorg133&S. Pochybovaorg135&P.L.M. Podesta-Lermaorg119&M.G.
Poghosyanorg85,org87&B. Polichtchoukorg111&N. Poljakorg129&W.
Poonsawatorg114&A. Poporg78&S. Porteboeuf-Houssaisorg70&J. Porterorg74&J.
Pospisilorg84&S.K. Prasadorg4&R. Preghenellaorg104,org35&F. Prinoorg110&C.A.
Pruneauorg134&I. Pshenichnovorg56&M. Puccioorg26&G. Pudduorg24&P.
Pujahariorg134&V. Puninorg99&J. Putschkeorg134&H. Qvigstadorg22&A.
Rachevskiorg109&S. Rahaorg4&S. Rajputorg91&J. Rakorg123&A.
Rakotozafindrabeorg15&L. Ramelloorg31&F. Ramiorg55&R. Raniwalaorg92&S.
Raniwalaorg92&S.S. Räsänenorg45&B.T. Rascanuorg53&D. Ratheeorg88&K.F.
Readorg85,org125&K. Redlichorg77&R.J. Reedorg134&A. Rehmanorg18&P.
Reicheltorg53&F. Reidtorg94,org35&X. Renorg7&R. Renfordtorg53&A.R.
Reolonorg72&A. Reshetinorg56&K. Reygersorg94&V. Riabovorg86&R.A. Ricciorg73&T.
Richertorg33&M. Richterorg22&P. Riedlerorg35&W. Rieglerorg35&F. Riggiorg28&C.
Risteaorg62&E. Roccoorg57&M. Rodríguez Cahuantziorg11,org2&A. Rodriguez
Mansoorg82&K. Røedorg22&E. Rogochayaorg66&D. Rohrorg42&D. Röhrichorg18&F.
Ronchettiorg72,org35&L. Ronfletteorg113&P. Rosnetorg70&A. Rossiorg35,org29&F.
Roukoutakisorg89&A. Royorg48&C. Royorg55&P. Royorg100&A.J. Rubio
Monteroorg10&R. Ruiorg25&R. Russoorg26&E. Ryabinkinorg80&Y. Ryabovorg86&A.
Rybickiorg117&S. Saarinenorg45&S. Sadhuorg132&S. Sadovskyorg111&K.
Šafaříkorg35&B. Sahlmullerorg53&P. Sahooorg48&R. Sahooorg48&S. Sahooorg61&P.K.
Sahuorg61&J. Sainiorg132&S. Sakaiorg74&M.A. Salehorg134&J. Salzwedelorg20&S.
Sambyalorg91&V. Samsonovorg86&L. Šándororg59&A. Sandovalorg64&M. Sanoorg128&D.
Sarkarorg132&N. Sarkarorg132&P. Sarmaorg44&E. Scapparoneorg104&F.
Scarlassaraorg29&C. Schiauaorg78&R. Schickerorg94&C. Schmidtorg97&H.R.
Schmidtorg34&S. Schuchmannorg53&J. Schukraftorg35&M. Schulcorg39&Y.
Schutzorg35,org113&K. Schwarzorg97&K. Schwedaorg97&G. Scioliorg27&E.
Scomparinorg110&R. Scottorg125&M. Šefčíkorg40&J.E. Segerorg87&Y.
Sekiguchiorg127&D. Sekihataorg46&I. Selyuzhenkovorg97&K. Senosiorg65&S.
Senyukovorg3,org35&E. Serradillaorg10,org64&A. Sevcencoorg62&A.
Shabanovorg56&A. Shabetaiorg113&O. Shaduraorg3&R. Shahoyanorg35&M.I.
Shahzadorg16&A. Shangaraevorg111&A. Sharmaorg91&M. Sharmaorg91&M.
Sharmaorg91&N. Sharmaorg125&A.I. Sheikhorg132&K. Shigakiorg46&Q. Shouorg7&K.
Shtejerorg26,org9&Y. Sibiriakorg80&S. Siddhantaorg105&K.M. Sielewiczorg35&T.
Siemiarczukorg77&D. Silvermyrorg33&C. Silvestreorg71&G. Simatovicorg129&G.
Simonettiorg35&R. Singarajuorg132&R. Singhorg79&S. Singhaorg132,org79&V.
Singhalorg132&B.C. Sinhaorg132&T. Sinhaorg100&B. Sitarorg38&M. Sittaorg31&T.B.
Skaaliorg22&M. Slupeckiorg123&N. Smirnovorg136&R.J.M. Snellingsorg57&T.W.
Snellmanorg123&J. Songorg96&M. Songorg137&Z. Songorg7&F. Soramelorg29&S.
Sorensenorg125&R.D.de Souzaorg121&F. Sozziorg97&M. Spacekorg39&E.
Spiritiorg72&I. Sputowskaorg117&M. Spyropoulou-Stassinakiorg89&J.
Stachelorg94&I. Stanorg62&P. Stankusorg85&E. Stenlundorg33&G. Steynorg65&J.H.
Stillerorg94&D. Stoccoorg113&P. Strmenorg38&A.A.P. Suaideorg120&T.
Sugitateorg46&C. Suireorg51&M. Suleymanovorg16&M. Suljicorg250&R.
Sultanovorg58&M. Šumberaorg84&S. Sumowidagdoorg49&A. Szaboorg38&A. Szanto de
Toledoorg1200&I. Szarkaorg38&A. Szczepankiewiczorg35&M. Szymanskiorg133&U.
Tabassamorg16&J. Takahashiorg121&G.J. Tambaveorg18&N. Tanakaorg128&M.
Tarhiniorg51&M. Tariqorg19&M.G. Tarzilaorg78&A. Tauroorg35&G. Tejeda
Muñozorg2&A. Telescaorg35&K. Terasakiorg127&C. Terrevoliorg29&B.
Teyssierorg130&J. Thäderorg74&D. Thakurorg48&D. Thomasorg118&R.
Tieulentorg130&A.R. Timminsorg122&A. Toiaorg53&S. Trogoloorg26&G.
Trombettaorg32&V. Trubnikovorg3&W.H. Trzaskaorg123&T. Tsujiorg127&A.
Tumkinorg99&R. Turrisiorg107&T.S. Tveterorg22&K. Ullalandorg18&A.
Urasorg130&G.L. Usaiorg24&A. Utrobicicorg129&M. Valaorg59&L. Valencia
Palomoorg70&S. Valleroorg26&J. Van Der Maarelorg57&J.W. Van Hoorneorg35&M. van
Leeuwenorg57&T. Vanatorg84&P. Vande Vyvreorg35&D. Vargaorg135&A. Vargasorg2&M.
Vargyasorg123&R. Varmaorg47&M. Vasileiouorg89&A. Vasilievorg80&A.
Vauthierorg71&V. Vecherninorg131&A.M. Veenorg57&M. Veldhoenorg57&A.
Velureorg18&E. Vercellinorg26&S. Vergara Limónorg2&R. Vernetorg8&M.
Verweijorg134&L. Vickovicorg116&G. Viestiorg290&J. Viinikainenorg123&Z.
Vilakaziorg126&O. Villalobos Baillieorg101&A. Villatoro Telloorg2&A.
Vinogradovorg80&L. Vinogradovorg131&Y. Vinogradovorg990&T. Virgiliorg30&V.
Vislaviciusorg33&Y.P. Viyogiorg132&A. Vodopyanovorg66&M.A. Völklorg94&K.
Voloshinorg58&S.A. Voloshinorg134&G. Volpeorg32,org135&B. von Hallerorg35&I.
Vorobyevorg36,org93&D. Vranicorg97,org35&J. Vrlákováorg40&B. Vulpescuorg70&B.
Wagnerorg18&J. Wagnerorg97&H. Wangorg57&M. Wangorg7,org113&D.
Watanabeorg128&Y. Watanabeorg127&M. Weberorg35,org112&S.G. Weberorg97&D.F.
Weiserorg94&J.P. Wesselsorg54&U. Westerhofforg54&A.M. Whiteheadorg90&J.
Wiechulaorg34&J. Wikneorg22&G. Wilkorg77&J. Wilkinsonorg94&M.C.S.
Williamsorg104&B. Windelbandorg94&M. Winnorg94&H. Yangorg57&P. Yangorg7&S.
Yanoorg46&Z. Yasinorg16&Z. Yinorg7&H. Yokoyamaorg128&I.-K. Yooorg96&J.H.
Yoonorg50&V. Yurchenkoorg3&I. Yushmanovorg80&A. Zaborowskaorg133&V.
Zaccoloorg81&A. Zamanorg16&C. Zampolliorg104,org35&H.J.C. Zanoliorg120&S.
Zaporozhetsorg66&N. Zardoshtiorg101&A. Zarochentsevorg131&P. Závadaorg60&N.
Zaviyalovorg99&H. Zbroszczykorg133&I.S. Zguraorg62&M. Zhalovorg86&H.
Zhangorg18&X. Zhangorg74,org7&Y. Zhangorg7&C. Zhangorg57&Z. Zhangorg7&C.
Zhaoorg22&N. Zhigarevaorg58&D. Zhouorg7&Y. Zhouorg81&Z. Zhouorg18&H.
Zhuorg18&J. Zhuorg7,org113&A. Zichichiorg27,org12&A. Zimmermannorg94&M.B.
Zimmermannorg54,org35&G. Zinovjevorg3&M. Zyzakorg42

