You are an expert at summarizing long articles. Proceed to summarize the following text: 
one of the main goal of the study of relativistic heavy ion collisions is to study the deconfined matter , known as quark - gluon plasma(qgp ) , which is expected to form at large densities . it has been suggested that the transition from hadronic to qgp state can be treated by percolation theory @xcite . the formulation of percolation problem is concerned with elementary geometrical objects placed on a random d - dimensional lattice . the objects have a well defined connectivity radius @xmath0 , and two objects can communicate if the distance between them is less than @xmath0 . several objects can form a cluster of communication . at certain density of the objects a infinite cluster appears which spans the entire system . this is defined by the dimensionless percolation parameter @xmath1 @xcite . percolation theory has been applied to several areas ranging from clustering in spin system to the formation of galaxies . figure 1 shows the transition from disconnected to connected system at high densities .    in nuclear collisions there is indeed , as a function of parton density , a sudden onset of large scale color connection . there is a critical density at which the elemental objects form one large cluster , loosing their independent existence . percolation would correspond to the onset of color deconfinement and it may be a prerequisite for any subsequent qgp formation . the determination of the eos of hot , strongly interacting matter is one of the main challenges of strong interaction physics ( hotqcd collaboration ) @xcite . recent lqcd calculations for the bulk thermodynamic observables , e.g. pressure , energy density , entropy density and for the sound velocity have been reported @xcite . a percolation model coupled with hydrodynamics has been utilized to calculate these quantities from the star data at rhic energies in central au+au collisions . multiparticle production at high energies is currently described in terms of color strings stretched between the projectile and target . hadronizing these strings produce the observed hadrons . the strings act as color sources of particles through the creation of @xmath2 pairs from the sea . the number of strings grows with the energy and with the number of nucleons of participating nuclei . color strings may be viewed as small discs in the transverse space filled with the color field created by colliding partons . particles are produced by the schwinger mechanisms @xcite . with growing energy and size of the colliding nuclei the number of strings grow and start to overlap to form clusters @xcite . at a critical density a macroscopic cluster appears that marks the percolation phase transition . 2d percolation is a non - thermal second order phase transition , but in cspm the schwinger barrier penetration mechanism for particle production and the fluctuations in the associated string tension due to the strong string interactions make it possible to define a temperature . consequently the particle spectrum is `` born '' with a thermal distribution @xcite . the percolation threshold at which the spanning cluster appear , a `` connected '' system of color sources , identifies the percolation phase transition .    with an increasing number of strings there is a progression from isolated individual strings to clusters and then to a large cluster which suddenly spans the area . in two dimensional percolation theory the relevant quantity is the dimensionless percolation density parameter given by @xcite @xmath3 where n is the number of strings formed in the collisions and @xmath4 is the transverse area of the single string and @xmath5 is the transverse nuclear overlap area . the critical cluster which spans @xmath5 , appears for @xmath6 1.2 @xcite . as @xmath1 increases the fraction of @xmath5 covered by this spanning cluster increases . the percolation theory governs the geometrical pattern of string clustering . it requires some dynamics to describe the interaction of several overlapping strings . it is assumed that a cluster behaves as a single string with a higher color field corresponding to the vectorial sum of the color charge of each individual string . knowing the color charge , one can calculate the multiplicity @xmath7 and the mean transverse momentum @xmath8 of the particles produced by a cluster of strings . one finds @xcite    @xmath9    @xmath10    where @xmath11 and @xmath12 are the mean multiplicity and average transverse momentum squared of particles produced by a single string with a transverse area @xmath13 . in the saturation limit , all the strings overlap into a single cluster that approximately occupies the whole interaction area , one gets the following universal scaling law    @xmath14    this scaling law shows a reasonable agreement with all the experimental data for all projectiles , targets and energies @xcite . in the limit of high density one obtains @xmath15 with @xmath16 being the color suppression factor . it follows that @xmath17 a similar scaling is found in the color glass condensate approach ( cgc)@xcite . the saturation scale @xmath18 in cgc corresponds to @xmath19 in cspm . the net effect due to @xmath20 is the reduction in hadron multiplicity and increase in the average transverse momentum of particles . the cspm model calculation for hadron multiplicities and momentum spectra was found to be in excellent agreement with experiment @xcite . within the framework of clustering of color sources , the elliptic flow , @xmath21 , and the dependence of the nuclear modification factor on the azimuthal angle show reasonable agreement with the rhic data @xcite . the critical density of percolation is related to the effective critical temperature and thus percolation may be the way to achieve deconfinement in the heavy ion collisions @xcite . an additional important check of this interacting string approach was provided by the measurement of long range forward backward multiplicity correlations ( lrc ) by the star group at rhic @xcite . to obtain the value of @xmath1 from data , a parameterization of p - p events at 200 gev is used to compute the @xmath22 distribution @xcite    @xmath23    where a , @xmath24 , and @xmath25 are parameters used to fit the data . this parameterization also can be used for nucleus - nucleus collisions to take into account the interactions of the strings@xcite @xmath26 where @xmath27 corresponds to the area occupied by the n overlapping strings . the thermodynamic limit , i.e. letting n and @xmath27 @xmath28 while keeping @xmath1 fixed , is used to evaluate @xmath20 given by eq.(5 ) . for nucleus - nucleus collisions eq.(8 ) becomes    @xmath29    in pp collisions @xmath30 @xmath31 1 due to the low string overlap probability . the star analysis of charged hadrons had presented the preliminary results for the percolation density parameter , @xmath1 at rhic for several collisions systems as a function of centrality@xcite . figure 2 shows @xmath1 as function of the number of participant nucleons(@xmath32 ) in au+au collisions at @xmath33 200 and 62.4 gev . these experimental @xmath1 values are used to get the bulk thermodynamic observables . the strong longitudinal chromo - electric fields produce schwinger - bialas @xcite like radiation with a thermal spectrum , in analogy with the hawking - unruh radiation @xcite . both the schwinger - bialas and hawking - unruh derivations lead to the same value of the maximum entropy temperature . above the critical value of @xmath1 , the qgp in cspm consists of massless constituents ( gluons ) . the percolation density parameter @xmath1 determines the cluster size distribution , the temperature t(@xmath1 ) and the transverse momentum in the collision @xcite . the connection between @xmath1 and the temperature @xmath34 involves the schwinger mechanism ( sm ) for particle production . in cspm the schwinger distribution for massless particles is expressed in terms of @xmath35 @xmath36 with the average value of string tension @xmath37 . gaussian fluctuations in the string tension around its mean value transforms sm into the thermal distribution @xcite @xmath38 with @xmath37 = @xmath39 . the temperature is given by @xmath40 the string percolation density parameter @xmath1 which characterizes the percolation clusters also determines the temperature of the system . in this way at @xmath41=1.2 the percolation phase transition at @xmath42 models the thermal deconfinement transition . in the determination of temperature using eq.(13 ) the value of @xmath20 is obtained using the experimental data @xcite . we will adopt the point of view that the experimentally determined chemical freeze - out temperature is a good measure of the phase transition temperature , @xmath43 @xcite . the single string average transverse momentum @xmath44 is calculated at @xmath41 = 1.2 with the universal chemical freeze - out temperature of 167.7 @xmath45 2.6 mev @xcite . this gives @xmath46 = 207.2 @xmath45 3.3 mev which is close to @xmath47200 mev used previously in the calculation of the percolation transition temperature @xcite . above @xmath41 = 1.2 the size and density of the spanning cluster increases . we use the measured value of @xmath1 = 2.88 to determine the temperature , before the expansion , @xmath48 = 193.6@xmath453.0 mev of the quark gluon plasma in reasonable agreement with @xmath48 = 221@xmath49 from the enhanced direct photon experiment measured by the phenix collaboration@xcite . among the most important and fundamental problems in finite - temperature qcd are the calculation of the bulk properties of hot qcd matter and characterization of the nature of the qcd phase transition . the qgp according to cspm is born in local thermal equilibrium because the temperature is determined at the string level . we use cspm coupled to hydrodynamics to calculate energy density , pressure , entropy density and sound velocity . as mentioned earlier the strings interact strongly to form clusters and produce the pressure at the early stages of the collisions , which is evident from the presence of elliptical flow in cspm @xcite . after the initial temperature @xmath50 the cspm perfect fluid may expand according to bjorken boost invariant 1d hydrodynamics @xcite @xmath51 where @xmath52 is the energy density , s the entropy density , @xmath53 the proper time , and @xmath54 the sound velocity . above the critical temperature only massless particles are present in cspm . the initial energy density @xmath55 above @xmath43 is given by @xcite @xmath56 to evaluate @xmath55 we use the charged pion multiplicity @xmath57 at midrapidity and @xmath27 values from star for 0 - 10% central au - au collisions with @xmath33200 gev @xcite . the factor 3/2 in eq.