Xavier Bressaud (Marseille) : Relaxation speed for a chain with complete connections

A chain with complete connections is a process on a discrete space described by means of its transition probabilities. It generalises Markov chains in the sense that it keeps memory of its whole past. It has a unique invariant measure if the dependence on the past decays fast enough (condition of summable decay). We estimate the relaxation speed of such chains with complete connections with summable decays by explicitly constructing a coupling between two chains with different histories. We show how to use this result to obtain mixing results for dynamical systems.

Antonio Galves (São Paulo) : Exponential approximations for hitting times of mixing sources

In this talk we shall discuss recent results on the law of the first visit time of a long string of symbols by a mixing stochastic source. After a brief presentation of former results, starting with the classical theorems by Doeblin and Harris, we shall compare recent sharp results on the exponential approximation for hittings times of stochastic processes, including Gibbsian sources and mixing dynamical systems. As an application we present a result on repetition times motivated by the Lempel-Ziv and related compression algorithms. This talk is mainly based on my recent joint work with Abadi, Collet and Schmitt. My original motivation to study the question was the so called pathwise approximation to metastability, introduced more than fifteen years ago by Cassandro, Olivieri, Vares and the speaker. Useless to say that our interest on metastability had its source in Joel Lebowitz's work on the field. This talk is dedicated to Joel on his 70^th birthday.

Mark Jerrum (Edinburgh) : Case studies in torpid mixing

A Markov chain is rapidly mixing if, loosely, it converges to equilibrium in a number of steps that is polynomial in the size of its description. Torpid mixing is the opposite: it describes a situation in which convergence requires an exponential, or at least super-polynomial number of steps. I'll cover some cases of provable torpidity, including the ``hard core gas'' or independent sets model (with Glauber dynamics and its relatives), and the Potts model (with Swendsen-Wang dynamics).

Roman Kotecký (Prague) : A general theory of Lee-Yang zeros in models with first-order phase transition

General theory allowing to derive positions and the density of Lee-Yang zeros for models with first-order phase transitions is introduced. Particular examples of Blume-Capel and Potts models are discussed, featuring noncircular loci of zeros with bifurcations.

Fabio Martinelli (Roma) : Conservative and non conservative stochastic spin dynamics for random systems in the griffiths phase

We will review the relaxation to equilibrium for Glauber (non conservative) and Kawasaki (conservative) spin dynamics reversible with respect to the grand canonical and canonical Gibbs measure of a lattice spin system with random interactions in the presence of Griffiths singularities.

Enzo Olivieri (Roma) : Metastability and nucleation for conservative dynamics

We discuss metastability and nucleation for a lattice gas with Kawasaki dynamics. Due to the conservative nature of the dynamics this problem turns out to me much more difficult than the analogous one in the case of non-conservative Glauber dynamics. Introducing some simplifying assumptions, we describe the asymptotics of the escape time from metastability as well as the typical nucleation mechanism.

Jesus Salas (Zaragoza) : Dynamic critical behavior of cluster algorithms for 2D Ashkin-Teller and Potts models

We study the dynamic critical behavior of two algorithms: 1) the Swendsen-Wang cluster algorithm for the 2D Potts model; and 2) a Swendsen-Wang-type algorithm for the 2D Ashkin-Teller model. In particular, we discuss the sharpness of the Li-Sokal bound $\tau_{{\rm int},{\cal E}} \geq {\rm const.} \times C_H$. We find that this bound is almost but not quite sharp for all the models considered in this study. The ratio $\tau_{{\rm int},{\cal E}}/C_H$ seems to diverge either with a small power ($0.05 \ltapprox p \ltapprox 0.012$) or as a logarithm.

Senya Shlosman (Marseille) : The Validity of the Weakly Gibbs Property at Low Temperatures

Alan Sokal (New-York) : A Personal List of Unsolved Problems Concerning Potts Models and Lattice Gases

Elke Thonnes (Göteborg) : Perfect Simulation in Stochastic Geometry

Simulation plays an important role in statistical physics and stochastic geometry, because many but the simplest models tend to be intractable to analysis. In particular, Markov Chain Monte Carlo methods deliver (approximate) samples of such models by simulating the equilibrium distribution of a Markov chain. The samples usually fail to be exact because the algorithm simulates the Markov chain for a long but finite time, and thus convergence to equilibrium may only be approximate. The seminal work by Propp and Wilson made an important contribution to simulation by proposing a coupling method, Coupling from the Past (CFTP), which delivers perfect, that is to say exact, samples of the Markov chain equilibrium distribution. This has been developed into a viable approach for simulation of Markov random fields as well as models in point process and object process theory. My talk will give an example-based introduction into Coupling from the Past and provide an overview of the latest developments in the area of perfect simulation.