## Affiliation notes

0Deceased

idp1779728Also at: Georgia State University, Atlanta, Georgia, United States

idp3139616Also at: Also at Department of Applied Physics, Aligarh Muslim
University, Aligarh, India

idp3840864Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn
Institute of Nuclear, Physics, Moscow, Russia

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

org4Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org5Budker Institute for Nuclear Physics, Novosibirsk, Russia

org6California Polytechnic State University, San Luis Obispo, California,
United States

org7Central China Normal University, Wuhan, China

org8Centre de Calcul de l’IN2P3, Villeurbanne, France

org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain

org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi”, Rome, Italy

org13Chicago State University, Chicago, Illinois, USA

org14China Institute of Atomic Energy, Beijing, China

org15Commissariat à l’Energie Atomique, IRFU, Saclay, France

org16COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan

org17Departamento de Física de Partículas and IGFAE, Universidad de Santiago
de Compostela, Santiago de Compostela, Spain

org18Department of Physics and Technology, University of Bergen, Bergen,
Norway

org19Department of Physics, Aligarh Muslim University, Aligarh, India

org20Department of Physics, Ohio State University, Columbus, Ohio, United
States

org21Department of Physics, Sejong University, Seoul, South Korea

org22Department of Physics, University of Oslo, Oslo, Norway

org23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN
Rome, Italy

org24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org30Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org31Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy

org32Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org33Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden

org34Eberhard Karls Universität Tübingen, Tübingen, Germany

org35European Organization for Nuclear Research (CERN), Geneva, Switzerland

org36Excellence Cluster Universe, Technische Universität München, Munich,
Germany

org37Faculty of Engineering, Bergen University College, Bergen, Norway

org38Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia

org39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org40Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org41Faculty of Technology, Buskerud and Vestfold University College,
Vestfold, Norway

org42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org43Gangneung-Wonju National University, Gangneung, South Korea

org44Gauhati University, Department of Physics, Guwahati, India

org45Helsinki Institute of Physics (HIP), Helsinki, Finland

org46Hiroshima University, Hiroshima, Japan

org47Indian Institute of Technology Bombay (IIT), Mumbai, India

org48Indian Institute of Technology Indore, Indore (IITI), India

org49Indonesian Institute of Sciences, Jakarta, Indonesia

org50Inha University, Incheon, South Korea

org51Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud,
CNRS-IN2P3, Orsay, France

org52Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org53Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org54Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
Münster, Germany

org55Institut Pluridisciplinaire Hubert Curien (IPHC), Université de
Strasbourg, CNRS-IN2P3, Strasbourg, France

org56Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org57Institute for Subatomic Physics of Utrecht University, Utrecht,
Netherlands

org58Institute for Theoretical and Experimental Physics, Moscow, Russia

org59Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org60Institute of Physics, Academy of Sciences of the Czech Republic, Prague,
Czech Republic

org61Institute of Physics, Bhubaneswar, India

org62Institute of Space Science (ISS), Bucharest, Romania

org63Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org64Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org65iThemba LABS, National Research Foundation, Somerset West, South Africa

org66Joint Institute for Nuclear Research (JINR), Dubna, Russia

org67Konkuk University, Seoul, South Korea

org68Korea Institute of Science and Technology Information, Daejeon, South
Korea

org69KTO Karatay University, Konya, Turkey

org70Laboratoire de Physique Corpusculaire (LPC), Clermont Université,
Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France

org71Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org72Laboratori Nazionali di Frascati, INFN, Frascati, Italy

org73Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy

org74Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org75Moscow Engineering Physics Institute, Moscow, Russia

