1. The Fyodorov–Bouchaud formula and Liouville conformal field theory
- Author
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Guillaume Remy, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
81T08 ,chaos ,General Mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Field (mathematics) ,Boundary Liouville field theory ,01 natural sciences ,Measure (mathematics) ,Gaussian multiplicative chaos ,Correlation function ,81T40 ,0103 physical sciences ,Gaussian free field ,FOS: Mathematics ,correlation function ,0101 mathematics ,circle ,Mathematical Physics ,Mathematical physics ,Mathematics ,60G60 ,field theory: conformal ,density ,Conformal field theory ,Probability (math.PR) ,010102 general mathematics ,Multiplicative function ,Mathematical Physics (math-ph) ,matrix model: random ,field theory: Liouville ,Unit circle ,60G15 ,60G57 ,010307 mathematical physics ,BPZ equations ,Random matrix ,Mathematics - Probability - 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., 27 pages
- Published
- 2020
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