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several years ago stillinger and helfand@xcite introduced a simple but nontrivial model of fluid demixing . their original model consists of a binary mixture of @xmath1 and @xmath2 particles . particles of the same type do not interact with one another , but @xmath1 and @xmath2 particles interact with a repulsive potential such that the mayer @xmath3-function is a gaussian . this choice for the @xmath4 potential , known as the gaussian molecule potential , greatly simplifies the calculation of virial coefficients and most work for this potential has been done using series methods@xcite . the main motivation for this work is to confirm ising universality for the critical exponents of continuum systems .    in this paper we study the  model and some of its generalizations using cluster monte carlo methods . where possible , we compare our results to the series analyses and to results for the ising - potts universality classes . although the gaussian molecule potential yields a more tractable virial expansion , it is easier to implement a cluster algorithm for the repulsive step potential . we also consider a generalization of the  model to @xmath0 species ( components ) , such that particles of the same species do not interact but particles of different species interact with a repulsive potential . we expect that this generalization will be in the same universality class as the @xmath0-state potts model for @xmath0 not too large and another motivation for this study is to confirm this correspondence . for example , we know that the two - dimensional ( 2d ) potts model for @xmath5 has a first - order transition . does the @xmath0-component 2d  model also have a first - order transition for @xmath5 ? in addition , we consider the effect of quenched disorder by randomly adding fixed scattering centers . there are general arguments@xcite that quenched disorder causes first - order transitions to become continuous . these arguments hold rigorously for 2d potts models@xcite , but have not been studied for continuum models .    in previous work@xcite cluster monte carlo methods were applied to the widom - rowlinson model@xcite . the widom - rowlinson and  models are closely related , the only difference is that the widom - rowlinson model has a hard - core interaction between different species . in this paper , the invaded cluster monte carlo method introduced in ref.@xcite is extended to soft - core repulsive potentials and is used to find the phase transition point for a given temperature without prior knowledge of the critical fugacity . the invaded cluster method has almost no critical slowing for the widom - rowlinson model , and we find that similar results hold for the  models studied here . we consider @xmath0-component ( @xmath6 ) fluids in @xmath7 dimensions with @xmath8 . the components ( species ) have no self - interaction but particles of one species interact with particles of all other species via an isotropic repulsive potential , @xmath9 . we consider two choices for @xmath9 ,    [ eqn : allpots ] @xmath10 @xmath11    the limit @xmath12 for the step potential corresponds to the widom - rowlinson model . for the gaussian molecule potential , the temperature @xmath13 plays no role because the boltzmann factor , @xmath14 is , by design , independent of @xmath13 . in general , each component of the @xmath0-component fluid may have a distinct fugacity ; however symmetry considerations dictate that a demixing transition occurs with all fugacities equal , and hence we set all the fugacities equal to a single value , @xmath15 . for sufficiently small @xmath15 , the @xmath0 species are mixed , while for large @xmath15 , there are @xmath0 distinct phases because different species repel one another , with each phase predominately composed of one species . if @xmath0 is not too large , there is expected to be a single demixing transition separating these regimes that is in the same universality class as the @xmath0-state potts model .    for very large @xmath0 , the correspondence between potts models and widom - rowlinson models must break down . although potts models have a single ordering transition , widom - rowlinson models can be presumed to have an intermediate crystalline phase for @xmath16 and large @xmath0 . to understand this phase , consider the limits @xmath17 and @xmath18 , with the product @xmath19 order unity . then non - overlapping particles appear with an effective fugacity of @xmath20 . however when two particles overlap , the cost is an additional factor of @xmath21 because the overlapping particles must be of the same species . hence the limiting model is precisely the hard sphere gas which we presume has a crystalline phase in @xmath16@xcite . it is therefore reasonable to assume that such a phase occurs in the widom - rowlinson models for large @xmath0 and @xmath15 of order @xmath22 . needless to say , for fixed @xmath0 , when the fugacity is sufficiently large , the model will demix , and thus the crystalline phase is an intermediate phase . we also expect an intermediate crystalline phase for large @xmath0 soft - core  models based on a mapping to a single component fluid with a repulsive soft - core potential . for example , the gaussian core model@xcite is known to crystallize in @xmath23 . although intermediate phases do not occur for the usual potts models , they are not a consequence of the continuum ; indeed such phases are known to occur on the lattice for the site - dilute ( annealed ) potts models@xcite as well as for the lattice version of the widom - rowlinson model@xcite . the algorithms used here are , broadly speaking , examples of cluster algorithms of the type first introduced by swendsen and wang@xcite . cluster algorithms have been found to be much more efficient than local algorithms such as the metropolis algorithm for simulating spin systems and lattice gases near critical points . cluster algorithms would be very useful for off - lattice systems , but no general cluster method has yet been developed ; indeed , the _ only _ off - lattice models for which highly efficient cluster methods are known are models of the stillinger - helfand and widom - rowlinson type . the distinguishing features of this class of models are that particles of the same species have no self - interaction and that there is a purely repulsive interaction between particles of different species . in this case , graphical representations and clusters algorithms are available@xcite and have been implemented for the widom - rowlinson model@xcite . cluster algorithms for spin systems work by identifying clusters of spins and then randomly flipping these clusters . cluster are defined by placing bonds between nearest neighbor aligned spins with a probability that depends on the temperature . for fluid systems , bonds are placed between particles of the same type with a probability that depends on the temperature and the interaction potential . instead of flipping spins , clusters of particles are removed from the system and then new particles are added via a nonuniform poisson process that depends on the fugacity and the potential due to the remaining particles . cluster algorithms are typically used with fixed values of the external parameters such as temperature or fugacity . however , when the location of the phase transition is not known , much computational effort in studying the transition is spent in locating the transition . to avoid this problem , invaded cluster methods can be used@xcite which automatically adjust a thermodynamic parameter ( for example , temperature or fugacity ) to its value at the phase transition . this adjustment is accomplished by using the fact ( proved for the @xmath24 case@xcite ) that the clusters just percolate at the transition . in invaded cluster algorithms , clusters are grown until a signature of percolation is observed . the value of the thermodynamic parameter at the transition is an output of the simulation obtained from the fraction of successful attempts to add particles or bonds to the system . the invaded cluster algorithm also may be used to distinguish first - order from continuous transitions as discussed in ref.@xcite for potts models . this method for distinguishing the order of the transition is discussed below and will be used in section  [ sec : stepq ] . we first describe the cluster algorithm discussed in section  3.5 of ref.@xcite for  models and then discuss how it can be modified to be an invaded cluster algorithm . we assume that we have a configuration consisting of particle positions and a set of bonds connecting some of the particles and describe how to obtain the next configuration :    1 . identify all clusters of particles defined by the bonds . a particle with no bonds is considered to be a singleton cluster . for each cluster , independently and with probability @xmath21 , label it a _ black _ cluster and with probability @xmath25 label it _ white_. 2 . remove all particles in black clusters . the remaining white particles are at a set of positions @xmath26 . 3 .   replenish the black particles via a poisson process with local intensity @xmath27 given by + @xmath28 + @xmath29 + where @xmath15 is the fugacity and @xmath9 the potential . 4 .   for each pair of black particles , place a new _ bond _ between them with probability @xmath30 given by @xmath31 where @xmath32 is the separation between the particles . note that @xmath30 is minus the mayer @xmath3-function for the potential . eliminate the white and black labels for the clusters . this procedure comprises one monte carlo step . given a configuration of particle positions and bonds without species labels , it is possible to obtain a full multicomponent configuration where each particle has a species label . this assignment is accomplished by identifying clusters and then randomly and independently assigning one of the @xmath0 species labels to each cluster . the species label of each particle is taken to be the species label of its cluster . this labeling of particles is only possible if @xmath0 is a positive integer . however , the algorithm makes sense for all @xmath6 , in analogy to the relation between potts models , which are defined for positive integer @xmath0 , and random cluster models which interpolate between them and are defined for all @xmath33 . it is instructive to consider the nature of the cluster configurations generated by the algorithm as a function