org76Nagasaki Institute of Applied Science, Nagasaki, Japan

org77National Centre for Nuclear Studies, Warsaw, Poland

org78National Institute for Physics and Nuclear Engineering, Bucharest,
Romania

org79National Institute of Science Education and Research, Bhubaneswar, India

org80National Research Centre Kurchatov Institute, Moscow, Russia

org81Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org82Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam,
Netherlands

org83Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org84Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež
u Prahy, Czech Republic

org85Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org86Petersburg Nuclear Physics Institute, Gatchina, Russia

org87Physics Department, Creighton University, Omaha, Nebraska, United States

org88Physics Department, Panjab University, Chandigarh, India

org89Physics Department, University of Athens, Athens, Greece

org90Physics Department, University of Cape Town, Cape Town, South Africa

org91Physics Department, University of Jammu, Jammu, India

org92Physics Department, University of Rajasthan, Jaipur, India

org93Physik Department, Technische Universität München, Munich, Germany

org94Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org95Purdue University, West Lafayette, Indiana, United States

org96Pusan National University, Pusan, South Korea

org97Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung, Darmstadt, Germany

org98Rudjer Bošković Institute, Zagreb, Croatia

org99Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org100Saha Institute of Nuclear Physics, Kolkata, India

org101School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org102Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org103Sezione INFN, Bari, Italy

org104Sezione INFN, Bologna, Italy

org105Sezione INFN, Cagliari, Italy

org106Sezione INFN, Catania, Italy

org107Sezione INFN, Padova, Italy

org108Sezione INFN, Rome, Italy

org109Sezione INFN, Trieste, Italy

org110Sezione INFN, Turin, Italy

org111SSC IHEP of NRC Kurchatov institute, Protvino, Russia

org112Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

org113SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France

org114Suranaree University of Technology, Nakhon Ratchasima, Thailand

org115Technical University of Košice, Košice, Slovakia

org116Technical University of Split FESB, Split, Croatia

org117The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org118The University of Texas at Austin, Physics Department, Austin, Texas,
USA

org119Universidad Autónoma de Sinaloa, Culiacán, Mexico

org120Universidade de São Paulo (USP), São Paulo, Brazil

org121Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org122University of Houston, Houston, Texas, United States

org123University of Jyväskylä, Jyväskylä, Finland

org124University of Liverpool, Liverpool, United Kingdom

org125University of Tennessee, Knoxville, Tennessee, United States

org126University of the Witwatersrand, Johannesburg, South Africa

org127University of Tokyo, Tokyo, Japan

org128University of Tsukuba, Tsukuba, Japan

org129University of Zagreb, Zagreb, Croatia

org130Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, France

org131V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia

org132Variable Energy Cyclotron Centre, Kolkata, India

org133Warsaw University of Technology, Warsaw, Poland

org134Wayne State University, Detroit, Michigan, United States

org135Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary

org136Yale University, New Haven, Connecticut, United States

org137Yonsei University, Seoul, South Korea

org138Zentrum für Technologietransfer und Telekommunikation (ZTT),
Fachhochschule Worms, Worms, Germany


††thanks:  0000-0001-9980-0363

# Percolation thresholds on triangular lattice for neighbourhoods containing
sites up-to the fifth coordination zone

Krzysztof Malarz<EMAIL_ADDRESS>AGH University of Science and Technology,
Faculty of Physics and Applied Computer Science,
al. Mickiewicza 30, 30-059 Kraków, Poland

###### Abstract

We determine thresholds $p_{c}$ for random-site percolation on a triangular
lattice for all available neighborhoods containing sites from the first to the
fifth coordination zones, including their complex combinations. There are 31
distinct neighbourhoods. The dependence of the value of the percolation
thresholds $p_{c}$ on the coordination number $z$ are tested against various
theoretical predictions. The newly proposed single scalar index
$\xi=\sum_{i}z_{i}r_{i}^{2}/i$ (depending on the coordination zone number $i$,
the neighbourhood coordination number $z$ and the square-distance $r^{2}$ to
sites in $i$-th coordination zone from the central site) allows to
differentiate among various neighbourhoods and relate $p_{c}$ to $\xi$. The
thresholds roughly follow a power law $p_{c}\propto\xi^{-\gamma}$ with
$\gamma\approx 0.710(19)$.

random site percolation; triangular lattice; complex and extended
neighborhoods; Newman–Ziff algorithm; Bastas et al. method; finite size
scaling hypothesis; analytical formulas for percolation thresholds

##  I Introduction

The concepts of site and bond percolation Stauffer and Aharony (1994); Wierman
(2014) introduced in middle fifties Broadbent and Hammersley (1957);
Hammersley (1957) and since then have been applied in various fields of
science ranging from agriculture Ramírez  _et al._ (2020) via studies of
polymer composites Zhang  _et al._ (2020), materials science Cheng  _et al._
(2020), forest fires Malarz  _et al._ (2002), oil and gas exploration
Ghanbarian  _et al._ (2020), quantifying urban areas Cao  _et al._ (2020),
Bitcoins transfer Bartolucci  _et al._ (2020), diseases propagation Ziff
(2021) to transportation networks Dong  _et al._ (2020) (see Refs. Li  _et
al._ , 2021; Saberi, 2015 for reviews).

Percolation is an example of phenomenon where a geometrical phase transition
(on $d$-dimensional lattice) takes place. The critical parameter (called
percolation threshold $p_{c}$ Frisch  _et al._ (1961)) separates two phases:
one for a low occupation probability $p<p_{c}$ and the other for $p>p_{c}$. In
the low-$p$ phase the system behaves as an insulator (without connectivity
path leading between system boundaries) while for the high-$p$ phase there is
a giant component spanning the system and connecting opposite boundaries; in
effect the system behaves as a conductor, when one refers to the electric
analogy.

The percolation thresholds were initially estimated for nearest neighbour
interactions Dean (1963); Dean and Bird (1967) but later also complex (or
extended) neighbourhoods were studied for 2D (square Dalton  _et al._ (1964);
Domb and Dalton (1966); Gouker and Family (1983); Malarz and Galam (2005);
Galam and Malarz (2005); Majewski and Malarz (2007); Xun  _et al._ (2021),
triangular Dalton  _et al._ (1964); Domb and Dalton (1966); d’Iribarne  _et
al._ (1999); Malarz (2020), honeycomb Dalton  _et al._ (1964)), 3D (simple
cubic Kurzawski and Malarz (2012); Malarz (2015); Xun  _et al._ (2021)) and 4D
(simple hyper-cubic Kotwica  _et al._ (2019)) lattices.

