Volume 41, Issue 3, March 2000
 REVIEW ARTICLES


Rigorous probabilistic analysis of equilibrium crystal shapes
View Description Hide DescriptionThe rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

Phase transitions on nonamenable graphs
View Description Hide DescriptionWe 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.

The Potts model and the Tutte polynomial
View Description Hide DescriptionThis is an invited survey on the relation between the partition function of the Potts model and the Tutte polynomial. On the assumption that the Potts model is more familiar we have concentrated on the latter and its interpretations. In particular we highlight the connections with Abelian sandpiles, counting problems on random graphs, error correcting codes, and the Ehrhart polynomial of a zonotope. Where possible we use the mean field and square lattice as illustrations. We also discuss in some detail the complexity issues involved.
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 EQUILIBRIUM CONSIDERATIONS

 Spin Systems and Other Statistical Mechanical Systems

Percolation and number of phases in the twodimensional Ising model
View Description Hide DescriptionWe reconsider the percolation approach of Russo, Aizenman, and Higuchi for showing that there exist only two phases in the Ising model on the square lattice. We give a fairly short alternative proof which is only based on stochastic monotonicity and avoids the use of symmetry inequalities originally needed for some background results. Our proof extends to the Ising model on other planar lattices such as the triangular and honeycomb lattice. We can also treat the Ising antiferromagnet in a homogeneous field and the hardcore lattice gas model on

Gibbs states of graphical representations of the Potts model with external fields
View Description Hide DescriptionWe consider the ferromagneticqstate Potts model, with each of the q spin values coupled to an external field. We also introduce a generalized random cluster model, which includes both the Potts model in arbitrary homogeneous external fields and the noninteger q random cluster model as special cases. We establish the FKG property, the finite energy condition, uniqueness of the infinite cluster, and Gibbsianness of limit states for this generalized model. Furthermore, we develop the theory of Gibbs states for the Edwards–Sokal representation of the Potts model in a field, and relate the phase structure in this representation to those in the spin and random cluster representations. Finally, we characterize the possible color(s) of the infinite cluster(s) and show that the correspondence between Edwards–Sokal Gibbs states and their random cluster marginals is bijective, once the color of the infinite cluster is fixed.

Absence of a wetting transition for a pinned harmonic crystal in dimensions three and larger
View Description Hide DescriptionWe consider a free lattice field (a harmonic crystal) with a hard wall condition and a weak pinning to the wall. We prove that in a weak sense the pinning always dominates the entropic repulsion of the hard wall condition when the dimension is a least three. This contrasts with the situation in dimension one, where there is a socalled wetting transition, as has been observed by Michael Fisher. The existence of a wetting transition in the delicate twodimensional case was recently proved by Caputo and Velenik.

Large deviations for ideal quantum systems
View Description Hide DescriptionWe consider a general ddimensional quantum system of noninteracting particles in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain Λ of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of Λ, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.
 Graphical and Probabilistic Models

The scaling limit of the incipient infinite cluster in highdimensional percolation. II. Integrated superBrownian excursion
View Description Hide DescriptionFor independent nearestneighbor bond percolation on with we prove that the incipient infinite cluster’s twopoint function and threepoint function converge to those of integrated superBrownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the twopoint function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of plus an error term of order with This is a strong version of the statement that the critical exponent δ is given by

On the effect of adding εBernoulli percolation to everywhere percolating subgraphs of
View Description Hide DescriptionWe show that adding εBernoulli percolation to an everywhere percolating subgraph of results in a graph which has large scale geometry similar to that of supercritical Bernoullipercolation, in various specific senses. We conjecture similar behavior in higher dimensions.

Some remarks on ABpercolation in high dimensions
View Description Hide DescriptionIn this paper we consider the ABpercolation model on and Let be the critical probability for ABpercolation on We show that If the probability of a site to be in state A is for some fixed then the probability that ABpercolation occurs converges as to the unique strictly positive solution of the equation We also find the limit for the analogous quantities for oriented ABpercolation on In particular, We further obtain a small extension to the two parameter problem in which even vertices of have probability of being in state A and odd vertices have probability of being in state B (but without relation between and The principal tools in the proofs are a method of Penrose (1993) for asymptotics of percolation on graphs with vertices of high degree and the second moment method.