of the fugacity for a fixed temperature . suppose that the fugacity and , hence , the density is very small . then @xmath30 is typically small because the particles are far apart , and most clusters are singletons . in step 2 , a fraction @xmath21 of the particles is removed . in step 3 , particles are replenished as a nearly ideal gas because the exponential factor in eq . ( [ eq : intensity ] ) is a small perturbation except in the vicinity of the remaining particles . the end result is a nearly ideal multi - component gas . in the limit of large fugacity and density , we expect a phase in which a single species is predominant with a small admixture of the other species . the bonds connecting particles of the dominant species are sufficiently dense that almost all members of this species are in a single large cluster . the minority species are almost all in widely scattered singleton clusters . when the majority species is white , as occurs in about @xmath21 of the monte carlo steps , the large cluster is removed and then replaced as a nearly ideal gas in a slightly perturbed background potential generated by the minority species . an important feature of this picture is that the clusters do not percolate at small fugacity and do percolate at large fugacity . at some intermediate value of the fugacity , there must be a percolation transition . as discussed in refs.@xcite , the percolation transition of the clusters coincides with the demixing transition of the fluid . the coincidence of the percolation transition and the demixing transition justifies an invaded cluster version of the above cluster algorithm . the invaded cluster algorithm is very similar to the fixed @xmath15 cluster algorithm described above except that steps 3 and 4 are modified as follows . instead of putting down new black particles as a poisson process at a fixed intensity , black particles are added to the system one at a time according to the potential @xmath34 ( see below ) . after each black particle is added , bonds between the new particle and all previously placed black particles are put down with probability @xmath35 . the black clusters defined by these bonds are monitored after each particle is added , and the process of adding particles is stopped when a _ stopping condition _ is satisfied . for simulating the phase transition , the stopping condition is that one cluster _ spans _ the system . for periodic boundary conditions , spanning is taken to mean that a cluster wraps around the system in at least one of the @xmath7 directions . the spanning condition insures that the algorithm simulates the phase transition@xcite .    in practice , a particle is added to the system according to the potential @xmath34 by the following procedure . a particle is tentatively placed at a random position @xmath36 . a random number @xmath32 is chosen in the interval @xmath37 , and the particle placement is accepted if @xmath38 otherwise the particle is rejected and another attempt is made to place a particle . let @xmath39 , where @xmath40 is the total number of attempted particle placements in a monte carlo step , including both accepted and rejected placements , @xmath41 is the system volume , and the brackets @xmath42 indicate an average over the simulation . because the intensity @xmath27 , defined in eq . ( [ eq : intensity ] ) , and the boltzmann factor @xmath43 governing particle placements differ by a factor of the fugacity , we conclude that @xmath44 is an estimator of @xmath45 , the value of the fugacity at the transition . note that if the fluctuations @xmath46 in @xmath47 are small , then the invaded cluster algorithm is essentially identical to the fixed fugacity algorithm operating at @xmath48 . this identification justifies the use of the invaded cluster method . a more complete discussion of the invaded cluster method and the use of @xmath44 as an estimator of a critical parameter is given in ref.@xcite . whenever the invaded cluster method simulates a system at its critical point , scaling methods can be used to obtain critical exponents from the size dependence of divergent thermodynamic quantities such as the compressibility or the susceptibility . to study the latter , we consider the quantity @xmath49 where @xmath50 is the number of particles in the @xmath51th cluster . we now show that @xmath52 is related to the usual susceptibility . consider , for simplicity , the discretized version of the  model on a lattice of linear dimension @xmath53 with spacing @xmath54 so that the total number of sites is @xmath55^d$ ] . the demixing order parameter at site @xmath36 is given by @xmath56 , where @xmath57 if there is a particle of type 1 at site @xmath36 and @xmath58 otherwise ; @xmath59 counts the presence of a particle of _ any _ type . the relevant susceptibility @xmath60 is defined by the second derivative of the pressure with respect to the ( ordering ) chemical potential : @xmath61 ( the reason that @xmath54 does not enter explicitly into eq . ( [ eq : chidef ] ) is that the derivatives are with respect to the log of the activity and it is the activity that is scaled by @xmath54 . ) for a given particle and bond configuration , averaging over assignments of species labels , it is clear that @xmath62 vanishes unless the sites @xmath36 and @xmath63 are both occupied and in the same cluster , in which case the result is @xmath64 . thus , for a fixed particle - bond configuration , we obtain the number of particles in the cluster at @xmath36 if we sum over @xmath63 . summing over @xmath36 yields the sum of the squares of the cluster sizes so that @xmath65 , and hence we conclude that @xmath52 is related to the usual susceptibility . finally , finite size scaling predicts that @xmath66 so that the scaling of @xmath52 with system size can be used to extract the magnetic exponent @xmath67 . cluster methods also may be used to distinguish first - order from continuous transitions . for this purpose , a fixed density stopping rule is used . black particles are added to the system until the density @xmath68 reaches a fixed value and then @xmath44 is measured . in this way the canonical ensemble is simulated rather than the grand canonical ensemble . this procedure is done for a range of densities near the transition . if the transition is continuous , the fugacity is a strictly increasing function of @xmath68 . however , if the transition is first - order , then the fugacity does not increase monotonically with increasing @xmath68 in the coexistence region . why does the nature of the @xmath44 versus @xmath68 curve signify whether a transition is continuous or first - order ? suppose that the demixing transition of a @xmath0-component system is first - order . at the transition , there is coexistence of @xmath69 phases ; @xmath0 demixed phases and one mixed phase . because the repulsive interaction is reduced for the demixed phases , these phases have a higher density than the mixed phase . thus , in the thermodynamic limit , there is a range of @xmath68 for which the fugacity is constant . let @xmath70 be the density of the mixed phase and @xmath71 the density of the demixed phase . because @xmath72 , where @xmath73 is the entropy density , we have that @xmath73 is a linear function of @xmath68 in the coexistence region . more specifically , @xmath74 is a linear combination of @xmath75 and @xmath76 , the entropy densities of the mixed and demixed phases . the linearity of @xmath74 applies in the thermodynamic limit . however , for a finite system , the entropy density is not linear in the coexistence region . consider a system with linear dimension @xmath53 and periodic boundary conditions at density @xmath68 . this system also can be viewed as an infinite system with periodic constraints on the particles . let @xmath77 be the entropy density of this periodically constrained system . now suppose the constraints are removed and the system comes to equilibrium . if @xmath78 , demixing will occur spontaneously so that @xmath79 with the equality holding only at the endpoints of the coexistence range . because @xmath80 , we must have that @xmath15 is non - monotone in the coexistence region . this approach for distinguishing the order of a transition is very similar to the microcanonical monte carlo method used in ref.@xcite . in section  [ sec : gaussian ] we present results for the 2d and 3d  gaussian molecule models . the two - component step potential model is discussed in section  [ sec : step2 ] and the @xmath0-component step potential is discussed in section  [ sec : stepq ] . we simulated the gaussian molecule model ( with the potential @xmath81 defined in eq .  ( [ eqn : sh ] ) ) using the invaded cluster method and the spanning rule described in section  [ sec : methods ] for a range of linear dimensions @xmath53 up to 140 in @xmath82 and 40 in @xmath23 . we choose units such that distances are measured in units of @xmath83 . we collected statistics for the number of particles in the spanning cluster @xmath84 , the critical density @xmath68 , the susceptibility @xmath52 , the estimator of the critical fugacity @xmath44 and its standard deviation @xmath46 , and the normalized autocorrelation function for the spanning cluster size , @xmath85 . for each value of @xmath53 we averaged over @xmath86 monte carlo steps . the estimator of the critical density is the average number of particles ( of any species ) per unit area ( volume ) when the spanning condition is fulfilled . although @xmath87 does not go to zero at finite @xmath32 , it becomes very small for larger @xmath32 and to speed the calculation , we set @xmath88 for @xmath89 . tables  [ table1 ] and [ table2 ] show the @xmath53 dependence of @xmath84 , @xmath68 , @xmath52 , @xmath44 , @xmath46 , and @xmath90 for the 2d and 3d  models , respectively . the integrated autocorrelation time @xmath90 is defined by @xmath91 this time is approximately the number of monte carlo steps between statistically independent configurations and enters into the error estimate for @xmath84 . in practice , @xmath92 becomes indistinguishable from the noise for @xmath93 monte carlo steps , and it is necessary to cut off the upper limit of the sum defining @xmath90 when the magnitude of @xmath85 becomes comparable to its error . note that the fluctuations @xmath46 in @xmath44 decrease with increasing @xmath53 and that @xmath90 is small and hardly increases with @xmath53 . these results demonstrate the validity and efficiency of the invaded cluster