Correlated Percolation

Definition of the Subject and Its Importance

Cluster concepts have been extremely useful in elucidating many problems in physics. Percolation theory (Grimmett 1989; Stauffer and Aharony 1994; Sahimi 1994) 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 (Mayer 1937; Mayer and Ackermann 1937; Mayer and Harrison 1938; Mayer and Mayer 1940) proposed a theory to describe the condensation from a gas to a liquid in...

Appendix – Random Cluster Model and Ising Droplets

In 1969, Fortuin and Kasteleyn (FK) (Kasteleyn and Fortuin 1969; Fortuin and Kasteleyn 1972) introduced a correlated bond percolation model, called random cluster model, and showed that the partition function of this percolation model was identical to the partition function of q–state Potts model. They also showed that the thermal quantities in the Potts model could be expressed in terms of connectivity properties of the random cluster model. Much later in 1980, Coniglio and Klein (1980) independently have used a different approach with the aim to define the proper droplets in the Ising model. It was only later that it was realized that the two approaches were related, although the meaning of the clusters in the two approaches is different. We will discuss these two approaches here, and show that their statistical properties are the same.

Random Cluster Model

Let us consider an Ising system of spins S_i = ± 1 on a lattice with nearest-neighbor interactions and, when needed, let us assume periodic boundary conditions in both directions. All interactions have strength J and the Hamiltonian is

$$ \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)=-\sum \limits_{<i,j>}J\left({S}_i{S}_j-1\right), $$

(57)

where {S_i} represents a spin configuration and the sum is over nn spins. The main point in the FK approach is to replace the original Ising Hamiltonian with an annealed diluted Hamiltonian

$$ {\mathrm{\mathscr{H}}}^{\prime}\left(\left\{{S}_i\right\}\right)=-\sum \limits_{<i,j>}{J}_{i\;j}^{\prime}\left({S}_i{S}_j-1\right), $$

(58)

where

$$ {J}_{i\;j}^{\prime }=\left\{\begin{array}{l}{J}^{\prime }\ \mathrm{with}\ \mathrm{probability}\ p\\ {}0\ \mathrm{with}\ \mathrm{probability}\ \left(1-p\right).\end{array}\right. $$

(59)

The parameter p is chosen such that the Boltzmann factor associated to an Ising configuration of the original model coincides with the weight associated to a spin configuration of the diluted Ising model

$$ {e}^{-\beta \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)}\equiv \prod \limits_{<i,j>}{e}^{\beta J\left({S}_i{S}_j-1\right)}=\prod \limits_{<i,j>}\left({pe}^{\beta {J}^{\prime}\left({S}_i{S}_j-1\right)}+\left(1-p\right)\right), $$

(60)

where β = 1/k_BT, k_B is the Boltzmann constant and T is the temperature. In order to satisfy (60), we must have

$$ {e}^{\beta J\left({S}_i\;{S}_j-1\right)}={pe}^{\beta\;{J}^{\prime}\left({S}_i\;{S}_i-1\right)}+\left(1-p\right). $$

(61)

We take now the limit J^′ ↦ ∞. In such a case, ${e}^{\beta\;{J}^{\prime}\left({S}_i\;{S}_j-1\right)}$ equals the Kronecker delta ${\delta}_{S_i{S}_j}$ and from (61) p is given by

$$ p=1-{e}^{-2\beta\;J}. $$

(62)

From (60), by performing the products we can write

$$ {e}^{-\beta \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)}=\sum \limits_C{W}_{F\;K}\left(\left\{{S}_i\right\},C\right), $$

(63)

where

$$ {W}_{F\;K}\left(\left\{{S}_i\right\},C\right)={p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid}\prod \limits_{<i,j>\in C}{\delta}_{S_i\;{S}_j}. $$

(64)

Here C is a configuration of interactions where |C| is the number of interactions of strength J^′ = ∞ and |A| the number of interactions of strength 0. |C| + |A| = |E|, where |E| is the total number of edges in the lattice. W_{F K}({S_i},C) is the statistical weight associated (a) to a spin configuration {S_i} and (b) to a set of interactions in the diluted model where |C| edges have ∞ strength interactions, while all the other edges have 0 strength interactions. The Kronecker delta indicates that two spins connected by an ∞ strength interaction must be in the same state. Therefore, the configuration C can be decomposed in clusters of parallel spins connected by infinite strength interactions.

