Guillaume Remy

The Fyodorov-Bouchaud formula and Liouville conformal field theory

Starting from the restriction of a 2d Gaussian free field (GFF) to the
unit circle 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. In our case the key
observation is that the negative moments of the total mass of GMC on the
circle determine its law and are equal to one-point correlation functions
of Liouville theory in the unit disk. Finally we will discuss applications
in random matrix theory, asymptotics of the maximum of the GFF, and tail
expansions of GMC.