Dimers Day in Cambridge

(4th Peter Whittle colloquium)

Times and room: All lectures will be given in MR12. Schedule:

Talk by Cedric Boutillier: An introduction to planar dimer models
A dimer model is a probability measure on perfect matchings on a graph. When the graph is planar, the so-called Kasteleyn theory allows one to compute many interesting probabilistic quantities about the model using determinants. Moreover, when the graph is also bipartite, the dimer model can be interpreted as a random surface model via a height function. After having presented some elements of Kasteleyn theory and the construction of the height function, we will discuss the connection of this model with spanning trees, by the Temperley bijection.

Talk by Sunil Chhita: The two-periodic Aztec diamond
Random domino tilings of the Aztec diamond shape exhibit interesting features and statistical properties related to random matrix theory. As a statistical mechanical model, it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models. These phases are often referred to as solid, liquid and gas, but are not physical states of matter. In this talk, we introduce the model and focus on the behaviour at the liquid-gas boundary. This is based on joint works with Vincent Beffara (Grenoble), Kurt Johansson (Stockholm) and Benjamin Young (Oregon).

Talk by Julien Dubédat :


Talk by Benoît Laslier : Universality of height fluctuations
On Z^2, Temperley's bijection gives a connection between a spanning tree and a dimer configuration. This bijection has a continuum analogue given by the coupling between SLE and gaussian free field of the imaginary geometry type. I will first present a new point of view on this coupling emphasizing the link with the discrete bijection. Then we will be able to connect the continuum coupling to the discrete bijection and study the dimer fluctuations on almost arbitrary superposition graphs or planar domains for lozenge tilings.

Talk by Fabio Toninelli : A class of (2+1)-dimensional growth process with explicit stationary measure
We introduce a class of (2 + 1)-dimensional random growth processes, that can be seen as asymmetric random dynamics of discrete interfaces. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. "Asymmetric" means that the interface has an average non-zero drift. When the asymmetry parameter p - 1/2 equals zero, the infinite-volume Gibbs measures pi_\rho (with given slope \rho) are stationary and reversible. When p\neq 1/2, \pi_\rho is not reversible any more but, remarkably, it is still stationary. In such stationary states, one finds that the height function at a given point x grows linearly with time t with a non-zero speed, E h_x (t) h_x(0) = vt, and the typical typical fluctuations of Q_x(t) are smaller than any power of t. For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" studied by A. Borodin and P. L. Ferrari.
If time allows, I will also mention ongoing related work with B. Laslier.