Random discrete interfaces and stochastic dynamics

Random discrete interfaces are a classical object in statistical mechanics: they provide for instance effective models for phase separation boundaries in spin systems. Similar objects arise also naturally in combinatorics: for instance, discrete random surfaces are associated naturally to random dimer coverings of bipartite graphs.

In recent years, a lot of activity has focused on studying equilibrium statistical properties of random surfaces, notably, proving convergence to the so-called "gaussian free field" for (2+1)-dimensional surfaces.

An equally interesting but mathematically much less developed topic is that of studying stochastic (Markov) dynamics of discrete interfaces. Motivations arise both from statistical physics (understanding the time evolution of phase boundaries) and from combinatorics/theoretical computer science (Markov Chain Monte Carlo algorithms providing an efficient way of counting and of uniformly sampling dimer coverings). In the "diffusive limit" where space and time are suitably rescaled, the stochastic interface dynamics is in many cases believed to converge to a deterministic evolution of mean curvature type. We will discuss some recent developments, the related mathemaical difficulties and some perspectives.