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 studythe behavior of the cluster distribution. In most cases the theory predicts a geometrical transition at the percolation threshold, characterized inthe percolative phase by the presence of a spanning cluster, which becomes infinite in the thermodynamic limit. Standard percolation usually dealswith the problem when the constitutive elements of the clusters are randomly distributed. However correlations cannot always be neglected. In this casecorrelated percolation is the appropriate theory to study such systems. The origin of correlated percolation could be dated back to 1937 whenMayer [76] proposed a theory to describe the condensation from a gas to a liquid interms of mathematical clusters (for a review of cluster theory in simple fluids see [88]). Thelocation for the divergence of the size of these clusters was...

Appendix: Random Cluster Model and Ising Droplets

In 1969 Kasteleyn and Fortuin (KF) [65] 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 [30] 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

$$ \mathcal{H}(\{S_{i}\})=-\sum_{\langle i,j \rangle}J(S_{i}S_{j}-1)\: , $$

(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

$$ \mathcal{H}^{\prime} (\{S_{i}\})=-\sum_{\langle i,j \rangle}J_{ij}^{\prime}(S_{i}S_{j}-1)\: , $$

(55)

where

$$ J_{ij}^{\prime}= \begin{cases} J^{\prime} & \text{with probability}\: p \\ 0 & \text{with probability}\: (1-p)\:. \end{cases} $$

(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

$$ \begin{aligned} \text{e}^{-\beta \mathcal{H}(\{S_i\})}& \equiv \prod_{\langle i,j \rangle} \text{e}^{\beta J(S_iS_j-1)}\\ &= \prod_{\langle i,j \rangle}\left(p\text{e}^{\beta J^{\prime}(S_iS_j-1)}+(1-p)\right) \: , \end{aligned} $$

(57)

where ${ \beta=1/k_\text{B}T }$, ${ k_\text{B} }$ is the Boltzmann constant and T is the temperature. In order to satisfy (57) we must have

$$ \text{e}^{\beta J(S_iS_j-1)}=p\text{e}^{\beta J^{\prime}(S_iS_j-1)}+(1-p)\:. $$

(58)

We take now the limit ${ J^{\prime}\mapsto\infty }$. In such a case ${ \text{e}^{\beta J^{\prime}(S_iS_j-1)} }$ equals the Kronecker delta ${ \delta_{S_iS_j} }$ and from (58) p is given by

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

(59)

From (57), by performing the products we can write

$$ \text{e}^{-\beta \mathcal{H}(\{S_i\})} = \sum_C W_{{\text{KF}}}(\{S_i\},C)\: , $$

(60)

where

$$ W_{{\text{KF}}}(\{S_i\},C)=p^{|C|}(1-p)^{|A|}\prod_{\langle i,j \rangle\in C}\delta_{S_iS_j}\:. $$

(61)

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

${ W_{{\text{KF}}}(\{S_i\},C) }$ is the statistical weight associated a) with a spin configuration ${ \{S_i\} }$ and b) with a set of interactions in the diluted model where ${ |C| }$ 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 (60) over all the spin configurations. Since each cluster in the configuration C gives a contribution of 2, we obtain:

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

(62)

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

In conclusion, in the KF formalism the partition function (62) is equivalent to the partition function of a correlated bond percolation model [62,65] where the weight of each bond configuration C is given by

$$ W(C) = \sum_{\{ S_i\}}W_{{\text{KF}}}(\{S_i\},C) = p^{|C|}(1-p)^{|A|} 2^{N_C} $$

