The last two decades have seen the emergence of a new area of probability theory concerned with certain random fractal structures characterized by their invariance under conformal transformations. The study of such structures has had deep repercussions on both mathematics and physics, generating tremendous progress in probability theory, statistical mechanics and conformal field theory. In this series of talks, I will give a personal perspective on some aspects of this new area, focusing for concreteness on three specific examples: the Ising model, percolation, Brownian loops. The three talks will be independent and self-contained.

In this talk, I will consider the scaling limit of critical site percolation on the triangular lattice. This was the first model where conformal invariance was proved rigorously, thanks to Smirnov’s celebrated proof of Cardy’s formula for the scaling limit of crossing probabilities (between boundary arcs of a bounded domain). Much progress followed swiftly, but the conformal covariance of connection probabilities (between points in the interior of a domain), expected by physicists since the 1980s and explicitly conjectured by Aizenman in the 1990s, remained open. I will discuss a recent proof of this conjecture based on the conformal invariance of the percolation full scaling limit constructed by Newman and myself in the early 2000s.

The last two decades have seen the emergence of a new area of probability theory concerned with certain random fractal structures characterized by their invariance under conformal transformations. The study of such structures has had deep repercussions on both mathematics and physics, generating tremendous progress in probability theory, statistical mechanics and conformal field theory. In this series of talks, I will give a personal perspective on some aspects of this new area, focusing for concreteness on three specific examples: the Ising model, percolation, Brownian loops. The three talks will be independent and self-contained.

In this talk, I will first introduce the Brownian loop soup (BLS), a conformally invariant Poissonian ensemble of loops in two dimensions whose intensity measure is proportional to the unique (up to a multiplicative constant) conformally invariant measure on simple planar loops. The BLS is closely related to the Schramm-Loewner evolution (SLE), conformal loop ensembles (CLE), and the scaling limit of various models of statistical mechanics, such as the Ising model, percolation, the loop O(n) model. I will then discuss several observable quantities of the BLS (e.g., the winding number of loops around a point) and show that, when properly rescaled, they behave like conformal primary fields in conformal field theory (CFT). These fields are the building blocks of any CFT, and our results suggest that we have identified a new family of CFTs with novel properties, but closely related to SLE and CLE and to some of the standard models of statistical mechanics. (Based on joint work with A. Gandolfi, V. Foit and M. Kleban.)