The Random-Cluster Model

By Geoffrey Grimmett. 2002

792 citations recorded by the source corpus.

The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two dimensions, with 1 ≤ q ≤ 4, converges when re-scaled to a stochastic Lowner evolution (SLE). Overall the emphasis is upon the random-cluster model for its own sake, rather than upon its applications to Ising and Potts systems.

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Citations in the corpus

Szegö's theorem and its probabilistic descendants (2011, background): “. Long-range dependence In various physical models, the property of long-range dependence (LRD) is important, particularly in connection with phase transitions (see e.g. [Si1], Ch. II, [Gri1], Ch. 9, [Gri2], Ch. 5), to which we return below. This is a spatial property, but applies also in time rathe”
Recent advances in percolation theory and its applications (2015, background): “4, they are closely related to each other [27, 28].”
Marginal triviality of the scaling limits of critical 4D Ising and 𝜑44 models (2019, unknown): “[28]).”
Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2013, methodology): “12) can be established for the nearest neighbor model on the square, hexagonal and triangular lattices through the Fortuin-Kasteleyn random cluster representation (see [Gri06] Theorem 6.”
Strong Data Processing Inequalities and $\Phi $ -Sobolev Inequalities for Discrete Channels (2014, methodology): “ xnvg: (5.13) Swendsen-Wang dynamics. This construction is based on a coupling of the q-Potts model and the so-called random-cluster (or Fortuin-Kasteleyn) model on G. The latter is dened as follows [65]. Let Y = 2E = fA : A Egand x a parameter p2(0;1). Then the random-cluster model is described by ”
Recent progress on the Random Conductance Model (2011, methodology): “A natural consequence of Theorem 5.4 is the result that was first proved by Grimmett, Kesten and Zhang [75] by rather different methods (see also Problem 1.16): Corollary 5.7.”
Sharp phase transition for the random-cluster and Potts models via decision trees (2017, unknown): “Exponential decay for the subcritical random-cluster model Since random-cluster models were introduced by Fortuin and Kasteleyn in 1969 [FK72], they have become the archetypal example of dependent percolation models and as such have played an important role in the study of phase transitions.”
The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1 (2010, methodology): “Since random-cluster models were introduced by Fortuin and Kasteleyn in 1969 [10], they have become an important tool in the study of phase transitions.”
Invariant percolation and harmonic Dirichlet functions (2004, methodology): “Kasteleyn [FK72] in relation with Ising and Potts models as explained for instance in [HJL02a, Prop.”
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model (2007, background): “For a general introduction to the Ising and random cluster models consult the books [1], [7], [16].”
Conformally invariant scaling limits: an overview and a collection of problems (2006, unknown): “The Fortuin-Kasteleyn [28] (FK) model (a.k.a. the random cluster model) is a probability measure on the collection of all subsets of the set of edges E of a finite graph G = (V,E).”
Influence and sharp-threshold theorems for monotonic measures (2005, unknown): “It is well known (see [8, 14]) that p p + q(1 − p) ≤ φp,q(Xe = 1) ≤ p, e ∈ E.”
MATHEMATICAL STATISTICAL MECHANICS (2006, unknown): “These methods can be combined with graphical representations of Edwards-Sokal [27] (or Fortuin-Kasteleyn [31]) to establish rather complicated phase diagrams, e.”
Walking, weak first-order transitions, and complex CFTs (2018, background): “Let us now discuss the cluster definition of the Potts model, due to Fortuin and Kasteleyn [14].”
Concentration for self-bounding functions and an inequality of Talagrand (2006, unknown): “6, [7] for additional background and references.”