Correlated Percolation

Definition of the Subject

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 (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 terms of mathematical clusters (for a review of cluster theory in simple fluids,...

Appendix: Random Cluster Model and Ising Droplets

In 1969, Kasteleyn and Fortuin (KF) (1969, Fortuin and Kasteleyn 1972) introduced a correlated bond percolation model, called the random cluster model, and showed that the partition function of this percolation model was identical to the partition function of the 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 ${S}_i=\pm 1$ 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

$$ \mathcal{H}\left(\left\{{S}_i\right\}\right)=-{\displaystyle \sum_{\left\langle i,j\right\rangle }J}\left({S}_i{S}_j-1\right) , $$

(54)

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

$$ {\mathcal{H}}^{\prime}\left(\left\{{S}_i\right\}\right)=-{\displaystyle \sum_{\left\langle i,j\right\rangle }{J}_{ij}^{\prime }}\left({S}_i{S}_j-1\right) , $$

(55)

where

$$ {J}_{ij}^{\prime }=\left\{\begin{array}{ll}{J}^{\prime}\hfill & \mathrm{with}\ \mathrm{probability} p\hfill \\ {}0\hfill & \mathrm{with}\ \mathrm{probability} \left(1-p\right) .\hfill \end{array}\right. $$

(56)

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

$$ \begin{array}{c}\hfill {e}^{-\beta \mathcal{H}\left(\left\{{S}_i\right\}\right)}\equiv {\displaystyle \prod_{\left\langle i,j\right\rangle }{\mathrm{e}}^{\beta J\left({S}_i{S}_j-1\right)}}\hfill \\ {}\hfill ={\displaystyle \prod_{\left\langle i,j\right\rangle}\left(p{\mathrm{e}}^{\beta {J}^{\prime}\left({S}_i{S}_j-1\right)}+\left(1-p\right)\right)} ,\hfill \end{array} $$

(57)

where $\beta =1/{k}_{\mathrm{B}}T$, k _B is the Boltzmann constant, and T is the temperature. In order to satisfy Eq. 57, we must have

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

(58)

We take now the limit ${J}^{\prime}\mapsto \infty$. In such a case, ${\mathrm{e}}^{\beta {J}^{\prime}\left({S}_i{S}_j-1\right)}$ equals the Kronecker delta ${\delta}_{S_i{S}_j}$, and from Eq. 58, p is given by

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

(59)

From Eq. 57, by performing the products we can write

$$ {\mathrm{e}}^{-\beta \mathcal{H}\left(\left\{{S}_i\right\}\right)}={\displaystyle \sum_C{W}_{\mathrm{KF}}}\left(\left\{{S}_i\right\},C\right) , $$

(60)

where

$$ {W}_{\mathrm{KF}}\left(\left\{{S}_i\right\},C\right)={p}^{\left|C\right|}{\left(1-p\right)}^{\left|A\right|}{\displaystyle \prod_{\left\langle i,j\right\rangle \in C}{\delta}_{S_i{S}_j}} . $$

(61)

Here, C is a configuration of interactions where $\left|C\right|$ is the number of interactions of strength ${J}^{\prime }=\infty$ and $\left|A\right|$ the number of interactions of strength 0. $\left|C\right|+\left|A\right|=\left|E\right|$, where $\left|E\right|$ is the total number of edges in the lattice.

W _KF({S _i }, C) is the statistical weight associated (a) with a spin configuration {S _i } and (b) with a set of interactions in the diluted model where $\left|C\right|$ 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 Eq. 60 over all the spin configurations. Since each cluster in the configuration C gives a contribution of 2, we obtain:

$$ Z={\displaystyle \sum_C{p}^{\left|C\right|}}{\left(1-p\right)}^{\left|A\right|}{2}^{N_C} , $$

(62)

where N _C is the number of clusters in the configuration C.

In conclusion, in the KF formalism, the partition function Eq. 62 is equivalent to the partition function of a correlated bond percolation model (Hu 1984, 1992; Hu and Mak 1989; Kasteleyn and Fortuin 1969; Fortuin and Kasteleyn 1972) where the weight of each bond configuration C is given by

$$ W(C)={\displaystyle \sum_{\left\{{S}_i\right\}}{W}_{\mathrm{KF}}}\left(\left\{{S}_i\right\},C\right)={p}^{\left|C\right|}{\left(1-p\right)}^{\left|A\right|}{2}^{N_C} $$

(63)

which coincides with the weight of the random percolation except for the extra factor ${2}^{N_C}$. Clearly, all percolation quantities in this correlated bond model are weighted according to Eq. 63 and coincide with the corresponding percolation quantities of the KF clusters made of parallel spins connected by an ∞ strength interaction, whose statistical weight is given by Eq. 61. Moreover, using Eqs. 61 and 60, Kasteleyn and Fortuin have proved that (Kasteleyn and Fortuin 1969; Fortuin and Kasteleyn 1972)

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

(64)

and

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

(65)

where 〈 … 〉 is the Boltzmann average and 〈 … 〉 _W is the average over bond configurations in the bond-correlated percolation with weights given by Eq. 63. Here, ${\gamma}_i^{\infty }(C)$ is equal to 1 if the spin at site i belongs to the spanning cluster, 0 otherwise; γ _ij (C) is equal to 1 if the spins at sites i and j belong to the same cluster, 0 otherwise.

