Editing Potts model (section)

=== Relation with the random cluster model ===

The Potts model has a close relation to the Fortuin-[[Pieter Kasteleyn|Kasteleyn]] [[random cluster model]], another model in [[statistical mechanics]]. Understanding this relationship has helped develop efficient [[Markov chain Monte Carlo]] methods for numerical exploration of the model at small <math>q</math>, and led to the rigorous proof of the critical temperature of the model.<ref name=":0" />

At the level of the partition function <math>Z_p = \sum_{\{s_i\}} e^{-H_p}</math>, the relation amounts to transforming the sum over spin configurations <math>\{s_i\}</math> into a sum over edge configurations <math>\omega=\Big\{(i,j)\Big|s_i=s_j\Big\}</math> i.e. sets of nearest neighbor pairs of the same color. The transformation is done using the identity<ref>{{cite book |last=Sokal |first=Alan D. |title=Surveys in Combinatorics 2005 |chapter=The multivariate Tutte polynomial (alias Potts model) for graphs and matroids |year=2005 |arxiv=math/0503607 |pages=173–226 |doi=10.1017/CBO9780511734885.009|isbn=9780521615235 |s2cid=17904893 }}</ref> 
: <math>
e^{J_p\delta(s_i,s_j)} = 1 + v \delta(s_i,s_j) \qquad \text{ with } \qquad v = e^{J_p}-1 \ . 
</math> 
This leads to rewriting the partition function as 
: <math>
Z_p = \sum_\omega v^{\#\text{edges}(\omega)} q^{\#\text{clusters}(\omega)}
</math>
where the ''' FK clusters''' are the connected components of the union of closed segments <math>\cup_{(i,j)\in\omega}[i,j]</math>. This is proportional to the partition function of the random cluster model with the open edge probability <math>p=\frac{v}{1+v}=1-e^{-J_p}</math>. An advantage of the random cluster formulation is that <math>q</math> can be an arbitrary complex number, rather than a natural integer.

Alternatively, instead of FK clusters, the model can be formulated in terms of '''spin clusters''', using the identity 
: <math>
e^{J_p\delta(s_i,s_j)} = (1 - \delta(s_i,s_j)) + e^{J_p} \delta(s_i,s_j)\ . 
</math>
A spin cluster is the union of neighbouring FK clusters with the same color: two neighbouring spin clusters have different colors, while two neighbouring FK clusters are colored independently.