Very recently, we have computed percolation thresholds for random site
percolation on triangular lattice with complex neighbourhoods with hexagonal
symmetry Malarz (2020). Here we supplement these results with 31 percolation
thresholds estimations for all neighbourhoods on triangular lattice containing
sites from the first, the second, the third, the fourth and the fifth
coordination zones (see Figure 1). Some of these neighbourhoods—those
containing sites from the fifth coordination zone—are presented in Figure A1
in Ref. Sup, . The lattice names follow convention proposed in Ref. Xun  _et
al._ , 2021 reflecting lattice topology (here tr, i.e. triangular lattice) and
numerical string specifying the coordination zones, where sites constituting
the neighbourhood come from.

Additionally—for triangular lattice and complex neighbourhoods—we test the
dependence of $p_{c}$ on the coordination number $z$, following the idea of
Ref. Xun  _et al._ , 2021. However, instead of values of the percolation
thresholds for selected (mainly compact) neighbourhoods, we use the mean
values $\bar{p}_{c}$ of percolation thresholds $p_{c}$ averaged over all
available neighbourhoods with given coordination number $z$. Unfortunately,
values of $\bar{p}_{c}$ do not follow any of dependencies proposed in Ref. Xun
_et al._ , 2021.

Finally, we propose a scalar quantity $\xi$ which may be helpful for
differentiating among various neighbourhoods. The quantity is based on the
coordination zone $i$, the sites number $z$ and the sites distances $r$ to the
central site in the neighbourhood. The dependency of $p_{c}$ on this newly
proposed index $\xi$ follows roughly a power law with an exponent close to
$-0.710(19)$.

(a)

(b)

(c)

(d)

(e)

Figure 1: (Color online). Basic neighbourhoods corresponding to subsequent
coordination zones $i=1,\cdots,5$ on the triangular lattice. The symbol $r$
stands for the Euclidean distance of black sites to the central one (in red)
and $z$ indicates the number of sites in the neighbourhood. Examples of
fifteen neighborhoods containing the next-next-next-next-nearest neighbors
(with sites from the 5-th coordination zone) on triangular lattice are
presented in Figure A1 in Ref. Sup, . (a) tr-1: $i=1$, $r^{2}=1$, $z=6$, (b)
tr-2: $i=2$, $r^{2}=3$, $z=6$, (c) tr-3: $i=3$, $r^{2}=4$, $z=6$, (d) tr-4:
$i=4$, $r^{2}=7$, $z=12$, (e) tr-5: $i=5$, $r^{2}=9$, $z=6$.

##  II Methods

In order to evaluate the percolation thresholds $p_{c}$ we follow the scheme
applied previously in Ref. Malarz, 2020. Namely, we combine Newman and Ziff
(2001), Bastas  _et al._ (2014) algorithms with finite size corrections
Stauffer and Aharony (1994); Privman (1990) to estimate $p_{c}$:

  1. 1.

Based on the Newman and Ziff (2001) algorithm we calculate the size of the
largest cluster $\langle\mathcal{S}_{\max}(n;N)\rangle$ on the number of
occupied sites $n$ (Figure 2(a)). The brackets $\langle\cdots\rangle$
represent averaging over $R=10^{5}$ lattice realizations. Applying a Gaussian
approximation to the Bernoulli distribution

$\mathcal{B}(n;N,p)=\binom{N}{n}p^{n}(1-p)^{N-n}$ (1)

one can calculate

$\mathcal{S}_{\max}(p;N)=\sum_{n=1}^{N}\langle\mathcal{S}_{\max}(n;N)\rangle\mathcal{B}(n;N,p)$
(2)

for different values of the occupation probability $p$ using the Gauss
function:

$\mathcal{B}(n;N,p)\approx\mathcal{G}(n;\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left(-\frac{(n-\mu)^{2}}{2\sigma^{2}}\right),$
(3)

with the expected value $\mu=pN$ and variance $\sigma^{2}=p(1-p)N$. The
dependence $\mathcal{S}_{\max}(p;N)$ yields a probability that an arbitrarily
chosen site belongs to the largest cluster

$\mathcal{P}_{\max}(p;L)=\mathcal{S}_{\max}(p;L)/N,$ (4)

for system with $N=L^{2}$ sites and the linear system size $L=64$, 128, 256,
512, 1024, 2048 and 4096 as presented in Figure 2(b);

  2. 2.

Using Bastas  _et al._ (2014) algorithm we minimize the pair-wise difference

$\lambda(p)=\sum_{i\neq j}[\mathcal{H}(p;N_{i})-\mathcal{H}(p;N_{j})]^{2}$ (5)

function (see Figure 2(c)) with
$\mathcal{H}(p;L)=L^{\beta/\nu}\cdot\mathcal{P}_{\max}(p;L)+1/[L^{\beta/\nu}\cdot\mathcal{P}_{\max}(p;L)]$
Bastas  _et al._ (2011) and with exponents $\beta=\frac{5}{36}$ and
$\nu=\frac{4}{3}$ (Stauffer and Aharony, 1994, p. 54). The minimum of
$\lambda(p)$ estimates the percolation threshold $\hat{p}_{c}$;

  3. 3.

Finally, in Figure 2(d) we plot the estimated values of percolation thresholds
$\hat{p}_{c}(L)$ for different ranges of summation in Equation 5—up to
$L=\max(L_{i,j})=512$, $1024$, $2048$ and $4096$. According to the standard
finite size scaling (Stauffer and Aharony, 1994, p. 77)

$\hat{p}_{c}(L)=p_{c}+a\cdot L^{-1/\nu},$ (6)

where $p_{c}$ is the percolation threshold for an infinitely large system. The
least squares linear fit to data presented in Figure 2(d) predicts $p_{c}$ and
its uncertainty $u(p_{c})$.

