Duke Mathematical Journal

The Fyodorov–Bouchaud formula and Liouville conformal field theory

Guillaume Remy

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In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (subcritical) 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 recently defined by Huang, Rhodes, and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (Belavin–Polyakov–Zamolodchikov 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.

Article information

Duke Math. J., Volume 169, Number 1 (2020), 177-211.

Received: 19 March 2018
Revised: 30 May 2019
First available in Project Euclid: 17 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G57: Random measures 60G60: Random fields 81T08: Constructive quantum field theory 81T40: Two-dimensional field theories, conformal field theories, etc.

Gaussian free field Gaussian multiplicative chaos boundary Liouville field theory conformal field theory BPZ equations


Remy, Guillaume. The Fyodorov–Bouchaud formula and Liouville conformal field theory. Duke Math. J. 169 (2020), no. 1, 177--211. doi:10.1215/00127094-2019-0045. https://projecteuclid.org/euclid.dmj/1576573213

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