This project proposes a 12-month mobility of the applicant F. Toninelli from the University of Lyon 1 to the Mathematics and Physics Department of Roma Tre. The applicant will be supervised by local scientistsF. Martinelli and A. Giuliani.The scientific project focuses on the probabilistic study of dynamical and equilibrium properties of discrete random interfaces, spin models and dimer models. One of the main goals is to derive, starting from a microscopic stochastic dynamics, a macroscopic deterministic evolution (often, an anisotropic mean curvature flow) in the scaling limit. Another important issue are equilibrium fluctuations of random discrete interfaces and dimer models. The proposed research has tight links with mathematical physics, discrete geometry and rigorous quantum field theory. Mathematical tools to be employed include Markov Chains, conformal invariance, random walks on graphs, rigorous Renormaliazation Group techniques... More specific problems that will be attacked include:1) Dynamics and equilibrium fluctuations of random dimer coverings (perfect matchings) of bipartite graphs: theseare a central object in combinatorics and discrete geometry and can be seen as discrete interfaces. How does their geometry evolve under stochastic dynamics? What is the dynamical consequence of the conformal invariance properties of their equilibrium fluctuations? How quickly can one sample a random perfect matching (mixing time)?2) Universality for two-dimensional statistical mechanics models. We will study the critical properties of weakly interacting dimers on the square lattice (the non-interacting case being exactly solvable). As a long-term goal, we wish to study the spin-spin correlation functions of two-dimensional, non-integrable critical Ising models.3) Discrete SOS model. What large-deviation equilibrium properties and the dynamical metastability phenomena of the two-dimensional Solid-on-Solid discrete interface model?