Abstract

Starting from the restriction of a 2d Gaussian Free Field (GFF) to the unit disk one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF . In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC . In this talk we will give a rigorous proof of this formula. Our method is inspired by the technology developed by Kupiainen, Rhodes and Vargas to derive the DOZZ formula in the context of Liouville conformal field theory on the Riemann sphere. The novel ingredients are the study of the Liouville theory on Riemann surfaces with a boundary and the key observations that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk. Finally we will discuss applications in random matrix theory, asymptotics of the maximum of the GFF , and tail expansions of GMC .