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 not connected with probability 1. 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)

$$ {W}_{\mathrm{CK}}\left(\left\{{S}_i\right\},C\right)={p}_{\mathrm{b}}^{\mid C\mid }{\left(1-{p}_{\mathrm{b}}\right)}^{\mid B\mid }{\mathrm{e}}^{-\beta \mathcal{H}\left(\left\{{S}_i\right\}\right)} , $$

(66)

where $\left|C\right|$ and $\left|B\right|$ 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 the Newton binomial rule, we have the following sum rule:

$$ {\displaystyle \sum_C{p}_{\mathrm{b}}^{\mid C\mid }}{\left(1-{p}_{\mathrm{b}}\right)}^{\mid B\mid }=1 . $$

(67)

From Eq. 67 follows that the Ising partition function, Z, may be obtained by summing Eq. 66 over all bond configurations and then over all spin configurations.

$$ Z={\displaystyle \sum_{\left\{{S}_i\right\}}{\displaystyle \sum_C{W}_{\mathrm{CK}}}}\left(\left\{{S}_i\right\},C\right)={\displaystyle \sum_{\left\{{S}_i\right\}}{\mathrm{e}}^{-\beta \mathcal{H}\left(\left\{{S}_i\right\}\right)}} . $$

(68)

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 ${p}_{\mathrm{b}}=1$, the clusters would coincide with nearest-neighbor parallel spins, while for ${p}_{\mathrm{b}}=0$, the clusters are reduced to single spins. By choosing the droplet bond probability ${p}_{\mathrm{b}}=1-{\mathrm{e}}^{-2\beta J} \equiv p$ and observing that ${\mathrm{e}}^{-\beta \mathcal{H}\left(\left\{{S}_i\right\}\right)}={\mathrm{e}}^{-2\beta J\left|D\right|}$, where $\left|D\right|$ is the number of antiparallel pairs of spins, the weight Eq. 66 simplifies and becomes

$$ {W}_{\mathrm{CK}}\left(\left\{{S}_i\right\},C\right)={p}^{\mid C\mid }{\left(1-p\right)}^{\mid A\mid} , $$

(69)

where $\left|A\right|=\left|B\right|+\left|D\right|=\left|E\right|-\left|C\right|$.

From Eq. 69, we can calculate the weight W(C) that a given configuration of connecting bonds C between nn parallel spins occurs. This configuration C can occur in many spin configurations. So we have to sum over all spin configurations compatible with the bond configuration C, namely:

$$ W(C)={\displaystyle \sum_{\left\{{S}_i\right\}}{W}_{\mathrm{CK}}}\left(\left\{{S}_i\right\},C\right){\displaystyle \prod_{\left\langle i,j\right\rangle \in C}{\delta}_{S_i{S}_j}} , $$

(70)

where, due to the product of the Kronecker delta, the sum is over all spin configurations compatible with the bond configuration C. From Eqs. 59 and 70, we have

$$ \begin{array}{l}W(C)={\displaystyle \sum_{\left\{{S}_i\right\}}{p}^{\mid C\mid }}{\left(1-p\right)}^{\mid A\mid }{\displaystyle \prod_{\left\langle i,j\right\rangle \in C}{\delta}_{S_i{S}_j}}\\ {} ={p}^{\left|C\right|}{\left(1-p\right)}^{\left|A\right|}{2}^{N_C} .\end{array} $$

(71)

Consequently in Eq. 68, by taking first the sum over all spins compatible with the configuration C, the partition function Z can be written as in the KF formalism Eq. 62.

$$ Z={\displaystyle \sum_C{p}^{\left|C\right|}}{\left(1-p\right)}^{\left|A\right|}{2}^{N_C} . $$

(72)

In spite of the strong analogies, the CK clusters and the KF 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 KF formalism, clusters are defined in the equivalent random cluster model. However, due to the equality of the weights in Eqs. 69 and 61, the statistical properties of both clusters are identical (Coniglio et al. 1989), and due to the relations between Eqs. 61 and 63, both coincide with those of the correlated bond percolation whose weight is given by Eq. 63. More precisely, any percolation quantity g(C) which depends only on the bond configuration has the same average:

$$ {\left\langle g(C)\right\rangle}_{\mathrm{KF}}={\left\langle g(C)\right\rangle}_{\mathrm{CK}}={\left\langle g(C)\right\rangle}_W , $$

(73)

where 〈 … 〉_KF and 〈 … 〉_CK are the average over spin and bond configurations with weights given by Eqs. 61 and 69, respectively, and 〈 … 〉 _W is the average over bond configurations in the bond-correlated percolation with weights given by Eq. 63. In view of Eq. 73, it follows (Coniglio et al. 1989)

$$ \left|\left\langle {S}_i\right\rangle \right|={\left\langle {\gamma}_i^{\infty}\right\rangle}_{\mathrm{CK}} $$

(74)

and

$$ \left\langle {S}_i{S}_j\right\rangle ={\left\langle {\gamma}_{ij}\right\rangle}_{\mathrm{CK}} . $$

(75)

We end this section 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 Eq. 59.