The Fyodorov–Bouchaud formula and Liouville conformal field theory
| Autor: | Guillaume Remy |
|---|---|
| Přispěvatelé: | 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), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2020 |
| Předmět: |
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 |
| Zdroj: | Duke Math.J. Duke Math.J., 2020, 169 (1), pp.177-211. ⟨10.1215/00127094-2019-0045⟩ Duke Math. J. 169, no. 1 (2020), 177-211 |
| ISSN: | 0012-7094 |
| DOI: | 10.1215/00127094-2019-0045 |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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