Random cluster model

Alexkyoung: /* See Also */

---

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 theory|percolation]] that is used to study [[Randomness|random]] [[Combinatorics|combinatorial]] structures, [[Electrical network|electrical networks]], etc.<ref name=":0">Liquid error: wrong number of arguments (1 for 2)</ref><ref name=":1"></ref>

== Definition ==
Let G be a [[Graph (mathematics)|graph]]. Suppose an edge <math>e \in E(G)</math> is open with probability p, wherein we say <math>\omega(e) = 1</math>, and is otherwise closed <math>\omega(e) = 0</math>. The probability of a given configuration is then

<math>\mu(\omega) = \prod_{e \in E(G)} p^{\omega(e)}(1-p)^{1-\omega(e)}</math>.

And this would give you the [[Erdős–Rényi model]] (independent edges, product [[Measure (mathematics)|measure]]). However, suppose you weight these in the following way. Let <math>C(\omega)</math> be the number or open clusters of the configuration (the number of [[Connected component (graph theory)|connected components]] in the subgraph of all edges with <math>\omega(e) = 1</math>). Let q be a positive real. Then define the new weighted measure as

<math>\mu(\omega) = \frac{1}{Z} q^{C(\omega)}\prod_{e \in E(G)} p^{\omega(e)}(1-p)^{1-\omega(e)} </math>.

Here Z is the [[Partition function (quantum field theory)|partition function]] or sum over all configurations:

<math>Z = \sum_{\omega \in \Omega} \{q^{C(\omega)}\prod_{e \in E(G)} p^{\omega(e)}(1-p)^{1-\omega(e)} \} </math>.

This resulting model is known as the '''random cluster model''' or '''RC''' for short.
== Relation to other models ==
Quoting Grimmet:<blockquote>Note the difference between the cases q ≤ 1 and q ≥ 1: the former favours fewer clusters, whereas the latter favours many clusters. When q = 1, edges are open/closed independently of one another. This very special case has been studied in detail under the titles percolation and random graphs; see [25, 71, 90]. Perhaps the most important values of q are the integers, since the random-cluster model with q ∈ {2, 3, . . . } corresponds [roughly] to the Potts model with q local states. The bulk of this review is devoted to the theory of random-cluster measures when q ≥ 1. The case q < 1 seems to be harder mathematically and less important physically. There is some interest in the limit as q ↓ 0.<ref name=":0" /></blockquote>
Regarding the Potts model:<blockquote>The family of random-cluster measures (that is, probability measures which govern random-cluster models) is not an extension of the Potts measures. The relationship is more sophisticated, and is such that correlations for Potts models correspond to connections in random-cluster models. Thus the [[Correlation function (statistical mechanics)|correlation structure]] of a Potts model may be studied via the [[stochastic geometry]] of a corresponding random-cluster model.<ref name=":0" /></blockquote>

== History ==
RC models were introduced in 1969 by Fortuin and Kasteleyn, mainly to solve combinatorial problems. Interest in and applications towards [[statistical physics]] restarted post 1987. Swendsen and Wang used RCM to propose an algorithm for the time-evolution of Potts models; Aizenman, et al used it to show discontinuity in long-range 1D Ising/Potts models.<ref name=":1" />

== See Also ==

* [[Tutte polynomial]]
* [[Ising model]]
* [[Random graph]]

*http://bit.ly/2vEl3yy

== Citations ==
<references />

[[Category:Physics]]
[[Category:Probability and statistics]]
[[Category:Graph theory]]
[[Category:Random graphs]]
[[Category:Percolation theory]]
[[Category:Statistical mechanics]]

from Wikipedia - New pages [en] http://bit.ly/2DM4Lb2
via IFTTT