Results 1  10
of
77
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... 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 ..."
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Cited by 406 (13 self)
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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...
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In Survey in Combinatorics, 2005, volume 327 of London Mathematical Society Lecture Notes
, 2005
"... and matroids ..."
GroupInvariant Percolation on Graphs
 Geom. Funct. Anal
, 1999
"... . 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 Ginvariant percolation processes on X , such as the number of infinite components, the expected degree, and the topology of the components. O ..."
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Cited by 84 (30 self)
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. 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 Ginvariant 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 masstransport technique that has been occasionally used elsewhere and is developed further here. Perhaps surprisingly, these investigations of groupinvariant 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 Ginvariant 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...
Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes
, 1996
"... The areainteraction process and the continuum randomcluster 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 simpl ..."
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Cited by 71 (6 self)
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The areainteraction process and the continuum randomcluster 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 twocomponent Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a SwendsenWang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.
Phase Transitions on Nonamenable Graphs
, 2000
"... 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. ..."
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Cited by 49 (8 self)
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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.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 47 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
The RandomCluster Model
, 2006
"... Abstract. The class of randomcluster 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, an ..."
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Cited by 43 (18 self)
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Abstract. The class of randomcluster 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 randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional 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 clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
Perfect simulation for interacting point processes, loss networks and Ising models
, 1999
"... 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 birthanddeath processes. Examples include area and perimeterinteracting point processes (with stochastic grains), invariant meas ..."
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Cited by 41 (12 self)
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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 birthanddeath processes. Examples include area and perimeterinteracting 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 infinitevolume measure. The algorithm is based on a twostep procedure: (i) a perfectsimulation scheme for (spacetime) marked Poisson processes (free birthanddeath 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 birthrate under the targe...
Exact Sampling From AntiMonotone Systems
 Statistica Neerlandica
, 1998
"... 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 ProppWilson algorithm requires this distribution to have a certain structure called monotonicity. ..."
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Cited by 39 (1 self)
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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 ProppWilson 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 antimonotonicity. As illustrating examples, simulations of the hardcore model and the randomcluster model are presented.
Chromatic roots are dense in the whole complex plane
 In preparation
, 2000
"... 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 pol ..."
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Cited by 37 (14 self)
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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 Pottsmodel partition functions) ZG(q,v) outside the disc q + v  < v. An immediate corollary is that the chromatic roots of notnecessarilyplanar 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,