(a)

(b)

(c)

(d)

Figure 2: (Color online). Subsequent steps of the percolation threshold estimation (example for tr-5 lattice). (a) The mean largest cluster size $\langle\mathcal{S}_{\max}\rangle$ vs. the number of occupied sites, normalized to the system size $N$. (b) The probability that an arbitrarily chosen site belongs to the largest cluster $\mathcal{P}_{\max}(p;L)\cdot L^{\beta/\nu}$ on the occupation probability $p$. (c) The dependence of $\lambda(p)$ on the occupation probability $p$. The minima give estimates of the percolation thresholds $\hat{p}_{c}$. (d) Finite size-scaling corrections to $\hat{p}_{c}$ vs. $L^{-1/\nu}$ according to Equation 6. Table 1: Estimated values of random site triangular lattice percolation thresholds $p_{c}$ for various complex neighbourhoods. The lattice name encodes the coordination zones $i$, to which sites in the neighbourhood belong. Also the coordination number $z$, the index $\xi$ and the value of $p$ at the minimum of $\lambda$ are presented. lattice | $z$ | $\xi$ |  $p$ at $\min(\lambda)$ | $p_{c}$ | $p_{c}$ | earlier estimations
---|---|---|---|---|---|---
|  |  | $(128\leq L\leq 4096)$ | $(128\leq L\leq 4096)$ | $(64\leq L\leq 4096)$ |
|  |  | with $\Delta p=10^{-6}$ |  |  |
tr-1,2,3,4,5 | 36 | 54.8 | 0.1157 40111$L\leq 2048$ | 0.1157370(74)11footnotemark: 1 | 0.1157399(58)11footnotemark: 1 | 0.115847(21) Malarz (2020)
tr-2,3,4,5 | 30 | 48.8 | 0.1174 4011footnotemark: 1 | 0.117467(15)11footnotemark: 1 | 0.117460(10)11footnotemark: 1 | 0.117579(41) Malarz (2020)
tr-1,3,4,5 | 30 | 45.8 | 0.1215 4811footnotemark: 1 | 0.121583(14)11footnotemark: 1 | 0.1215730(83)11footnotemark: 1 |
tr-1,2,4,5 | 30 | 46.8 | 0.1225 9311footnotemark: 1 | 0.1226215(68)11footnotemark: 1 | 0.1226119(30)11footnotemark: 1 |
tr-1,2,3,5 | 24 | 33.8 | 0.1522 59 | 0.152297(17) | 0.152282(10) |
tr-3,4,5 | 24 | 39.8 | 0.1255 48 | 0.1255511(43) | 0.1255483(43) |
tr-2,4,5 | 24 | 40.8 | 0.1266 22 | 0.126653(11) | 0.1266400(40) |
tr-2,3,5 | 18 | 27.8 | 0.1616 45 | 0.161664(15) | 0.161653(15) |
tr-1,4,5 | 24 | 37.8 | 0.1316 69 | 0.1316677(13) | 0.13166484(66) | 0.131792(58) Malarz (2020)
tr-1,3,5 | 18 | 24.8 | 0.1700 42 | 0.1700473(87) | 0.170039(10) |
tr-1,2,5 | 18 | 25.8 | 0.1762 42 | 0.176263(11) | 0.1762610(92) |
tr-4,5 | 18 | 31.8 | 0.1402 43 | 0.1402453(79) | 0.1402382(92) | 0.140286(5) Malarz (2020)
tr-3,5 | 12 | 18.8 | 0.1957 03 | 0.1956981(14) | 0.1957039(24) |
tr-2,5222equivalent of TR-1,2 | 12 | 19.8 | 0.2902 68 | 0.290280(20) | 0.290279(17) |
tr-1,5 | 12 | 16.8 | 0.2095 62 | 0.209563(13) | 0.209561(10) |
tr-5333equivalent of TR-1 | 6 | 10.8 | 0.5000 00 | 0.5000029(40) | 0.4999961(55) |  $\frac{1}{2}$ Malarz (2020)
tr-1,2,3,4 | 30 | 44 | 0.1358 1311footnotemark: 1 | 0.135817(29)11footnotemark: 1 | 0.135823(27)11footnotemark: 1 |
tr-2,3,4 | 24 | 38 | 0.1391 15 | 0.1391118(33) | 0.1391117(26) |
tr-1,3,4 | 24 | 35 | 0.1443 07 | 0.1443064(38) | 0.1443074(32) |
tr-1,2,4 | 24 | 36 | 0.1489 78 | 0.1489791(88) | 0.1489757(74) |
tr-3,4 | 18 | 29 | 0.1519 32 | 0.1519532(26) | 0.1519393(35) |
tr-2,4 | 18 | 30 | 0.1584 53 | 0.1584634(54) | 0.1584620(43) |
tr-1,4 | 18 | 27 | 0.1651 88 | 0.165186(14) | 0.165186(12) |
tr-4 | 12 | 21 | 0.1924 37 | 0.1924356(68) | 0.1924428(50) | 0.192410(43) Malarz (2020)
tr-1,2,3 | 18 | 23 | 0.2154 62 | 0.21546261(91) | 0.2154657(17) | 0.215484(19) Malarz (2020), 0.215 d’Iribarne  _et al._ (1999)
tr-2,3 | 12 | 17 | 0.2320 12 | 0.232019(23) | 0.232020(20) | 0.232008(38) Malarz (2020)
tr-1,3 | 12 | 14 | 0.2645 25 | 0.264545(25) | 0.264539(21) |
tr-333footnotemark: 3 | 6 | 8 | 0.5000 24 | 0.500027(31) | 0.500013(23) |  $\frac{1}{2}$ Malarz (2020)
tr-1,2 | 12 | 15 | 0.2902 67 | 0.290261(22) | 0.290258(19) | 0.295 Dalton  _et al._ (1964)
tr-233footnotemark: 3 | 6 | 9 | 0.4999 85 | 0.499987(20) | 0.499978(20) |  $\frac{1}{2}$ Malarz (2020)
tr-1 | 6 | 6 | 0.4999 93 | 0.499994(17) | 0.499996(14) |  $\frac{1}{2}$ (Stauffer and Aharony, 1994, p. 17)

##  III Results

In Figure 2 we present examples of results used to predict the percolation
thresholds $p_{c}$ as described in Section II.

In Figure 2(a) the dependence of $\langle\mathcal{S}_{\max}(n;L)\rangle$ on
the number of occupied sites $n$ (normalized to the system size $N$) are
presented. The dependence for all discussed neighbourhoods is presented in
Figure A2 in Ref. Sup, . With the increase of the system size $N=L^{2}$ the
curves describing this dependence become steeper and steeper. For infinitely
large systems the function $d\langle\mathcal{S}_{\max}(n;L)\rangle/dn$ becomes
discontinuous at $p=p_{c}$.

In Figure 2(b) we show the dependence of the normalized probability of a
randomly chosen site belonging to the largest cluster
$\mathcal{P}_{\max}(p;L)\cdot L^{\beta/\nu}$ on the sites occupation
probability $p$ for various linear system sizes $L$. The curves representing
this dependence for all discussed neighbourhoods are presented in Figure A3 in
Ref. Sup, . The abscissas of the points where curves intercept each other
estimate the percolation thresholds $\hat{p}_{c}$.

