Random cluster model

In physics, probability theory, graph theory, etc. the random cluster model is a random graph that generalizes and unifies the Ising model, Potts model, and percolation model. It is used to study random combinatorial structures, electrical networks, etc. It is also referred to as the RC model or sometimes the FK representation after its founders Kees Fortuin and Piet Kasteleyn.

Definition

Let $G=(V,E)$ be a graph, and $\omega :E\to \{0,1\}$ be a bond configuration on the graph that maps each edge to a value of either 0 or 1. We say that a bond is closed on edge $e\in E$ if $\omega (e)=0$, and open if $\omega (e)=1$. If we let $A(\omega )=\{e\in E:\omega (e)=1\}$ be the set of open bonds, then an open cluster is any connected component in $A(\omega )$. Note that an open cluster can be a single vertex (if that vertex is not incident to any open bonds).

Suppose an edge is open independently with probability $p$ and closed otherwise, then this is just the standard Bernoulli percolation process. The probability measure of a configuration $\omega$ is given as

$\mu (\omega )=\prod _{e\in E}p^{\omega (e)}(1-p)^{1-\omega (e)}.$

The RC model is a generalization of percolation, where each cluster is weighted by a factor of $q$. Given a configuration $\omega$, we let $C(\omega )$ be the number of open clusters, or alternatively the number of connected components formed by the open bonds. Then for any $q>0$, the probability measure of a configuration $\omega$ is given as

$\mu (\omega )={\frac {1}{Z}}q^{C(\omega )}\prod _{e\in E}p^{\omega (e)}(1-p)^{1-\omega (e)}.$

Z is the partition function, or the sum over the unnormalized weights of all configurations,

$Z=\sum _{\omega \in \Omega }\left\{q^{C(\omega )}\prod _{e\in E(G)}p^{\omega (e)}(1-p)^{1-\omega (e)}\right\}.$

Note that we can recover the percolation model by setting $q=1$, in which case $Z=1$.

Relation to other models

The random cluster model is equivalent to the q-state Potts model for $q\in \mathbb {Z} ^{+}$ (with the $q=1$ case being Bernoulli percolation and the $q=2$ case being the Ising model). In general, $q$ can be any positive real number, with $q\leq 1$ favoring the formation of fewer clusters and $q\geq 1$ favoring the formation of more clusters (in comparison to percolation). The $q\to 0$ limit describes linear resistance networks. The partition function of the RC model is a specialization of the Tutte polynomial.

History and applications

RC models were introduced in 1969 by Fortuin and Kasteleyn, mainly to solve combinatorial problems. After their founders, it is sometimes referred to as FK models. In 1971 they used it to obtain the FKG inequality. Post 1987, interest in the model and applications in statistical physics reignited. It became the inspiration for the Swendsen–Wang algorithm describing the time-evolution of Potts models. Michael Aizenman, et al. used it to study the phase boundaries in 1D Ising and Potts models.

See also

• Tutte polynomial
• Ising model
• Random graph
• Swendsen–Wang algorithm
• FKG inequality

External links

• Random-Cluster Model – Wolfram MathWorld