algorithm . the decrease in @xmath46 shows that as @xmath53 increases , the invaded cluster becomes essentially equivalent to a fixed parameter cluster algorithm for which detailed balance can be proven . the error estimates for all quantities in tables  [ table1 ] and [ table2 ] except @xmath90 were obtained by computing the standard deviation of the quantity of interest and dividing by the square root of the number of measurements . this error estimate does not take into account correlations between successive monte carlo steps . to account for correlations , the error estimates in the tables for an observable @xmath94 must be multiplied by @xmath95 , where @xmath96 is the integrated autocorrelation time for @xmath94 . the statistical errors for quantities derived from fits such as @xmath97 and @xmath98 include the factor @xmath99 except that @xmath96 is replaced by @xmath90 . figure  [ fig : rho2d ] shows the results for @xmath100 versus @xmath101 for the 2d  model . the value of @xmath68 in the limit of @xmath102 is extrapolated from the finite size data by doing a linear least squares fit omitting the values for @xmath103 and @xmath104 yielding the result , @xmath105 . a similar extrapolation for the critical fugacity yields @xmath106 . similarly , extrapolating the result for the 3d  model using the data for all available @xmath53 yields @xmath107 and @xmath108 . our error values for these critical parameters are one standard deviation from the linear least squares fit of the fugacity or density versus @xmath101 ; no effort has been made to estimate systematic errors . all of the fits have acceptable goodness - of - fit probability values @xmath109 . our 3d value for the critical density is consistent with the series result of lai and fisher@xcite , @xmath110 ( eq .  ( 36 ) of ref.@xcite ) but our critical fugacity is somewhat larger than their value , @xmath111 ( eq .  ( 44 ) of ref.@xcite ) . note that lai and fisher report results using a different convention so that their values of @xmath97 and @xmath112 must be divided by @xmath113 to compare with our values . the exponent ratio @xmath98 can be obtained from the scaling of the susceptibility @xmath52 with @xmath53 according to eq . ( [ eq : chifss ] ) . figure  [ fig : chi2 ] shows a log - log plot of @xmath52 versus @xmath53 for the 2d gaussian molecule model . a least squares fit of all the data to a simple power law does not yield an acceptable goodness of fit value @xmath109 . if the smallest value of @xmath53 is omitted , we obtain @xmath114 with @xmath115 , @xmath116 , and @xmath117 ( degrees of freedom ) . the @xmath109 value indicates a reasonable fit to a simple power law , but the fitted value of @xmath98 is @xmath118 from the 2d ising value of @xmath119 . a reasonable explanation of this result is that the 2d gaussian molecule is , indeed , in the 2d ising universality class , but that there are relatively slowly varying corrections to scaling . a least squares fit to all the data for the susceptibility @xmath52 for the 3d gaussian molecule model yields @xmath120 with @xmath121 , @xmath122 , and @xmath123 . the @xmath109 value near unity suggests that the data is well fit to a pure power law . recent high precision monte carlo studies of the 3d ising model@xcite yield @xmath124 which is consistent with our results . our results add weight to the hypothesis that the  model is in the ising universality class for both 2d and 3d . the relatively high precision results for @xmath98 from the gaussian molecule model suggests that models of the  type may be useful for high precision studies of the 3d ising universality class . the isotropy of the interaction and absence of an underlying lattice might make for smaller corrections to scaling in  models compared to lattice spin models . table  [ table3 ] summarizes our results for the 2d  model with the step potential given by eq . ( [ eqn : step ] ) and temperature @xmath125 ( measured in units of @xmath126 ) . for each value of @xmath53 we averaged over @xmath127 monte carlo steps . the results are qualitatively similar to the gaussian molecule model . table  [ table4 ] shows the temperature dependence of the measured quantities for @xmath53 fixed at @xmath103 . the values of @xmath68 and @xmath44 at low temperatures should reduce to the widom - rowlinson model . in ref.@xcite we measured the critical parameters of the widom - rowlinson model using the invaded cluster method . for @xmath128 ( the smallest size measured ) we obtained @xmath129 and @xmath130 , values that are close to the values of @xmath68 and @xmath44 for the two lowest temperatures in table  [ table4 ] this agreement confirms that the step potential is continuously connected to the hard core potential . if the @xmath103 data point is omitted , we obtain from a least squares fit to the data for the susceptibility @xmath52 , @xmath131 with @xmath132 , @xmath133 , and @xmath134 . [ sec : stepq ] the critical properties of the @xmath0-component  model are expected to be closely related to the @xmath0-state potts model . one of the features of the @xmath0-state potts model is that the transition is continuous for small @xmath0 and is first - order for @xmath135 , where @xmath136 and @xmath137 . we have