Olle Häggström

The random geometry of equilibrium phases

Publisher Summary This chapter discusses the random geometry of equilibrium phases. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc. will be useful for describing certain geometric features of equilibrium phases, allowing characterizations of phases in percolation terms. Examples are presented where the (thermal) phase transition goes hand in hand with a phase transition in an associated percolation process. Percolation techniques can be used to obtain specific information about the phase diagram of the system. For example, equilibrium correlation functions are sometimes dominated by connectivity functions in an associated percolation problem which is easier to investigate. Further, representations in terms of percolation models yield explicit relations between certain observables in equilibrium models and some corresponding percolation quantities.

Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes

The area-interaction process and the continuum random-cluster model are characterized 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.

Nearest neighbor and hard sphere models in continuum percolation

Consider a Poisson process X in Rd with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). 0 1996 John Wiley & Sons, Inc.

Phase transition in continuum Potts models

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with supperstable interaction.

Random-cluster measures and uniform spanning trees

Consider the random-cluster model on the integer lattice with parameters p and q. As p, q --> 0 in such a way that q/p --> 0, the random-cluster measures converge weakly to the uniform spanning tree measure of Pemantle (1991).

Exact sampling from anti‐monotone systems

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.

Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models

We give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontanoous symmetry breaking appears or disappears. The two systems are the hard-core model and the Widom–Rowlinson model, and both examples take place on a variation of the Cayley tree (Bethe lattice) devised by Schonmann and Tanaka. We obtain similar, though less constructive, examples of nonmonotonicity via certain local modifications of any graph, e.g., the square lattice, which is known to have a critical point for either model. En route we discuss the critical behavior of the Widom–Rowlinson model on the ordinary Cayley tree. Some results about monotonicity of the phase transition phenomenon relative to graph structure are also given.

The random-cluster model on a homogeneous tree

SummaryThe random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1≦q≦2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=pc(q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Zd.

Percolation transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness

Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasi-transitive, there is another critical value p u ∈ [p c , 1] such that the number of infinite clusters is a.s. ∞ for p ∈ (p c , p u ), and a.s. one for p > p u . We prove a simultaneous version of this result in the canonical coupling of the percolation processes for all p ∈ [0, 1]. Simultaneously for all p ∈ (p c , p u ), we also prove that each infinite cluster has uncountably many ends. For p > p c we prove that all infinite clusters are indistinguishable by robust properties. Under the additional assumption that G is unimodular, we prove that a.s. for all p 1 < p 2 in (p c , p u ), every infinite cluster at level p 2 contains infinitely many infinite clusters at level p 1. We also show that any Cartesian product G of d infinite connected graphs of bounded degree satisfies p u (G) ≤ p c (Z d ).

Uniqueness and Non-uniqueness in Percolation Theory

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on