(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) 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 (61). Moreover using (61) and (60) Kasteleyn and Fortuin have proved that [65]

$$ | \langle S_i \rangle | = \langle \gamma_i^{\infty}\rangle_{W} $$

(64)

and

$$ \langle S_iS_j\rangle=\langle\gamma_{ij}\rangle_{W} \: , $$

(65)

where ${ \langle \dots \rangle }$ is the Boltzmann average and ${ \langle \dots\rangle _{W} }$ is the average over bond configurations in the bond correlated percolation with weights given by (63). Here ${ \gamma_i^{\infty}(C) }$ is equal to 1 if the spin at site i belongs to the spanning cluster, 0 otherwise; ${ \gamma_{ij}(C) }$ 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 [30], given a configuration of spins, one introduces at random connecting bonds between nn parallel spins with probability ${ p_{\text{b}} }$, 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 [28,40]

$$ W_{{\text{CK}}}(\{S_i\},C) = p_{\text{b}}^{\mid C\mid}(1-p_{\text{b}})^{\mid B\mid}\text{e}^{-\beta\mathcal{H} (\{S_i\})}\: , $$

(66)

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 the Newton binomial rule, we have the following sum rule

$$ \sum_{C} p_{\text{b}}^{\mid C\mid}(1-p_{\text{b}})^{\mid B\mid}=1\:. $$

(67)

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

$$ Z=\sum_{\{S_i\}}\sum_C W_{{\text{CK}}}(\{S_i\},C) = \sum_{\{S_i\}} \text{e}^{-\beta\mathcal{H}(\{S_i\})}\:. $$

(68)

The partition function of course does not depend on the value of ${ p_{\text{b}} }$ which controls the bond density. By tuning ${ p_{\text{b}} }$ instead it is possible to tune the size of the clusters. For example by taking ${ p_{\text{b}}=1 }$ the clusters would coincide with nearest‐neighbor parallel spins, while for ${ p_{\text{b}}=0 }$ the clusters are reduced to single spins. By choosing the droplet bond probability ${ p_{\text{b}}=1-\text{e}^{-2\beta J}\equiv p }$ and observing that ${ \text{e}^{-\beta\mathcal{H}(\{S_i\})}=\text{e}^{-2\beta J|D|} }$, where ${ |D| }$ is the number of antiparallel pairs of spins, the weight (66) simplifies and becomes:

$$ W_{{\text{CK}}}(\{S_i\},C) = p^{\mid C\mid}(1-p)^{\mid A\mid}\: , $$

(69)

where ${ |A|=|B|+|D|=|E|-|C| }$.

From (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

$$ W(C)=\sum_{\{S_{i}\}}W_{{\text{CK}}}(\{S_{i}\},C)\prod_{\langle i,j \rangle \in C}\delta_{S_{i}S_{j}}\: , $$

(70)

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

$$ \begin{aligned} W(C) &=\sum_{\{S_{i}\}}p^{\mid C\mid} (1-p)^{\mid A\mid}\prod_{\langle i,j \rangle \in C}\delta_{S_{i}S_{j}}\\ &=p^{|C|}(1-p)^{|A|}2^{N_{C}}\:. \end{aligned} $$

(71)

Consequently in (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 (62).

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

(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 (69) and (61) the statistical properties of both clusters are identical [40] and due to the relations between (61) and (63) both coincide with those of the correlated bond percolation whose weight is given by (63). More precisely, any percolation quantity g(C) which depends only on the bond configuration has the same average

$$ \langle g(C)\rangle _{{\text{KF}}}=\langle g(C)\rangle _{{\text{CK}}}=\langle g(C)\rangle _{W}\: , $$

(73)

where ${ \langle \dots\rangle _{{\text{KF}}} }$, ${ \langle \dots\rangle _{{\text{CK}}} }$ are the average over spin and bond configurations with weights given by (61) and (69) respectively and ${ \langle \dots\rangle _{W} }$ is the average over bond configurations in the bond correlated percolation with weights given by (63). In view of (73) it follows [40]

$$ | \langle S_i \rangle | = \langle \gamma_i^{\infty}\rangle_{{\text{CK}}} $$

(74)

and

$$ \langle S_iS_j\rangle=\langle\gamma_{ij}\rangle_{{\text{CK}}} \:. $$

(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 (59).