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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. 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 model, after its founders.^{[4]}
Contents
Definition
Let G be a 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)}(1p)^{1\omega(e)}.</math>
And this would give you the Erdős–Rényi model (independent edges, product 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 components in the subgraph of all open edges <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)}(1p)^{1\omega(e)}. </math>
Here Z is the partition function or sum over all configurations:
 <math>Z = \sum_{\omega \in \Omega} \left\{q^{C(\omega)}\prod_{e \in E(G)} p^{\omega(e)}(1p)^{1\omega(e)} \right\}. </math>
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]}^{[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 timeevolution 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
References
 ↑ ^{1.0} ^{1.1} ^{1.2} Fortuin; Kasteleyn. "On the randomcluster model: I. Introduction and relation to other models". doi:10.1016/00318914(72)900456. https://www.sciencedirect.com/science/article/pii/0031891472900456?via%3Dihub.
 ↑ ^{2.0} ^{2.1} Grimmett. "Random cluster models". https://arxiv.org/pdf/math/0205237.pdf.
 ↑ ^{3.0} ^{3.1} Grimmett. The random cluster model. http://www.statslab.cam.ac.uk/~grg/books/rcm11.pdf.
 ↑ ^{4.0} ^{4.1} NEWMAN, CHARLES M.. "DISORDERED ISING SYSTEMS AND RANDOM CLUSTER REPRESENTATIONS". https://www.math.nyu.edu/faculty/newman/disordered_ising_systems.pdf.
 ↑ 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. http://adsabs.harvard.edu/abs/1969PSJJS..26...11K.
 ↑ Swendsen, Robert H.; Wang, JianSheng (19870112). "Nonuniversal critical dynamics in Monte Carlo simulations". Physical Review Letters 58 (2): 86–88. doi:10.1103/PhysRevLett.58.86. https://link.aps.org/doi/10.1103/PhysRevLett.58.86.
 ↑ 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. doi:10.1088/03054470/20/5/010. ISSN 03054470. https://doi.org/10.1088%2F03054470%2F20%2F5%2F010.