For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

. Let G be a closed group of automorphisms of a graph X . We relate geometric properties of G and X , such as amenability and unimodularity, to properties of G-invariant percolation processes on X , such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new mass-transport technique that has been occasionally used elsewhere and is developed further here. Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group. We show that G is amenable iff for all ff ! 1, there is a G-invariant site percolation process ! on X with P[x 2 !] ? ff for all vertices x and with no infinite components. When G is not amenable, a threshold ff ! 1 app...

The area-interaction process and the continuum random-cluster model are characterised in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen-Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.

and matroids

Abstract. 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

A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yields samples which have exactly the desired distribution. The Propp-Wilson algorithm requires this distribution to have a certain structure called monotonicity. In this paper an idea of Kendall is applied to show how the algorithm can be extended to the case where monotonicity is replaced by anti-monotonicity. As illustrating examples, simulations of the hard-core model and the random-cluster model are presented.

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.

We present a perfect simulation algorithm for measures that are absolutely continuous with respect to some Poisson process and can be obtained as invariant measures of birth-and-death processes. Examples include area- and perimeter-interacting point processes (with stochastic grains), invariant measures of loss networks, and the Ising contour and random cluster models. The algorithm does not involve any coupling ---hence it is not tied up to monotonicity requirements--- and it directly provides samples of the infinite-volume measure. The algorithm is based on a two-step procedure: (i) a perfect-simulation scheme for (space-time) marked Poisson processes (free birth-and-death process, free loss networks), and (ii) a "cleaning" algorithm that trims out this process according to the interaction rules of the target process. The first step involves the perfect generation of "ancestors" of a given object, that is of predecessors 1 that may have an influence on the birth-rate under the targe...

. We show that independent percolation on any Cayley graph of a nonamenable group has no innite components at the critical parameter. This result was obtained in Benjamini, Lyons, Peres, and Schramm (1999) as a corollary of a general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasitransitive graphs with a unimodular automorphism group. The key tool is a \mass-transport" method, which is a technique of averaging in nonamenable settings. x1. Introduction. The main long-standing open question in percolation theory is to show that critical percolation in Z d has no innite components for all d 2. The work of Harris (1960) and Kesten (1980) established the two-dimensional case; Hara and Slade (1994) proved it for d 19. Recently, a study of percolation on other graphs, such as Cayley graphs, was initiated. The relevant denitions appear below. Here is our main theorem. Theorem 1.1. Let X be a Cayley...

to appear in Combinatorics, Probability and Computing I show that the zeros of the chromatic polynomials PG(q) for the generalized theta graphs Θ (s,p) are, taken together, dense in the whole complex plane with the possible exception of the disc |q − 1 | < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) ZG(q,v) outside the disc |q + v | < |v|. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha–Kahane–Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof. KEY WORDS: Graph, chromatic polynomial, dichromatic polynomial, Whitney rank function, Tutte polynomial, Potts model, Fortuin–Kasteleyn representation,