The Fyodorov-Bouchaud formula and Liouville conformal field theory

TL;DR

This paper proves Fyodorov and Bouchaud's conjectured formula for the density of the total mass of Gaussian multiplicative chaos on the circle, linking it to Liouville conformal field theory and providing explicit probability densities.

Contribution

It provides the first rigorous proof of an explicit density for GMC total mass, connecting probabilistic methods with Liouville CFT and computing negative moments via conformal field theory techniques.

Findings

Proof of Fyodorov-Bouchaud formula for GMC density

Explicit probability density for GMC total mass

Applications to random matrix theory and GFF maxima

Abstract

In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is 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 defined by Huang, Rhodes and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (BPZ equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF and tail expansions of GMC.