The random-cluster model is defined as a model for phase transitions and other phenomena in lattice systems, or more generally in systems with a graph structure. The model is characterized by a (probability) measure on a graph and a real parameter κ. By specifying the value of κ to 1, 2, 3, 4, … is shown that the model covers the percolation model, the Ising model, the Ashkin-Teller-Potts model with 3, 4, … states per atom, respectively, and thereby, contains information on graph-colouring problems; in the limit κ ↓ 0 it describes linear resistance networks. It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
%0 Journal Article
%1 Fortuin1972Randomcluster
%A Fortuin, C. M.
%A Kasteleyn, P. W.
%D 1972
%J Physica
%K random-cluster-model percolation critical-phenomena lattice-models
%N 4
%P 536--564
%R 10.1016/0031-8914(72)90045-6
%T On the random-cluster model
%U http://dx.doi.org/10.1016/0031-8914(72)90045-6
%V 57
%X The random-cluster model is defined as a model for phase transitions and other phenomena in lattice systems, or more generally in systems with a graph structure. The model is characterized by a (probability) measure on a graph and a real parameter κ. By specifying the value of κ to 1, 2, 3, 4, … is shown that the model covers the percolation model, the Ising model, the Ashkin-Teller-Potts model with 3, 4, … states per atom, respectively, and thereby, contains information on graph-colouring problems; in the limit κ ↓ 0 it describes linear resistance networks. It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
@article{Fortuin1972Randomcluster,
abstract = {{The random-cluster model is defined as a model for phase transitions and other phenomena in lattice systems, or more generally in systems with a graph structure. The model is characterized by a (probability) measure on a graph and a real parameter \^{I}º. By specifying the value of \^{I}º to 1, 2, 3, 4, … is shown that the model covers the percolation model, the Ising model, the Ashkin-Teller-Potts model with 3, 4, … states per atom, respectively, and thereby, contains information on graph-colouring problems; in the limit \^{I}º ↓ 0 it describes linear resistance networks. It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Fortuin, C. M. and Kasteleyn, P. W.},
biburl = {https://www.bibsonomy.org/bibtex/227127ddca323724c8e45f8d8b0d277b4/nonancourt},
citeulike-article-id = {1918577},
citeulike-linkout-0 = {http://dx.doi.org/10.1016/0031-8914(72)90045-6},
citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6X42-46BVYM3-6N/2/7e5582678de82b52f0eb71b8c1689471},
day = 15,
doi = {10.1016/0031-8914(72)90045-6},
interhash = {b537118d31b057718a92a4956ccab9f5},
intrahash = {27127ddca323724c8e45f8d8b0d277b4},
issn = {00318914},
journal = {Physica},
keywords = {random-cluster-model percolation critical-phenomena lattice-models},
month = feb,
number = 4,
pages = {536--564},
posted-at = {2011-05-23 18:18:59},
priority = {2},
timestamp = {2019-08-01T15:36:09.000+0200},
title = {{On the random-cluster model}},
url = {http://dx.doi.org/10.1016/0031-8914(72)90045-6},
volume = 57,
year = 1972
}