Anisotropic selfavoiding walks
View Description Hide DescriptionWe consider a model of selfavoiding walks on the lattice with different weights for steps in each of the lattice directions. We find that the directiondependent mass for the twopoint function of this model has three phases: mass positive in all directions; mass identically −∞; and masses of different signs in different directions. The final possibility can only occur if the weights are asymmetric, i.e., in at least one coordinate the weight in the positive direction differs from the weight in the negative direction. The boundaries of these phases are determined exactly. We also prove that if the weights are asymmetric then a typical Nstep selfavoiding walk has order N distance between its endpoints.

Longrange properties of spanning trees
View Description Hide DescriptionWe compute some largescale properties of the uniform spanning tree process on bounded regions in In particular, we compute the distribution of the meeting point of the branches of the tree issued from three boundary points. We also compute the crossing probabilities of branches of the tree on rectangular and annular regions, as well as the winding number of the branches of the tree.

Geometric variational problems of statistical mechanics and of combinatorics
View Description Hide DescriptionWe present the geometric solutions of the various extremal problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.

Towards a theory of negative dependence
View Description Hide DescriptionThe FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.
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 DYNAMICAL AND NONEQUILIBRIUM CONSIDERATIONS

 Spin Systems and Other Statistical Mechanical Systems

On the spectral gap of Kawasaki dynamics under a mixing condition revisited
View Description Hide DescriptionWe consider a conservative stochastic spin exchange dynamics which is reversible with respect to the canonical Gibbs measure of a lattice gas model. We assume that the corresponding grand canonical measure satisfies a suitable strong mixing condition. We give an alternative and quite natural, from the physical point of view, proof of the famous Lu–Yau result which states that the relaxation time in a box of side L scales like We then show how to use such an estimate to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and for for

Metastability and nucleation for conservative dynamics
View Description Hide DescriptionIn this paper we study metastability and nucleation for a local version of the twodimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let be the inverse temperature and let be two finite boxes. Particles perform independent random walks on and inside feel exclusion as well as a binding energy with particles at neighboring sites, i.e., inside the dynamics follows a Metropolis algorithm with an attractive lattice gas Hamiltonian. The initial configuration is chosen such that is empty, while a total of particles is distributed randomly over with no exclusion. That is to say, initially the system is in equilibrium with particle density ρ conditioned on being empty. For large β, the system in equilibrium has fully occupied because of the binding energy. We consider the case where for some and investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet and the time of its creation in the limit as for fixed Λ and In addition, we obtain some information on the typical trajectory of the system prior to the creation of the critical droplet. The choice corresponds to the situation where the critical droplet has side length i.e., the system is metastable. The side length of must be much larger than and independent of β, but is otherwise arbitrary. Because particles are conserved under Kawasaki dynamics, the analysis of metastability and nucleation is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas in can be treated as a reservoir that creates particles with rate ρ at sites on the interior boundary of and annihilates particles with rate 1 at sites on the exterior boundary of Once this approximation has been achieved, the problem reduces to understanding the local metastable behavior inside in the presence of a nonconservative boundary. The dynamics inside is still conservative and this difficulty has to be handled via local geometric arguments. Here it turns out that the Kawasaki dynamics has its own peculiarities. For instance, rectangular droplets tend to become square through a movement of particles along the border of the droplet. This is different from the behavior under the Glauber dynamics, where subcritical rectangular droplets are attracted by the maximal square contained in the interior, while supercritical rectangular droplets tend to grow uniformly in all directions (at least for not too long a time) without being attracted by a square.

Mixing properties of the Swendsen–Wang process on the complete graph and narrow grids
View Description Hide DescriptionWe consider the mixing properties of the Swendsen–Wang process for the twostate Potts model or Ising model, on the complete n vertex graph and for the Qstate model on an grid where a is bounded as

On the definition of entropy production, via examples
View Description Hide DescriptionWe present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which “the density” is determined with respect to the timereversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem and it is illustrated via a number of examples.
 Graphical and Combinatorial Models

Improved bounds for sampling colorings
View Description Hide DescriptionWe consider the problem of sampling uniformly at random from the set of proper kcolorings of a graph with maximum degree Δ. Our main result is the design of a simple Markov chain that converges in time to the desired distribution when