The random-cluster model. (English) Zbl 1122.60087

Grundlehren der Mathematischen Wissenschaften 333. Berlin: Springer (ISBN 3-540-32890-4/hbk; 978-3-642-06943-7/pbk; 978-3-540-32891-9/ebook). xiii, 377 p. (2006).
The so-called Fortuin-Kasteleyn representation of Ising and Potts spin systems gives rise to a unifying class of random geometric models that can be most naturally seen as generalizations of percolation. Thus, very naturally, the author [“Percolation”, Berlin: Springer (1999; Zbl 0926.60004)] presents here the comprehensive treatise on these random cluster models. The link to percolation created by the random cluster models has had an extraordinary impact on the development of the theory of spin systems. Numerous results have been achieved solely due to the fact that the FK representation has made the application of percolation techniques and, in particular, monotonicity arguments possible. As only one example of many, one may think of the extension of the Wulff-construction in the 2D-Ising model up to the critical point.
The book gives a very comprehensive treaty of this model and the techniques, ideas and applications surrounding it. The short first chapter presents the basic definitions of the model. The following two chapters are devoted to the fundamental comparison methods: stochastic ordering of measures, positive and negative association, and differential inequalities.
Chapter 4 presents the construction of the random cluster measures in infinite volume. This is done, as in the case of spin systems, through weak limits of finite volume measures, and via the DLR constructions through the specification of conditional probabilities. Chapter 5 then discusses issues of uniqueness of infinite volume random cluster measures, and that of the phase transition in the associated spin systems. Note that unlike the Gibbs measure of the spin systems, the random cluster measure is (proven, resp., conjectured) to be unique except at a the critical point. The phase transition of the spin measure is related by a percolation transition in the random cluster measure. The analysis of the supercritical phase in the random cluster model is one of the strong points of the method, and in Section 5.7 the application of this analysis in the proof of the Wulff-construction up to the critical point is explained.
The two-dimensional case is in many respects special and the interest in it is highlighted by the recent results on conformal invariance and the stochastic Löwner evolution. Chapter 6 is devoted to this case. The use of duality methods in higher dimension is the main topic of Chapter 7, together with applications such as the proof of first-order phase transitions in the Potts model with \(q\) large and the existence of non-translation invariant (Dobrushin) states in three and more dimensions.
Chapter 8 deals with the Glauber dynamics of random cluster models. This is an important topic, as the random cluster representation is a crucial ingredient in the Swendson-Wang algorithm, a method for Monte-Carlo simulation of Ising and Potts models that in many cases can (partially) avoid the problem of critical slowing down in simulations near the critical temperature.
Chapter 9 discusses ways to extend the random current representation of the Ising model which was used in proving triviality in more than 4 dimensions to \(q\)-state Potts models. Chapter 10 discusses the random cluster models on graphs other than the lattice \(\mathbb Z^d\), in particular the complete graph and non-amenable graphs. Chapter 11 describes five spin systems to which the FK representation can be applied, including the disordered Potts model and the Ising spin glass model. A very interesting appendix describes the history of the random cluster model. This contains two letters by Piet Kasteleyn to the author.
This book constitutes an authoritative treatise of the random cluster model and its applications. It will be indispensable for anyone working on this and related subjects and can provide the basis for a graduate course or seminar.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60-02 Research exposition (monographs, survey articles) pertaining to probability theory


Zbl 0926.60004
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