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.[1][2][3] It is also referred to as the RC model or sometimes the FK representation after its founders Kees Fortuin and Piet Kasteleyn.[4]

Definition

Let G be a graph. Suppose an edge ${\displaystyle e\in E(G)}$ is open with probability p, wherein we say ${\displaystyle \omega (e)=1}$, and is otherwise closed ${\displaystyle \omega (e)=0}$. The probability of a given configuration is then

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

And this would give you the Erdős–Rényi model (independent edges, product measure). However, suppose you weight these in the following way. Let ${\displaystyle C(\omega )}$ be the number of open clusters of the configuration (the number of connected components in the subgraph of all open edges ${\displaystyle \omega (e)=1}$). Let q be a positive real. Then define the new weighted measure as

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

Here Z is the partition function or sum over all configurations:

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

This resulting model is known as the random cluster model or RCM for short.

Relation to other models

There are two cases: q ≤ 1 and q ≥ 1. The former favors fewer clusters, whereas the latter favors many clusters. When q = 1, edges are open and closed independently of one another, and the model reduces to percolation and random graphs.[2]

It is a generalization of the Tutte polynomial. The limit as q ↓ 0 describes linear resistance networks.[1]

It is a special case of the exponential random graph models.

History and applications

RC models were introduced in 1969 by Fortuin and Kasteleyn, mainly to solve combinatorial problems.[1][3][5] After their founders, it is sometimes referred to as FK models.[4] 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.[6] Michael Aizenman, et al. used it to study the phase boundaries in 1D Ising and Potts models.[7][3]

See also

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

References

1. Fortuin; Kasteleyn (1972). "On the random-cluster model: I. Introduction and relation to other models". Physica. 57 (4): 536. Bibcode:1972Phy....57..536F. doi:10.1016/0031-8914(72)90045-6.
2. Grimmett (2002). "Random cluster models". arXiv:math/0205237.
3. Grimmett. The random cluster model. http://www.statslab.cam.ac.uk/~grg/books/rcm1-1.pdf.CS1 maint: location (link)
4. NEWMAN, CHARLES M. "DISORDERED ISING SYSTEMS AND RANDOM CLUSTER REPRESENTATIONS" (PDF).
5. Kasteleyn, P. W.; Fortuin, C. M. (1969). "Phase Transitions in Lattice Systems with Random Local Properties". Physical Society of Japan Journal Supplement, Vol. 26. Proceedings of the International Conference on Statistical Mechanics Held 9–14 September 1968 in Koyto., P.11. 26: 11. Bibcode:1969PSJJS..26...11K.
6. Swendsen, Robert H.; Wang, Jian-Sheng (1987-01-12). "Nonuniversal critical dynamics in Monte Carlo simulations". Physical Review Letters. 58 (2): 86–88. Bibcode:1987PhRvL..58...86S. doi:10.1103/PhysRevLett.58.86. PMID 10034599.
7. Aizenman, M.; Chayes, J. T.; Chayes, L.; Newman, C. M. (April 1987). "The phase boundary in dilute and random Ising and Potts ferromagnets". Journal of Physics A: Mathematical and General. 20 (5): L313–L318. Bibcode:1987JPhA...20L.313A. doi:10.1088/0305-4470/20/5/010. ISSN 0305-4470.

External links

• Random-Cluster Model – Wolfram MathWorld