In Figure 2(c) the dependence of $\lambda(p)$ on the occupation probability
$p$ is presented. The curves representing this dependence for all discussed
neighbourhoods are presented in Figure A4 in Ref. Sup, . The minima of
$\lambda(p)$ give estimates of the percolation thresholds $\hat{p}_{c}$.

In Figure 2(d) finite size corrections to $\hat{p}_{c}$ according to Equation
6 are presented. The curves representing this dependence for all discussed
neighbourhoods are presented in Figure A5 in Ref. Sup, . The initial value of
the linear fit function predicts the percolation threshold values $p_{c}$.

The obtained percolation thresholds $p_{c}$ together with their uncertainties
and earlier estimates are gathered in Table 1. We used all lattice sizes
$64\leq L\leq 4096$ for the estimation of $p_{c}$ (they are collected in the
sixth column of Table 1), while estimations given in the fifth column rely on
systems with $128\leq L\leq 4096$ (i.e. the smallest systems, for $L=64$, have
been excluded from calculations). For neighbourhoods containing 30 or more
sites we were able to carry out simulations for lattices up to $L=2048$. To
check how the minimisation of pair-wise difference function $\lambda$ (5)
reduces the finite size effects we present also values of $p$ for which
$\lambda$ reaches the minimum when $p$ is scanned with $\Delta p=10^{-6}$
accuracy (see the fourth column of Table 1).

##  IV Discussion

(a)

(b)

Figure 3: (Color online). Percolation thresholds for complex neighbourhoods on
triangular lattice. (a) Degenerated dependence of the percolation threshold
$p_{c}$ on the coordination number $z$. The maxima of $p_{c}$ for fixed $z$
follow Equation 10. The power is $\delta\approx 0.811$. (b) Power fit (12) of
the percolation thresholds $p_{c}$ vs. $\xi$ for complex neighbourhoods. The
exponent $\gamma$ in Equation 10 is $0.710(19)$. The $p_{c}$ values for
equivalents of tr-1 and tr-1,2 neighbourhoods (marked with crosses) are
excluded from fitting.

In Ref. Xun  _et al._ , 2021 several analytical formulas for the dependence of
the percolation threshold $p_{c}$ on the coordination number $z$ were tested,
with

$p_{c}=c/(z+b),$ (7) $p_{c}=1-\exp(d/z)$ (8)

among the others. These formulas work fine for “compact” neighbourhoods (for
instance tr-1, tr-1,2,3 and tr-1,2,3,4,5, here).

Unfortunately, for complex neighbourhoods these formulas must fail as the
dependence $p_{c}$ on $z$ is “degenerated”, i.e. several values of $p_{c}$ are
associated with the same number $z$ of sites in the neighbourhood (see Figure
3(a), and also Figure 4b in Ref. Majewski and Malarz, 2007 for the square
lattice). These degeneration is also observed for basic neighbourhoods what
brought some brickbats van der Marck (1997) on the possibility of existing
universal formulas for the percolation threshold (depending solely on the
spatial dimension $d$ of the system and the coordination number $z$), as
proposed by Galam and Mauger (1996).

To remove this degeneracy we tested formulas (7) and (8) for the mean values
$\bar{p}_{c}$ of percolation thresholds. The averaging, denoted by the bar,
goes over percolation thresholds for neighbourhoods for fixed number $z$. For
instance $\bar{p}_{c}(z=6)$ is the mean value of $p_{c}$ for tr-1, tr-2, tr-3,
tr-5 lattices, while $\bar{p}_{c}(z=30)$ is based on values of $p_{c}$ for
tr-1,2,3,4, tr-1,2,4,5, tr-1,3,4,5 and tr-2,3,4,5 lattices, respectively.
Additionally, we checked the quality of such fits for the Galam–Mauger formula
Galam and Mauger (1996), which for fixed network topology reduces to the
following power-law dependence

$p_{c}\propto(z-1)^{-a}.$ (9)

Unfortunately, the formulas $\bar{p}_{c}(z)$ describing the dependence on $z$
are not consistent with either (7) or (8) or (9). However, we notice that the
maximum value of $p_{c}$ for the fixed value of coordination number $z$
follows a power law

$\max_{z=\text{const}}p_{c}\propto z^{-\delta}$ (10)

with the power $\delta\approx 0.811(20)$ (see Figure 3(a)).

As the combination of $d$ and $z$ seems to be insufficient to differentiate
among various lattices with assumed neighbourhoods we are looking for another
index $\xi$ which may be useful for both: deriving a formula $p_{c}(\xi)$ and
removing the $p_{c}$-degeneracy. The attempts for such index identification
were undertaken also in graph theory (for topological invariants for trees
Piec  _et al._ (2005)) and in organic chemistry (for molecular topological
index Gutman (1994); Schultz (1989); Wiener (1947)).

The primary reason why the formula (9) does not work in our case seems to be
related to the fact that it does not take into account spatial distribution of
neighbors. We expect some corrections depending on the distance of neighbors
from the central node. To implement this basic intuition, we found, using
trial and error method, that a relatively simple heuristic formula

$\xi=\sum_{i}z_{i}r_{i}^{2}/i$ (11)

does the job, in the sense that the percolation thresholds for complex
neighborhoods in the studied model are indeed very well described by the
scaling

$p_{c}\propto\xi^{-\gamma}$ (12)

with $\gamma\approx 0.710(19)$, except of a single value of $p_{c}$ for tr-2,5
(see Figure 3(b)). It remains to check in other models of this type how
universal is the scaling of the percolation thresholds in the effective
weighted coordination number $\xi$ (11).

The values of $\xi$ are presented in Table 1.

Please note that
$p_{c}(\textsc{tr}\text{-1})=p_{c}(\textsc{tr}\text{-2})=p_{c}(\textsc{tr}\text{-3})=p_{c}(\textsc{tr}\text{-5})$
are exactly the same and equal to $\frac{1}{2}$ as these neighbourhoods are
equivalent to each other as described in Ref. Malarz, 2020. Thus $p_{c}$
values for tr-2, tr-3, tr-5 neighbourhoods are excluded from fitting. The same
occurs for tr-2,5 which is equivalent to tr-1,2 as presented in Figure 4. In
general, in case of neighbourhood equivalence the system should be
characterized by $\xi$ value associated with this neighbourhood which has the
smallest indices that characterize the connection.

(a)

(b)

Figure 4: (Color online). Topological equivalence of the neighbourhoods tr-1,2
and tr-2,5. The distances to the central site in tr-2,5 are $\sqrt{3}$ times
longer than in tr-1,2 lattice. (a) tr-1,2: $r^{2}=1$ and 3. (b) tr-2,5:
$r^{2}=3$ and 9.

