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15 January 2020 The Fyodorov–Bouchaud formula and Liouville conformal field theory
Guillaume Remy
Duke Math. J. 169(1): 177-211 (15 January 2020). DOI: 10.1215/00127094-2019-0045


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.


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Guillaume Remy. "The Fyodorov–Bouchaud formula and Liouville conformal field theory." Duke Math. J. 169 (1) 177 - 211, 15 January 2020.


Received: 19 March 2018; Revised: 30 May 2019; Published: 15 January 2020
First available in Project Euclid: 17 December 2019

zbMATH: 07198457
MathSciNet: MR4047550
Digital Object Identifier: 10.1215/00127094-2019-0045

Primary: 60G15
Secondary: 60G57 , 60G60 , 81T08 , 81T40

Keywords: boundary Liouville field theory , BPZ equations , conformal field theory , Gaussian free field , Gaussian multiplicative chaos

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 1 • 15 January 2020
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