The Fyodorov-Bouchaud conjecture and Liouville conformal field theory

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 .

• Speaker: Guillaume Remy (ENS Paris)
• Tuesday 13 February 2018, 16:15–17:15
• Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
• Series: Probability; organiser: Perla Sousi.