Abstract : The dimer model is a system from statistical mechanics modelizing the adsorption of diatomic molecules on the surface of a crystal, represented by a bipartite biperiodic planar graph. An energy is assigned to every type of edge. For such a distribution of energy, there exists a two-parameter family of Gibbs measures on configurations, the behaviour of which is classified into phases: gaseous, liquid or solid.
In the first part, we study the behaviour of such systems near the liquid-solid transition. Considering first the case of the honeycomb lattice, we exhibit two sorts of limit behaviours. The first one is a collection of non-colliding random paths. The second one, called the bead model, is a point random field on ZxR. They both have marginals given by the determinantal random field on R with the sine kernel, describing also the eigenvalues of large random matrices of the GUE ensemble. The bead model is universal: we prove that it is the limit of any dimer model on a bipartite planar periodic graph.
In the second part, we study the statistics of patterns made of dimers. We prove that the fluctuations of pattern density converge in the scaling limit to a Gaussian random field. When the measure is liquid, the limiting object is a derivative of the free field plus an independent white noise. For a gaseous measure, the limit is a white noise.
In the last chapter, we solve a counting problem of paths on the ladder graph. This problem is related to the asymptotics of the heat kernel on the lamplighter's group, as well as to spectral theory of Schrödinger operators with random potential.