2017/10/31 by Guillaume Remy
Mathematics · Physics and Astronomy · #Conformal field theory #Conformal map #Degenerate energy levels #Direct proof #Field (mathematics) #Gaussian #Gaussian free field #Mathematical Dynamics and Fractals #Multiplicative function #Random Matrices and Applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.1215/00127094-2019-0045
published as Duke Math. J. 169, no. 1 (2020), 177-211 · 27 pages
openalex created_date 2017/11/10 · arxiv created 2019/11/04 · openalex publication_date 2019/12/17 · arxiv updated 2020/02/12 · openalex updated_date 2026/08/05
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.