##  V Conclusions

Concluding, in this paper we estimated percolation thresholds $p_{c}$ for
random site triangular lattice percolation and for neighborhoods containing
sites from the first to fifth coordination zone. The estimated values of
percolation thresholds are collected in Table 1.

We note that the method Bastas  _et al._ (2014) allows (at least partially) to
get rid of finite size effects. The minima of the pairwise difference
$\lambda$ function presented in Table 1 are consistent with a five digit
accuracy with the percolation threshold $p_{c}$ obtained for infinite lattice
according to Equation 6. The five digit accuracy seems to outperform by at
least one order of magnitude practical requirements on experimenters in any
field of science where percolation theory may be applied.

Percolation thresholds for extended neighbourhoods were utilised in studies
ranging from agriculture Ramírez  _et al._ (2020), materials chemistry Alguero
_et al._ (2020), magnetic Ueland  _et al._ (2018); Keeney  _et al._ (2017);
Buczek  _et al._ (2016) and electronic Jeong  _et al._ (2018) properties of
solids, nanoengineering Xu  _et al._ (2014), etc. These applications utilised
results for square or simple cubic lattices. However, also finding percolation
thresholds for extended neighbourhoods on honeycomb lattice may be attractive
for better understanding some phenomena as proposed for instance in Ref.
Castro  _et al._ , 2006. Thus application of the results for triangular
lattice presented here cannot be excluded.

As the percolation thresholds $p_{c}(z)$ are multiply degenerated, we propose
the weighted square distance $\xi$ to differentiate among various
neighbourhoods. This index seems to be effective in this respect (at least for
neighbourhoods investigated here).

Finally, the $p_{c}(\xi)$ dependence follows roughly a power law (12). The
obtained results may be useful in further searching for the universal formula
on percolation thresholds Lebrecht  _et al._ (2021).

