Thierry Bodineau (Palaiseau), Béatrice de Tilière (Paris), Benoit Laslier (Cambridge), Oren Louidor (Haifa), Fabio Martinelli (Roma), Ron Peled (Tel Aviv).

Random surface models (discrete or continuous) are ubiquitous in statistical physics, arising for instance as models for interfaces between thermodynamic phases or as height functions associated to discrete models (dimer models, etc). In this introduction, that should serve as a motivation for the rest of the day, I will give a non-systematic overview of some open challenges in this field, both about equilibrium fluctuations and about dynamical evolution of random interfaces. In some of these directions there were important advances and exciting activity recently, as witnessed by the following talks

Thierry Bodineau (Palaiseau): Interface motion in disordered media. pdf

We will first review the return to equilibrium of the Ising model when a small external field is applied. The relaxation time is extremely long and can be estimated as the time needed to create critical droplets of the stable phase which will invade the whole system. We will then discuss the impact of disorder on this metastable behavior and show that for Ising model with random interactions (dilution of the couplings) the relaxation time is much faster as the disorder acts as a catalyst. In the last part of the talk, we will focus on the droplet growth and study a toy model describing interface motion in disordered media.

Béatrice de Tilière (Paris): Height representation of XOR-Ising loops via bipartite dimers. pdf

The XOR-Ising model is constructed from two independent Ising models. We show in an explicit way that loops separating clusters of spins of XOR-Ising configurations have the same law as loops in a bipartite dimer model. As a consequence, XOR-loops have the same law as level lines of a height function which, at the critical point, converges weakly in distribution to a Gaussian free field. This is joint work with Cédric Boutillier.

Benoit Laslier (Cambridge): The Glauber dynamics on lozenge tilings and other dimer models. pdf

We study the so called Glauber dynamics on dimer model where the only allowed moves are rotations of all the dimers among a single face. We will show in a relatively general setup that the time needed for the system to reach equilibrium is of order L^(2+o(1)), where L is the typical length scale of the system. The exponent 2 is optimal. More precisely, for surfaces attached to a curve drawn in some plane, we will control the mixing time for several models, including lozenge tilings and domino tilings. For surfaces attached to a general curve, we will only work with lozenges and use a weaker notion of "macroscopic" convergence.

Oren Louidor (Haifa): The full extremal process of the discrete Gaussian free field in 2D.

We show the existence of the limit of the full extremal process of the discrete Gaussian free field in 2D with zero boundary conditions. The limit is a clustered Poisson point process with a random intensity measure, which is conjecturally related to the critical Liouiville quantum gravity measure w.r.t. the continuous Gaussian free field. Several corollaries follow directly, e.g. a natural construction for the super-critical Gaussian multiplicative chaos and Poisson-Dirichlet statistics for the limiting Gibbs measure - both w.r.t. the CGFF. The proof is based on a novel concentric decomposition of the DGFF which effectively reduces the problem to that of finding asymptotics for the probability of a decorated non-homogenous random-walk required to stay positive. Entropic repulsion plays a key in the analysis. Joint work with M. Biskup (UCLA).

Fabio Martinelli (Roma): Harmonic pinnacles in the discrete Gaussian model. pdf

The 2D Discrete Gaussian model gives each height function eta: Z^2 to Z a probability proportional to exp[-beta*H(eta)], where beta is the inverse temperature and H(eta)=sum(eta_x-eta_y)^2 sums over nearest-neighbor bonds. We consider the model at large fixed beta, where it is flat unlike its continuous analog (the Gaussian Free Field). We first establish that the maximum height in an LxL box with 0 boundary conditions concentrates on two integers M,M+1 with M ~ [(2/pi*beta)logL(loglogL)]^{1/2}. The key is a large deviation estimate for the height at the origin in Z^2, dominated by "harmonic pinnacles", integer approximations of a harmonic variational problem. Second, in this model conditioned on eta>=0 (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H,H+1 where H ~ M/sqrt(2). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Fröhlich (1986). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices. Joint work with Eyal Lubetzky and Allan Sly.

Ron Peled (Tel Aviv): Delocalization of two-dimensional random surfaces with hard-core constraints.

We study the fluctuations of random surfaces on a two-dimensional discrete torus. The random surfaces we consider are defined via a nearest-neighbor pair potential which we require to be twice continuously differentiable on a (possibly infinite) interval and infinity outside of this interval. No convexity assumption is made and we include the case of the so-called hammock potential, when the random surface is uniformly chosen from the set of all surfaces satisfying a Lipschitz constraint. Our main result is that these surfaces delocalize, having fluctuations whose variance is at least of order log n, where n is the side length of the torus. We also show that the expected maximum of such surfaces is of order at least log n. The main tool in our analysis is an adaptation to the lattice setting of an algorithm of Richthammer, who developed a variant of a Mermin-Wagner type argument applicable to hard-core constraints. We rely also on the reflection positivity of the random surface model. The result answers a question mentioned by Brascamp, Lieb and Lebowitz on the hammock potential and a question of Velenik. Joint work with Piotr Miłoś.