MARK KAC SEMINAR

The East model is a linear chain of spins, each labeled either 0 or 1, evolving according to a very simple rule:

1. with rate one and independently for each vertex, a new value 1/0 is proposed with probability 1-q and q respectively;
2. the proposed value is accepted iff the spin immediately to the left is labeled "0".

The above is just an example of a general class of interacting particle systems, known as "Kinetically constrained spin models" (KCM) in which the local update of a spin occurs only in the presence of a "facilitating" configuration at neighboring vertices. Although the i.i.d. Bernoulli(1-q) distribution remains a reversible stationary measure, the relaxation to equilibrium of KCMs can be extremely complex, featuring dynamical phase transitions, metastability, dynamical heterogeneities and universality. KCMs have been introduced in the physics literature in the '80s in order to reproduce some of the most characteristic features of the glass transition for supercooled liquids, while keeping a very simple reversible equilibrium distribution. In this context small values of the density q of the facilitating spins correspond to low temperatures.

Mathematically KCMs pose several interesting new challenges because of the constraints. After the pioneering work by Aldous and Diaconis in 2001 on the East chain, several further mathematical progresses have been achieved which improved and sometimes corrected quite basic assumptions made in the physics literature. In this talk I will first survey some of the basic features of this class of models and I'll present some techniques which have been developed to analyze ergodicity, the scaling of spectral gap and mixing times for small values of the parameter q. I'll conclude by presenting some challenging open problems.

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, "large" lattice loops from the random walk loop soup converge to "large" loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soups and CLEs goes via cluster boundaries. In this lecture, we study the scaling limit of clusters of "large" lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of "large" lattice loops converges to CLE. This is joint work with Federico Camia and Marcin Lis.