used the method described in section  [ sec : methods ] to determine the order of the transition as a function of @xmath0 for the @xmath0-component  step potential model . figure  [ fig : fig4 ] shows the fugacity @xmath44 as a function of @xmath68 for @xmath138 for @xmath128 and @xmath125 . note that for @xmath139 the curve is clearly monotonically increasing , which implies a continuous transition . for @xmath140 the curves are clearly non - monotonic , which implies a first - order transition . for @xmath141 the curve is essentially flat within the error bars ( whose size is approximately that of the symbols ) . although the effective value of @xmath142 is expected to vary with @xmath53 , these results are consistent with the hypothesis that @xmath136 for the 2d  step potential model . figure  [ fig : fig5 ] shows @xmath44 as a function of @xmath68 for the 3d  step potential model for @xmath103 and @xmath125 . the @xmath24 curve is clearly monotonically increasing while the @xmath139 is clearly not , implying that @xmath137 as for the 3d potts model . finally , we have studied the effect of quenched impurities on the nature of the transition for the @xmath139  step potential model . the impurities consist of randomly placed scatterers that interact with all the fluid particles via the same repulsive step potential that exists between different components . figure  [ fig : fig6 ] shows a plot of @xmath44 versus @xmath68 for the @xmath139  step potential model in 3d for four impurity densities ranging from 0.025 to 0.0625 . for each of the 10 impurity configurations considered for a given density , data from @xmath143 monte carlo steps are collected . for the two lowest impurity concentrations , the @xmath44 versus @xmath68 curve is non - monotonic as is the case for the pure system , while for the two highest impurity concentrations , the curve is monotonic indicating a crossover to a continuous transition . this behavior is in accord with general arguments@xcite that the presence of quenched impurities should cause a first - order transition to become continuous . it is not clear from our data whether there is a critical value of the disorder below which the transition remains continuous or whether the crossover at finite disorder strength is a finite size effect and that any strength of disorder is sufficient to make the transition continuous in the thermodynamic limit . we have studied the  model and several generalizations using the invaded cluster algorithm . our results for @xmath0-component models with @xmath144 are consistent with the hypothesis that these models are in the same universality class as the corresponding potts models . in addition , we have shown that the addition of quenched disorder causes the demixing transition to change from first - order to continous for those values of @xmath0 for which the pure system transition is first - order . for the case @xmath24 and @xmath138 , our results for the magnetic exponent are outside the statistical error bars of the exact ising value . however , we believe that this difference is most likely the result of slowly varying corrections to scaling . it would be useful to consider larger system to confirm ising universality . it would also be interesting to consider larger values of @xmath0 to explore the possibility of an intermediate crystalline phase in  models . this work was supported by nsf grants phy-9801878 ( rs ) , dmr-9633385 ( hg ) , dmr-9978233 ( jm ) , dms-9971016 ( lc ) , and nsa grant mda904 - 98 - 1 - 0518 ( lc ) . we thank gregory johnson for useful discussions .                                                    .the @xmath53 dependence of the number of particles in the spanning cluster @xmath84 , the critical density @xmath68 , the susceptibility @xmath52 , the estimator of the critical fugacity @xmath44 , its standard deviation @xmath46 , and decorrelation time @xmath90 for the 2d gaussian molecule potential . the averages are over @xmath146 spanning clusters . the error estimates were obtained by computing the standard deviation of the quantity of interest and dividing by the square root of the number of measurements . the error estimates for @xmath90 are obtained from the variation of @xmath147 with the upper limit in the summation of eq . ( [ eq : tau ] ) and hence represent an estimate of the systematic error . the autocorrelation function @xmath92 is distinguishable from the noise for @xmath148 monte carlo steps . [ cols=">,^,^,^,^,^,^ " , ]
phase transitions of fluid mixtures of the type introduced by stillinger and helfand are studied using a continuum version of the invaded cluster algorithm . 
 particles of the same species do not interact , but particles of different types interact with each other via a repulsive potential . 
 examples of interactions include the gaussian molecule potential and a repulsive step potential . 
 accurate values of the critical density , fugacity and magnetic exponent are found in two and three dimensions for the two - species model . 
 the effect of varying the number of species and of introducing quenched impurities is also investigated . in all the cases studied , mixtures of @xmath0-species are found to have properties similar to @xmath0-state potts models .