The second approach is more a brute force procedure, since it aims to nd an Ising-like spin model, with just spin-spin interactions, which reproduces the congurations of the signs of the Polyakov loops. The new method leads to good results both in the strong coupling case we had examined with the rst approach (
N
= 2) and if we move towards the weak coupling limit (N
= 4). However, forN
= 4 the approximation looks worse than forN
= 2. More precisely, the value of the exponents' ratio=
seems to shift slightly towards the random percolation value, even if it is still in agreement with the Ising ratio. This can mean that the procedure is not reliable for higher values ofN
. As a matter of fact, we have to recognize that our ansatz for the Hamiltonian of the eective model is probably too restrictive, and that multispin couplings may become important for bigN
's. Moreover, the precision of the method decreases the more spin-spin operators we introduce. In fact, if we analyze any time the same number ofSU
(2) congurations, the errors on the nal couplings of the eective theory are of the same order, no matter how many couplings we have. Consequently, the corresponding uncertainty on the model is the greater the more the couplings are. The Polyakov loop clusters, which are built by using the bond weights calculated from the couplings of the eective theory, become thus less and less dened. In order not to lose accuracy, one must lower the error on each single coupling, and that is possible only if we increase the number ofSU
(2) congurations to analyze, which can lead to prohibitively lengthy simulations.In conclusion, the second approach has certainly some drawbacks. Nevertheless, it allowed us to dene some Polyakov loop clusters which have, with good approximation, the properties of the physical droplets of
SU
(2) we were looking for, also in a case which approaches the weak coupling limit (N
= 4). For this reason, the second approach is to be preferred to the rst one, which strongly depends on a special lattice regularization ofSU
(2).From our investigations it is not possible to argue whether the critical behaviour of other eld theories can be described by means of percolation. The strict relationship between
SU
(N
) gauge theories andZ
(N
) spin models can represent a useful tool to devise suitable percolation pictures for the gauge theories starting from results known for the simpler spin models. In principle, that is exactly what we have done in our case, exploiting the analogy betweenSU
(2) and the Ising model. In practice, the task gets more complicated forSU
(N
), whenN >
2. For example,SU
(3) gauge theory is certainly the most interesting case of all, because it involves the "real"gluons. In two space dimensions,
SU
(3) undergoes a second order phase transition, like the three states Potts model. Very recently [66] it was shown that the 2-dimensional three states Potts model admits an equivalent percolation formulation, which could thus be used forSU
(3).However,
SU
(3) in two space dimensions is rather an academic model. One is surely more interested in the realistic 3-dimensional case. The fact that theSU
(3) phase transition in three space dimensions is rst order poses an essential problem concerning the relationship between percolation and rst order phase transitions.The situation gets even more involved when one considers the case of full
QCD
, i. e.SU
(3) plus dynamical quarks, since the transition from connement to deconnement is probably a crossover, i.e. it takes place without any singularity in the partition function. We have seen in Section 2.6 that there are cases in which geometrical properties can change abruptly without a corresponding discontinuity in the thermal variables. This could provide a criterion to deneSummary 103 dierent phases and the relative transition in an extended sense [67]. Work in this direction is in progress.
We conclude our summary with some general remarks concerning the method we have chosen to study correlated percolation, i. e. Monte Carlo simulations. There is, in fact, basically no literature about this subject, as most of the known results are based on analytical proofs, and the few numerical studies rely on series expansions.
We point out the importance of the percolation cumulant, from which one can derive a precise estimate of the critical point. Besides, the scaling of the percolation cumulant curves allows to get the value of the critical exponent
, with 4?5% accuracy for the lattice sizes we have considered.The accuracy can be increased by analyzing larger lattices. Anyway, better estimates of
can be obtained by using standard nite size scaling techniques, like the scaling of the pseudocritical points (see end of Section 1.5).We remark that, for equal statistics, the errors on the percolation variables are much smaller than the errors on the corresponding thermal variables. The latter seems to be a general feature of site-bond percolation, because the clusters depend as well on the bonds' distribution. This introduces a further random element which contributes to reduce sensibly the correlation of the percolation measurements with respect to the thermal counterparts, which depend only on the spin congurations. We found that the data of the percolation strength
P
are always more correlated than the corresponding data of the average cluster sizeS
.For a study of the thermal transition variables like the susceptibility
or the Binder cumulantg
r are necessary. Such quantities cannot be determined directly from measurements on the spin congurations, but are calculated by means of averages of powers of the order parameter.That usually leads to big error bars on the nal results of
andg
r. Instead, the percolation counterparts ofandg
r, i.e. the average cluster size and the percolation cumulant, are calculated directly from the clusters' congurations, so that their errors are rather small.Hence, in order to get the same accuracy on the average values, the thermal investigation of a model would require more
CPU
time than the relative percolation study. Nevertheless we have to point out that the errors on the thermal variables can be considerably reduced by means of reweighting techniques like theDSM
[45], which we have often used in our studies, whereas similar interpolation methods do not exist for correlated percolationz. In this work we were thus forced to use directly the data points in the nite size scaling ts. We think that the Fortuin-Kasteleyn-Swendsen-Wang model we have discussed in Section 2.4 could be used to implement an ecient method for the interpolation of Fortuin-Kasteleyn percolation data relative to theq
-state Potts model.From the nite size scaling analysis, it turns out that the scaling behaviour of the percolation variables is rather pure: that is clearly shown by the precision of the scaling of the percolation
zFor random percolation a reweighting method was recently proposed [68, 69]; the role of the energy is carried out by the probability of having a conguration in correspondence of a value pof the density of occupied sites (bonds).
cumulants we have performed many times in this work. In particular, in all our analyses, corrections to scaling seem negligible, and nite size eects disappear already for relatively small lattice sizes. This is quite impressive, especially when one makes comparisons with the thermal variables, which are normally strongly aected by such perturbations. Nevertheless, we have to keep in mind that the accuracy on our evaluation of the critical exponents has always been about 1?2%, which is good for our purposes x but not exceptional. Moreover, the percolation data of our
SU
(2) studies are already aected by the approximations involved in the determination of the eective theory, which are by far more important than eventual corrections to scaling. On the other hand, if we want to obtain more accurate estimates of the results for models which admit an exact percolation formulation, like the continuous spin models of Chapter 3, corrections to scaling may become important: in high precision numerical studies of random percolation that seems indeed to be the case [23]. We remind that we have almost always adopted free boundary conditions for the cluster identication. The results onO
(n
) spin models, however, suggest that the situation could be further on improved by using periodic boundary conditions (see Section 3.3).xWe remind that for the systems we investigated we had to check whether the critical exponents of the perco-lation transition agree with the thermal exponents of the system or rather with the ones of random percoperco-lation.
The thermal exponents of all the models we have considered dier from the random percolation exponents of about 10?20%, so that our accuracy is good enough to distinguish the two cases.
Appendix A