Abstract

About 20 years ago, Kontsevich & Suhov conjectured the existence and uniqueness of a family of measures on the set of Jordan curves, characterised by conformal invariance and a restriction-type property. This conjecture was motivated by (seemingly unrelated) works of Schramm, Lawler & Werner on Schramm—Loewner evolutions (SLE), and Malliavin, Airault & Thalmaier on “unitarising measures”. The existence of this family was settled by works of Werner—Kemppainen and Zhan, using a loop version of SLE . The uniqueness was recently obtained in a joint work with Baverez. I will start by reviewing the different notions involved before giving some ideas of our proof of uniqueness: in a nutshell, we construct a family of “orthogonal polynomials” which completely characterises the measure. I will discuss the broader context in which our construction fits, namely the conformal field theory associated with SLE .