Finally, the partition function of the Ising model Z is obtained by summing the Boltzmann factor (63), over all the spin configurations. Since each cluster in the configuration C gives a contribution of 2, we obtain:

$$ Z=\sum \limits_{\left\{{S}_i\right\}}{W}_{F\;K}\left(\left\{{S}_i\right\},C\right)=\sum \limits_C{p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid }{2}^{N_C}, $$

(65)

where N_C is the number of clusters in the configuration C.

In conclusion, in the FK formalism the partition function of the Ising model (65) is equivalent to the partition function of a correlated bond percolation model (Kasteleyn and Fortuin 1969; Fortuin and Kasteleyn 1972; Hu 1984, 1992; Hu and Mak 1989), where the weight of each bond configuration C is given by

$$ W(C)={p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid }{2}^{N_C}, $$

(66)

which coincides with the weight of the random percolation except for the extra factor ${2}^{N_C}$. The correlation is due to the presence of this extra factor. Clearly all percolation quantities in this correlated bond percolation model can be obtained by using the weight given by Eq. (66) Interestingly, using (64) and (63) Fortuin and Kasteleyn have proved that (Kasteleyn and Fortuin (1969), Fortuin and Kasteleyn (1972))

$$ \left|\left\langle {S}_i\right\rangle \right|={\left\langle {\gamma}_i^{\infty}\right\rangle}_W $$

(67)

and

$$ \left\langle {S}_i{S}_j\right\rangle ={\left\langle {\gamma}_{i\;j}\right\rangle}_W, $$

(68)

where 〈…〉 is the Boltzmann average of the standard Ising model (57) while 〈…〉_W is the average over bond configurations in the bond correlated percolation, with weights given by (66). Here ${\gamma}_i^{\infty }(C)$ is equal to 1 if the spin at site i belongs to the spanning cluster in such bond correlated percolation, 0 otherwise; γ_i j(C) is equal to 1 if the spins at sites i and j belong to the same cluster, 0 otherwise. Eqs. (67, and 68) link the connectivity properties of the correlated percolation with weights given by (66), with the Boltzmann average of thermal quantities.

Connection Between the Ising Droplets and the Random Cluster Model

In the approach followed by Coniglio and Klein (1980), given a configuration of spins, one introduces at random connecting bonds between nn parallel spins with probability p_b, antiparallel spins are never considered connected. Clusters are defined as maximal sets of parallel spins connected by bonds. The bonds here are fictitious, they are introduced only to define the clusters and do not modify the interaction energy as in the FK approach. For a given realization of bonds, we distinguish the subsets C and B of nn parallel spins, respectively, connected and not connected by bonds and the subset D of nn antiparallel spins. The union of C, B, and D coincides with the total set of nn pair of spins E. The statistical weight of a configuration of spins and bonds is (Coniglio 1990; Coniglio et al. 1989)

$$ {W}_{C\;K}\left(\left\{{S}_i\right\},C\right)={p}_b^{\mid C\mid }{\left(1-{p}_b\right)}^{\mid B\mid }{e}^{-\beta \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)}, $$

(69)

where |C| and |B| are the number of nn pairs of parallel spins respectively in the subset C and B not connected by bonds. For a given spin configuration, using Newton binomial rule, we have the following sum rule

$$ \sum \limits_C{p}_b^{\mid C\mid }{\left(1-{p}_b\right)}^{\mid B\mid }=1. $$

(70)

From Eq. (70) follows that the Ising partition function, Z, may be obtained by summing (69) over all bond configurations and then over all spin configurations.

$$ Z=\sum \limits_{\left\{{S}_i\right\}}\sum \limits_C{W}_{C\;K}\left(\left\{{S}_i\right\},C\right)=\sum \limits_{\left\{{S}_i\right\}}{e}^{-\beta \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)}. $$

