T. Bodineau (Paris), F. Camia (Amsterdam), A. Járai (Bath), J.L. Lebowitz (Rutgers), O. Riordan (Oxford), S. Smirnov (Genève), B. Tóth (Budapest)

Equivalence of ensembles plays a crucial role in equilibrium statistical mechanics and in the justification of the Gibbs measures. We will first review some facts on the equivalence of ensembles and then address the question of its validity for non-equilibrium statistical mechanics.

Federico Camia (Amsterdam): Ising Euclidean Fields and (Conformal) Measure Ensembles

The two-dimensional Ising model is one of the most studied models of statistical mechanics and has played a fundamental role in the theory of phase transitions. In this talk, I will focus on the magnetization field, which describes the spatial fluctuations of the local magnetic field generated by the spins and is one of the main objects in the Ising field theory.
In the scaling limit, as the lattice spacing is sent to zero, the magnetization field can be desribed as a random generalized function. Above the critical temperature, for instance, the scaling limit of the lattice magnetization field is Gaussian white noise. The situation is more interesting at the critical point where thermal fluctuations extend over all scales, leading to scale invariance and a conformally covariant field. One also expects cluster boundaries to converge in the scaling limit to SLE-type curves, and a proof of this fact has been recently announced by S. Smirnov.
I will introduce a representation for the magnetization field which leads to a simple proof of the existence of subsequential scaling limits, provides a connection between the field-theoretic and the SLE approach, and can be used to prove uniqueness of the scaling limit and its expected conformal covariance properties. The key ingredient is an ensemble of measures of fractal support coming from the scaling limit of the rescaled areas of critical FK clusters.
(Based on Joint work with C.M. Newman and with C. Garban and Newman.)

Antal A. Jarai (Bath): Abelian sandpiles: an overview and results on transitive graphs.

The Abelian sandpile was introduced in the physics literature as a model for self-organized criticality. Its dynamics is defined in terms of simple local rules that result, due to a separation of time scales, in a non-local dynamics. The stationary distribution of the model has a description in terms of the uniform spanning tree. In this talk I will give an introduction to the sandpile model, explain the connection to the uniform spanning tree, and consider the infinite volume limit on certain transitive graphs.

Joel L. Lebowitz (Rutgers University): Properties of systems with non reflection invariant interactions: variations on the ABC model.

We consider q-component systems with q greater or equal to three on a d-dimensional lattice. The energy of the system is given by the sum of pair-interactions which are not invariant under spatial reflections, e.g. in one dimension the interaction between an A and B particle depends on whether A is to the left or right of B. We have obtained the exact phase diagram for such a system when the interactions are of mean field type, Arxiv: 0905.4849. It is very different from the standard mean field model with reflection symmetric interactions. Various generalization of this model will be discussed.

Oliver Riordan (Oxford): The generalized triangle-triangle transformation for percolation.

One of the main aims in the theory of percolation is to find the `critical probability' above which long range connections emerge from random local connections with a given pattern and certain individual probabilities. The quintessential example is Kesten's result from 1980 that if the edges of the square lattice are selected independently with probability $p$, then long range connections appear if and only if $p>1/2$. The starting point is a certain self-duality property, observed already in the early 60s; the difficulty is not in this observation, but in proving that self-duality does imply criticality in this setting.
Since Kesten's result, more complicated duality properties have been used to determine a variety of other critical probabilities. Recently, Scullard and Ziff have described a very general class of self-dual planar percolation models; we show that for the entire class (in fact, a larger class), self-duality does imply criticality.

Stanislav Smirnov (Genève): On the scaling limits of 2D lattice models.

Balint Tóth (Budapest): Long-time asymptotic behaviour of self-repelling random processes.

I will present a survey of recent results about the long-time asymptotic behaviour of random processes with long memory due to some rather natural local self-interaction (self-repellence) of the trajectories. Typical examples are the so-called myopic (or "true") self-avoiding random walk and the self-repelling Brownian polymer models. The long-time asymptotics of the displacement is expected to be robust (not depending on some microscopic details), but dimension-dependent. It is expected that: in 1d, the motion is strongly superdiffusive with time-to-the-two-thirds scaling; in 2d, the motion is marginally superdiffusive with logarithmic multiplicative correction in the scaling; in three and more dimensions the displacement is diffusive. Some of these have been recently proved, at least for some particular models.