## References

  * Stauffer and Aharony (1994) D. Stauffer and A. Aharony,  _Introduction to Percolation Theory_, 2nd ed. (Taylor and Francis, London, 1994).
  * Wierman (2014) J. Wierman, “Percolation theory,” in  _Wiley StatsRef: Statistics Reference Online_ (American Cancer Society, 2014) pp. 1–9.
  * Broadbent and Hammersley (1957) S. R. Broadbent and J. M. Hammersley, “Percolation processes: I. Crystals and mazes,” Mathematical Proceedings of the Cambridge Philosophical Society 53, 629–641 (1957).
  * Hammersley (1957) J. M. Hammersley, “Percolation processes: II. The connective constant,” Mathematical Proceedings of the Cambridge Philosophical Society 53, 642–645 (1957).
  * Ramírez  _et al._ (2020) J. E. Ramírez, C. Pajares, M. I. Martínez, R. Rodríguez Fernández, E. Molina-Gayosso, J. Lozada-Lechuga, and A. Fernández Téllez, “Site-bond percolation solution to preventing the propagation of _Phytophthora zoospores_ on plantations,” Physical Review E 101, 032301 (2020).
  * Zhang  _et al._ (2020) Q. Zhang, B.-Y. Zhang, B.-H. Guo, Z.-X. Guo, and J. Yu, “High-temperature polymer conductors with self-assembled conductive pathways,” Composites Part B—Engineering 192, 107989 (2020).
  * Cheng  _et al._ (2020) L. Cheng, P. Yan, X. Yang, H. Zou, H. Yang, and H. Liang, “High conductivity, percolation behavior and dielectric relaxation of hybrid ZIF-8/CNT composites,” Journal of Alloys and Compounds 825, 154132 (2020).
  * Malarz  _et al._ (2002) K. Malarz, S. Kaczanowska, and K. Kułakowski, “Are forest fires predictable?” International Journal of Modern Physics C 13, 1017–1031 (2002).
  * Ghanbarian  _et al._ (2020) B. Ghanbarian, F. Liang, and H.-H. Liu, “Modeling gas relative permeability in shales and tight porous rocks,” Fuel 272, 117686 (2020).
  * Cao  _et al._ (2020) W. Cao, L. Dong, L. Wu, and Y. Liu, “Quantifying urban areas with multi-source data based on percolation theory,” Remote Sensing of Environment 241, 111730 (2020).
  * Bartolucci  _et al._ (2020) S. Bartolucci, F. Caccioli, and P. Vivo, “A percolation model for the emergence of the Bitcoin Lightning Network,” Scientific Reports 10, 4488 (2020).
  * Ziff (2021) R. M. Ziff, “Percolation and the pandemic,” Physica A: Statistical Mechanics and its Applications 568, 125723 (2021).
  * Dong  _et al._ (2020) S. Dong, A. Mostafizi, H. Wang, J. Gao, and X. Li, “Measuring the topological robustness of transportation networks to disaster-induced failures: A percolation approach,” Journal of Infrastructure Systems 26, 04020009 (2020).
  * Li  _et al._ (2021) M. Li, R.-R. Liu, L. Lü, M.-B. Hu, S. Xu, and Y.-C. Zhang, “Percolation on complex networks: Theory and application,” Physics Reports (2021), 10.1016/j.physrep.2020.12.003.
  * Saberi (2015) A. A. Saberi, “Recent advances in percolation theory and its applications,” Physics Reports 578, 1–32 (2015).
  * Frisch  _et al._ (1961) H. L. Frisch, E. Sonnenblick, V. A. Vyssotsky, and J. M. Hammersley, “Critical percolation probabilities (site problem),” Physical Review 124, 1021–1022 (1961).
  * Dean (1963) P. Dean, “A new Monte Carlo method for percolation problems on a lattice,” Mathematical Proceedings of the Cambridge Philosophical Society 59, 397–410 (1963).
  * Dean and Bird (1967) P. Dean and N. F. Bird, “Monte Carlo estimates of critical percolation probabilities,” Mathematical Proceedings of the Cambridge Philosophical Society 63, 477–479 (1967).
  * Dalton  _et al._ (1964) N. W. Dalton, C. Domb, and M. F. Sykes, “Dependence of critical concentration of a dilute ferromagnet on the range of interaction,” Proceedings of the Physical Society 83, 496–498 (1964).
  * Domb and Dalton (1966) C. Domb and N. W. Dalton, “Crystal statistics with long-range forces: I. The equivalent neighbour model,” Proceedings of the Physical Society 89, 859–871 (1966).
  * Gouker and Family (1983) M. Gouker and F. Family, “Evidence for classical critical behavior in long-range site percolation,” Physical Review B 28, 1449–1452 (1983).
  * Malarz and Galam (2005) K. Malarz and S. Galam, “Square-lattice site percolation at increasing ranges of neighbor bonds,” Physical Review E 71, 016125 (2005).
  * Galam and Malarz (2005) S. Galam and K. Malarz, “Restoring site percolation on damaged square lattices,” Physical Review E 72, 027103 (2005).
  * Majewski and Malarz (2007) M. Majewski and K. Malarz, “Square lattice site percolation thresholds for complex neighbourhoods,” Acta Physica Polonica B 38, 2191–2199 (2007).
  * Xun  _et al._ (2021) Z. Xun, D. Hao, and R. M. Ziff, “Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit,” Physical Review E 103, 022126 (2021).
  * d’Iribarne  _et al._ (1999) C. d’Iribarne, M. Rasigni, and G. Rasigni, “From lattice long-range percolation to the continuum one,” Physics Letters A 263, 65–69 (1999).
  * Malarz (2020) K. Malarz, “Site percolation thresholds on triangular lattice with complex neighborhoods,” Chaos 30, 123123 (2020).
  * Kurzawski and Malarz (2012) Ł. Kurzawski and K. Malarz, “Simple cubic random-site percolation thresholds for complex neighbourhoods,” Reports on Mathematical Physics 70, 163–169 (2012).
  * Malarz (2015) K. Malarz, “Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors,” Physical Review E 91, 043301 (2015).
  * Kotwica  _et al._ (2019) M. Kotwica, P. Gronek, and K. Malarz, “Efficient space virtualisation for Hoshen–Kopelman algorithm,” International Journal of Modern Physics C 30, 1950055 (2019).
  * (31) See Supplemental Material at [URL will be inserted by publisher]. In Figure A1 neighbourhoods containing sites from $i=5$ coordination zone are presented. In Figure A2 the mean largest cluster size $\langle\mathcal{S}_{\max}\rangle$ vs. the number $n$ of occupied sites, normalized to the system size $N$ are presented. Figure A3 shows the probability $\mathcal{P}_{\max}$ that an arbitrarily chosen site belongs to the largest cluster vs. occupation probability $p$. Sums of the pairwise difference $\lambda$ vs. the occupation probability $p$ are presented in Figure A4. In Figure A5 finite size corrections to $p_{c}$ are presented.
  * Newman and Ziff (2001) M. E. J. Newman and R. M. Ziff, “Fast Monte Carlo algorithm for site or bond percolation,” Physical Review E 64, 016706 (2001).
  * Bastas  _et al._ (2014) N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, “Method for estimating critical exponents in percolation processes with low sampling,” Physical Review E 90, 062101 (2014).
  * Privman (1990) V. Privman, “Finite-size scaling theory,” in  _Finite size scaling and numerical simulation of statistical systems_, edited by V. Privman (World Scientific, Singapore, 1990) pp. 1–98.
  * Bastas  _et al._ (2011) N. Bastas, K. Kosmidis, and P. Argyrakis, “Explosive site percolation and finite-size hysteresis,” Physical Review E 84, 066112 (2011).
  * van der Marck (1997) S. C. van der Marck, “Universal formulas for percolation thresholds — Comment,” Physical Review E 55, 1228–1229 (1997).
  * Galam and Mauger (1996) S. Galam and A. Mauger, “Universal formulas for percolation thresholds,” Physical Review E 53, 2177–2181 (1996).
  * Piec  _et al._ (2005) S. Piec, K. Malarz, and K. Kułakowski, “How to count trees?” International Journal of Modern Physics C 16, 1527–1534 (2005).
  * Gutman (1994) I. Gutman, “Selected properties of the Schultz molecular topological index,” Journal of Chemical Information and Computer Sciences 34, 1087–1089 (1994).
  * Schultz (1989) H. P. Schultz, “Topological organic chemistry. 1. Graph theory and topological indices of alkanes,” Journal of Chemical Information and Computer Sciences 29, 227–228 (1989).
  * Wiener (1947) H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society 69, 17–20 (1947).
  * Alguero  _et al._ (2020) M. Alguero, M. Perez-Cerdan, R. P. del Real, J. Ricote, and A. Castro, “Novel Aurivillius Bi4Ti3-2xNbxFexO12 phases with increasing magnetic-cation fraction until percolation: A novel approach for room temperature multiferroism,” Journal of Materials Chemistry C 8, 12457–12469 (2020).
  * Ueland  _et al._ (2018) B. G. Ueland, N. H. Jo, A. Sapkota, W. Tian, M. Masters, H. Hodovanets, S. S. Downing, C. Schmidt, R. J. McQueeney, S. L. Bud’ko, A. Kreyssig, P. C. Canfield, and A. I. Goldman, “Reduction of the ordered magnetic moment and its relationship to Kondo coherence in Ce1-xLaxCu2Ge2,” Physical Review B 97, 165121 (2018).
  * Keeney  _et al._ (2017) L. Keeney, C. Downing, M. Schmidt, M. E. Pemble, V. Nicolosi, and R. W. Whatmore, “Direct atomic scale determination of magnetic ion partition in a room temperature multiferroic material,” Scientific Reports 7, 1737 (2017).
  * Buczek  _et al._ (2016) P. Buczek, L. M. Sandratskii, N. Buczek, S. Thomas, G. Vignale, and A. Ernst, “Magnons in disordered nonstoichiometric low-dimensional magnets,” Physical Review B 94, 054407 (2016).
  * Jeong  _et al._ (2018) J. Jeong, K. J. Park, E.-J. Cho, H.-J. Noh, S. B. Kim, and H.-D. Kim, “Electronic structure change of NiS2-xSex in the metal-insulator transition probed by X-ray absorption spectroscopy,” Journal of the Korean Physical Society 72, 111–115 (2018).
  * Xu  _et al._ (2014) F. Xu, Z. Xu, and B. I. Yakobson, “Site-percolation threshold of carbon nanotube fibers—Fast inspection of percolation with Markov stochastic theory,” Physica A 407, 341–349 (2014).
  * Castro  _et al._ (2006) E. V. Castro, N. M. R. Peres, K. S. D. Beach, and A. W. Sandvik, “Site dilution of quantum spins in the honeycomb lattice,” Physical Review B 73, 054422 (2006).
  * Lebrecht  _et al._ (2021) W. Lebrecht, P. M. Centres, and A. J. Ramirez-Pastor, “Empirical formula for site and bond percolation thresholds on Archimedean and 2-uniform lattices,” Physica A: Statistical Mechanics and its Applications 569, 125802 (2021).