By Federico Camia, NYU Abu Dhabi

Several classical lattice models of statistical mechanics, such as percolation and the Ising and Potts models, can be described in terms of clusters. In the last fifteen years, there has been tremendous progress in the study of the geometric properties of such models in two dimensions in the scaling limit, when the lattice spacing is sent to zero. Much of that work has focused on cluster boundaries, using the Schramm-Loewner Evolution (SLE), introduced by Oded Schramm, and collections of SLE loops called Conformal Loop Ensembles (CLEs). In this talk I will discuss the scaling limit of the clusters themselves and their ''areas'' in the case of percolation and the Ising model. This leads to the study of rescaled counting measures and to the concept of Conformal Measure Ensembles (first introduced in joint work with Chuck Newman), with interesting applications to two-dimensional critical percolation and the two-dimensional critical Ising model.

Based on joint work with Rene Conijn and Demeter Kiss.