### The Fyodorov-Bouchaud formula and Liouville conformal field theory

Mathematical Physics Seminar

19th January 2018, 2:00 pm – 3:00 pm

Howard House, 4th Floor Seminar Room

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.

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