(19 ) accounts for the neutral pions . the dynamics of massless particle production has been studied in qe2 quantum electrodynamics . qe2 can be scaled from electrodynamics to quantum chromodynamics using the ratio of the coupling constants @xcite . the production time @xmath58 for a boson ( gluon ) is @xcite @xmath59 in cspm the total transverse energy is proportional to @xmath1 . from the measured value of @xmath1 and @xmath52 it is found that @xmath52 is proportional to @xmath1 for the range @xmath60 , @xmath61 @xmath1 gev/@xmath62 @xcite . this relationship has been extrapolated to below @xmath63 and above @xmath64 for the energy and entropy density calculations . figure 3 shows @xmath65 as obtained from cspm along with the lqcd calculations @xcite and the cspm pressure @xmath66 . for an ideal gas of massless constituents , the energy density and pressure are related by @xmath67 . in lqcd the basic quantity is the interaction measure @xmath68 , which is also known as the trace anomaly . in cspm @xmath69 reaches zero for t @xmath70 while in case of lqcd it is strongly interacting even for t @xmath71 @xcite . the sound velocity requires the evaluation of s and @xmath72 , which can be expressed in terms of @xmath1 and @xmath20 . with @xmath73 = @xmath20 one obtains :    @xmath74    then @xmath75 becomes : @xmath76 for @xmath77 , an analytic function of @xmath1 for the equation of state of the qgp for t @xmath78 . figure 4 shows the comparison of @xmath75 from cspm and lqcd . the lqcd values were obtained using the eos of 2 + 1 flavor qcd at finite temperature with physical strange quark mass and almost physical light quark masses @xcite . at @xmath79=1 the cspm and lqcd agree with the @xmath75 value of the physical hadron gas with resonance mass truncation m @xmath80 2.5 gev @xcite .    the entropy density @xmath81 is obtained from energy density and speed of sound as shown in fig.5 along with the lqcd results . cspm is in excellent agreement with the lqcd calculations in the phase transition region for @xmath821.5 . the percolation analysis of the color sources applied to star data at rhic provides a compelling argument that the qgp is formed in central au - au collisions at @xmath33 200 gev . it also suggests that the qgp is produced in all soft high energy high multiplicity collisions when the string density exceeds the percolation transition . the results are also in agreement with lattice qcd in the phase transition region , when the results are plotted with respect to @xmath83 and @xmath84 . the value of @xmath75=0.14 is in agreement with the physical resonance gas value at the critical temperature . thus clustering and percolation can provide a conceptual basis for the qcd phase diagram which is more general than symmetry breaking @xcite . this research was supported by the office of nuclear physics within the u.s . department of energy office of science under grant no . de - fg02 - 88er40412 . 00 t. celik , f. karsch , h. satz , phys . b*97 * , 128 ( 1980 ) . m. b. isichenko , rev . phys . * 64 * , 961 ( 1992 ) . a. bazavov et al . , phys . rev . d*80 * , 014504 ( 2009 ) . j. schwinger , phys . rev . * 82 * , 664 ( 1951 ) . m. a. braun , c. pajares , eu . j. c*16 * , 349 ( 2000 ) . m. a. braun , f. del moral , c. pajares , phys . c * 65 * , 024907 ( 2002 ) . j. dias de deus , c. pajares , phys . b*642 * , 455 ( 2006 ) . a. bialas , phys . b*466 * , 301 ( 1999 ) . h. satz , rep . . phys . * 63 * , 1511 ( 2000 ) . j. dias de deus , e. g. ferreiro , c. pajares , r. ugoccioni , phys . b*581 * , 156 ( 2004 ) . j. schaffner - bielich , d. kharzeev , l. mclerran , r. venugopalan , nucl . a*705 * , 494 ( 2002 ) . j. dias de deus , e. g. ferreiro , c. pajares , r. ugoccioni , eur . j. c*40 * , 229 ( 2005 ) . p. brogueira , j. dias de deus , j. g. milhano , nucl . a. * 832 * , 76 ( 2010 ) . i. bautista , l. cunqueiro , j. dias de deus , c. pajares , j. phys . g*37 * , 015103 ( 2010 ) . b. i. abelev et al . , ( star collaboration ) , phys . * 103 * , 172301 ( 2009 ) . b. k. srivastava , r. p. scharenberg , t. tarnowsky , ( star collaboration ) , nukleonika * 51 * , s109 ( 2006 ) . s. w. hawking , commun . phys . * 43 * , 199 ( 1975 ) . w. g. unruh , phys . d*14 * , 870 ( 1976 ) . m. k. parikh , f. wilczek , phys . . lett . * 85 * , 5042 ( 2000 ) . d. kharzeev , k. tuchin , nucl . a*753 * , 316 ( 2005 ) . d. kharzeev , e. levin , k. tuchin , phys . c*75 * , 044903 ( 2007 ) . p. castorina , d. kharzeev , h. satz , eur . j. c*52 * , 187 ( 2007 ) . braun - munzinger , j. stachel , christof wetterich , phys . b*596 * , 61 ( 2004 ) . f. becattini , p. castorina , a. milov , h. satz , eur . j. c*66 * , 377 ( 2010 ) . a. adare et al . , ( phenix collaboration ) , phys . 104 * , 132301 ( 2010 ) . j. d. bjorken , phys . rev . d*27 * , 140 ( 1983 ) . b. i. abelev et al . , ( star collaboration ) , phys . c*79 * , 34909 ( 2009 ) . c. y. wong , introduction to high energy heavy ion collisions ( world scientific,1994 ) . j. schwinger , phys . rev . * 128 * , 2425 ( 1962 ) . p. castorina , j. cleymans , d. e. miller , h. satz , arxiv : hep - ph/0906.2289v1 . h. satz , quantum field theory in extreme environments , paris , april 2009 .
possible phase transition of strongly interacting matter from hadron to a quark - gluon plasma ( qgp ) state have in the past received considerable interest . 
 it has been suggested that this problem might be treated by percolation theory . 
 the color string percolation model ( cspm ) is used to determine the equation of state ( eos ) of the qgp produced in central au - au collisions at rhic energies . 
 the bulk thermodynamic quantities- energy density , entropy density and the sound velocity- are obtained in the framework of cspm . 
 it is shown that the results are in excellent agreement with the recent lattice qcd calculations(lqcd ) . 
 relativistic heavy - ion collisions , percolation , qgp , eos