Filippo Colomo (Firenze), Sylvie Corteel (Paris), Thomas Fernique (Villetaneuse), Patrik Ferrari (Bonn), Alessandro Giuliani (Roma), Leonid Petrov (Charlottesville).

Counting domino or rhombus tilings of a planar domain is a classical combinatorial problem which received historical contributions from MacMahon, Kasteleyn, Temperley and Fisher, etc. It is intimately connected with statistical physics (it models the adsorption of diatomic molecules on a surface) and probability theory (how does a large uniform random tiling look like ?). In this introductive talk, I will review some basic facts about tilings and related objects, paving the way to the more advanced talks of the day.

Filippo Colomo (Firenze): Arctic curves of the six-vertex model. pdf

The six-vertex model with domain wall boundary conditions can be regarded as an `interacting' generalization of the famous `free-fermion' problem of domino tilings of the Aztec Diamond. In a suitable scaling limit, it is known to exhibit spatial phase separations, with the emergence of various regions of order and disorder, sharply separated by a smooth curve, called Arctic curve. Here we review the state of the art on the model, in particular: the determination of the explicit analytic form of the Arctic curve for arbitrary weights; the generalization of the treatment to domains of generic shape; the third-order phase transition induced by suitable modification of the aspect ratio of the domain.

Sylvie Corteel (Paris): Dimers on Rail Yard Graphs.

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call Rail Yard Graphs (RYG). Using a transfer matrix approach and the celebrated boson-fermion correspondence, the model can be reformulated as a Schur process (i.e. a random sequence of integer partitions). We obtain explicit expressions for the partition function and for the inverse Kasteleyn matrix, which yields all dimer correlation functions. Plane partitions, domino tilings of the Aztec diamond and pyramid partitions arise as particular cases of our model. This is joint work with Cédric Boutillier (Paris 6), Jérémie Bouttier (CEA), Guillaume Chapuy (CNRS) and Sanjay Ramassamy (Brown U.).

Thomas Fernique (Villetaneuse): From random to quasiperiodic tilings. pdf

We shall discuss questions that arise when modelling so-called quasicrystals by tilings, in particular rhombus tilings of the plane. Those can indeed be easily seen as surfaces in a higher dimensional space. Tilings modelling quasicrystals obtained by quenching then correspond to maximal entropy random surfaces, while the more recent and nicer annealed quasicrystals correspond to irrational planes. This rises various theoretical questions (ranging from Markov chain mixing through calculability and combinatorics), most of which are open.

Patrik Ferrari (Bonn): From a 2+1 dimensional particle system to random tilings and random matrices.

We will describe a 2+1 dimensional system of interacting particle system system that includes simultaneously a 1+1 dimensional particle system (the totally asymmetric simple exclusion process) and a random tiling model. In a special case in a discrete time setting, the latter is equivalent to the Aztec diamond. Further, under diffusion scaling limit one recovers the GUE minor measure of random matrices. This talk is based on the construction made with Alexei Borodin in arXiv:0804.3035 .

Alessandro Giuliani (Roma): Height fluctuations in interacting dimers. pdf

Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances. The proof is based on a combination of multiscale methods with the path-independence properties of the height function. Joint work with V. Mastropietro and F. Toninelli.

Leonid Petrov (Charlottesville): Dynamics of random surfaces and interacting particle systems via spectral properties.

Many interesting ensembles of random surfaces are "integrable", i.e., their distributions and asymptotics can be studied by algebraic methods. Often this integrability is triggered by remarkable algebraic identities, such as the Cauchy identity for summation of Schur (or, more generally, Macdonald) symmetric polynomials. With the help of these identities, integrable stochastic Markov dynamics on random surfaces (and also related interacting particle systems) can be modeled through multiplication operators in a "spectral" space. This approach also leads to discovery of new integrable Markov dynamics.