(71)

The partition function of course does not depend on the value of p_b which controls the bond density. By tuning p_b instead it is possible to tune the size of the clusters. For example, by taking p_b = 1 the clusters would coincide with nearest neighbor parallel spins, while for p_b = 0 the clusters are reduced to single spins. By choosing the droplet bond probability p_b = 1 − e^−2βJ ≡ p and observing that ${e}^{-\beta \mathrm{\mathscr{H}}\left(\left\{{S}_i\right\}\right)}={e}^{-2\beta J\mid D\mid }$, where |D| is the number of antiparallel pairs of spins, the weight (69) simplifies and becomes:

$$ {W}_{C\;K}\left(\left\{{S}_i\right\},C\right)={p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid}\prod \limits_{<i,j>\in C}{\delta}_{S_i\;{S}_j}, $$

(72)

where |A| = |B| + |D| = |E| – |C| and the product of the Kronecker delta implies that the spin configurations are compatible with the bond configuration C. This result shows that the weights of the CK droplets coincide with the weight of the FK clusters.

From (72), by summing over all the spin configurations we can calculate the weight W(C) that a given configuration of connecting bonds C between nn parallel spins occurs namely

$$ W(C)=\sum \limits_{\left\{{S}_i\right\}}{p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid}\prod \limits_{<i,j>\in C}{\delta}_{S_i\;{S}_j}={p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid }{2}^{N_C}. $$

(73)

Consequently in (71) by taking first the sum over all spins compatible with the configuration C, the partition function Z can be written as in the FK formalism (65).

$$ Z=\sum \limits_C{p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid }{2}^{N_C}. $$

(74)

In spite of the strong analogies the CK clusters and the FK clusters have a different meaning. In the CK formalism the clusters are defined directly in a given configuration of the Ising model as parallel spin connected by fictitious bonds, while in the FK formalism clusters are defined in the equivalent random cluster model. However, due to the equality of the weights (72) and (64) the statistical properties of both clusters are identical (Coniglio et al. 1989) and due to the relations between (64) and (66), both coincide with those of the correlated bond percolation whose weight is given by (66). More precisely, any percolation quantity g(C) which depends only on the bond configuration has the same average

$$ {\left\langle g(C)\right\rangle}_{F\;K}={\left\langle g(C)\right\rangle}_{CK}={\left\langle g(C)\right\rangle}_W, $$

(75)

where 〈…〉_F K, 〈…〉_CK are the average over spin and bond configurations with weights given by (64) and (72), respectively, and 〈…〉_W is the average over bond configurations in the bond correlated percolation with weights given by (66). In view of (75) it follows (Coniglio et al. 1989)

$$ {\left\langle {\gamma}_i^{\infty}\right\rangle}_{F\;K}={\left\langle {\gamma}_i^{\infty}\right\rangle}_{CK}={\left\langle {\gamma}_i^{\infty}\right\rangle}_W $$

(76)

and

$$ {\left\langle {\gamma}_{i\;j}\right\rangle}_{F\;K}={\left\langle {\gamma}_{i\;j}\right\rangle}_{C\;K}={\left\langle {\gamma}_{i\;j}\right\rangle}_{C\;K}. $$

(77)

where 〈…〉_F K, 〈…〉_CK, and 〈…〉_W are the average over spin and bond configurations with weights given by (64), (72), and (66), respectively. Finally, in view of (67) and (68),

$$ \left|\left\langle {S}_i\right\rangle \right|={\left\langle {\gamma}_i^{\infty}\right\rangle}_{C\;K} $$

(78)

and

$$ \left\langle {S}_i\;{S}_j\right\rangle ={\left\langle {\gamma}_{i\;j}\right\rangle}_{C\;K}. $$

(79)

The results of this Appendix can be extended immediately to relate q-state Potts droplets to the random cluster model.

We conclude by noting that in order to generate an equilibrium CK droplet configuration in a computer simulation, it is enough to equilibrate a spin configuration of the Ising model and then introduce at random fictitious bonds between parallel spins with a probability given by (62).