Correlated Percolation

Cluster concepts have been extremely useful in elucidating many problems in physics. Percolation theory provides a generic framework to study the behavior of the cluster distribution. In most cases the theory predicts a geometrical transition at the percolation threshold, characterized in the percolative phase by the presence of a spanning cluster, which becomes infinite in the thermodynamic limit. Standard percolation usually deals with the problem when the constitutive elements of the clusters are randomly distributed. However correlations cannot always be neglected. In this case correlated percolation is the appropriate theory to study such systems. The origin of correlated percolation could be dated back to 1937 when Mayer [76] proposed a theory to describe the condensation from a gas to a liquid in terms of mathematical clusters (for a review of cluster theory in simple fluids see [ 88]). The location for the divergence of the size of these clusters was interpreted as the condensation transition from a gas to a liquid. One of the major drawbacks of the theory was that the cluster number for some values of thermodynamical parameters could become negative. As a consequence the clusters did not have any physical interpretation [ 50]. This theory was followed by Frenkel's phenomenological model [54], in which the fluid was considered as made of non interacting physical clusters with a given free energy. This model was later improved by Fisher [50], who proposed a different free energy for the clusters, now called droplets, and consequently a different scaling form for the droplet size distribution. This distribution, which depends on two geometrical parameters, ? and ?, has the nice feature that the mean droplet size exhibits a divergence at the liquid-gas critical point. Interestingly the critical exponents of the liquid gas critical point can be expressed in terms of the two parameters, ? and ?, and are found to satisfy the standard scaling relations proposed at that time in the theory of critical phenomena.