M. Biskup (Los Angeles), B. Duplantier (Saclay), N. Kurt (Berlin), C. Le Bris (Marne la Vallée), S. Luckhaus (Leipzig), J. Miller (Stanford).

Recently the study of gradients fields has attained a lot of attention because they are space-time analogy of Brownian motions, and are connected to the Schramm-Loewner evolution. The corresponding discrete versions arise in equilibrium statistical mechanics, e.g., as approximations of critical systems and as effective interface models. The latter models - seen as gradient fields - enable one to study effective descriptions of phase coexistence. Gradient fields have a continuous symmetry and coexistence of different phases breaks this symmetry. In the probabilistic setting gradient fields involve the study of strongly correlated random variables. As a result the asymptotic behavior (free energy, measures) depends on the boundary constraint (enforced tilt). Main challenge is the question of uniqueness of Gibbs measures and the strict convexity of the free energy (surface tension) for any non-convex interaction potential. We present in the talk the first break through for low temperatures using Gaussian measures and renormalization group techniques yielding an analysis in terms of dynamical systems. Our main input is a finite range decomposition for a family of Gaussian measures depending on non isotropic tuning parameters. We outline also the connection to the Cauchy-Born rule which states that the deformation on the atomistic level is locally given by an affine deformation at the boundary (Work in cooperation with R. Kotecky and S. Mueller).

Marek Biskup (Los Angeles): New phenomena for gradient models at intermediate temperatures.

I will review some of my recent attempts in the area of gradient models with non-convex potentials paying special attention to the regime that corresponds (explicitly or implicitly) to intermediate temperatures. All my talk will be focused on the question, and consequences, of the strictness of convexity of the surface tension as a function of the tilt. In particular, I will demonstrate a failure of strict convexity in a specific model of gradient interaction studied, earlier, in joint works with R. Kotecky and H. Spohn. Then I will discuss what consequences one can/should expect and/or prove under the assumption that the surface tension is (locally) strictly convex, regardless of the convexity type of the interaction potential.

Bertrand Duplantier (Saclay): Duality and KPZ in Liouville Quantum Gravity.

Liouville quantum gravity in two dimensions is described by the "random Riemannian manifold" obtained by changing the Lebesgue measure $dz$ in the plane by a random conformal factor $\exp [\gamma h(z)]$, where $h(z)$ is a random function called the Gaussian Free Field, and $\gamma$ a real parameter.
For $\gamma < 2$, this "random surface" is believed to be the continuum scaling limit of certain discretized random surfaces that can be studied with combinatorics and random matrix theory. The case $\gamma' >2$ is believed to describe random surfaces with a proliferation of ``baby-universes''. A duality $\gamma \gamma'=4$ appears to relate the two domains.
A famous formula, due to Knizhnik, Polyakov and Zamolodchikov in '88, relates standard critical exponents in the Euclidean plane to their counterparts on the random surfaces mentioned above. We describe a recent proof of the KPZ formula in the probabilistic setting given above, applicable to both dual phases. (joint work with Scott Sheffield, MIT).

Noemi Kurt (Berlin): Entropic repulsion for gradient and Laplacian interface models.

Related to gradient models is another class of interface models, where the gradient interaction is replaced by the discrete Laplacian. There are many similarities, but also some differences between the two classes of models. We will present the main properties of the Laplacian model, and discuss the effect of a hard wall on the interface, with a particular focus on the differences in the mathematical methods used for the two types of models. Generally speaking, the hard wall constraint, which forces the interface to be positive in a certain region of the domain, has a repulsive effect. In the Gaussian case (in sufficiently high dimensions), quite precise results on the repulsion height are obtained.

Claude Le Bris (Marne la Vallée): Random media and related problems.

The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization, for the modelling of random materials. In particular, some variants of the theory of classical stochastic homogenization will be introduced. The relation between such homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence in stochastic homogenization (representative volume element, variance reduction issues, etc) as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. The talk is based upon a series of joint works with X. Blanc (CEA, Paris), PL. Lions (College de France, Paris), and F. Legoll, A. Anantharaman, R. Costaouec, F. Thomines (ENPC, Paris).

Stephan Luckhaus (Leipzig): Lattice based Hamiltonians in elasticity, large deviations and two scale convergence.

We look at elastically deformed crystals in the framework of equilibrium statistical mechanics.The state space is given by the particle positions relative to a reference lattice. The potentials are k-nearest neighbour interactions invariant under rigid motions. Under growth and nondegeneracy assumptions we are able to show a large deviations result where the rate functional is a quasiconvex elastic free energy. So the macroscopic shape is given as the solution to a variational problem. Moreover we obtain a two scale convergence result describing the fluctuations at a macroscopic position via a gradient Young Gibbs measure whose slope is given by the macroscopic deformation gradient.

Jason Miller (Stanford): Fluctuations for the Ginzburg-Landau Grad-Phi Interface Model on Bounded Domains.

The object of our study is the massless field with strictly convex nearest neighbor interaction on lattice approximations of a bounded, smooth, planar domain D with boundary conditions given by the restriction of a continuous function f. This is a general model for a (2+1) effective interface. We prove that the mean of the random height function h converges to the harmonic extension of f from the boundary of D to D and that linear functionals of h converge to the Gaussian free field on D, a conformally invariant random distribution.