The distribution of Gaussian multiplicative chaos on the unit interval

Autor: Tunan Zhu, Guillaume Remy
Přispěvatelé: Département de Mathématiques et Applications - ENS Paris (DMA), 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:
Statistics and Probability
Pure mathematics
60G70
chaos
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
Log-correlated field
Holomorphic function
FOS: Physical sciences
Field (mathematics)
conformal field theory
holomorphic
integrability
01 natural sciences
Measure (mathematics)
Gaussian multiplicative chaos
010104 statistics & probability
FOS: Mathematics
0101 mathematics
Mathematical Physics
Mathematics
60G60
Liouville quantum gravity
field theory: conformal
Conformal field theory
integrable probability
Probability (math.PR)
010102 general mathematics
Multiplicative function
Mathematical Physics (math-ph)
Auxiliary function
field theory: Liouville
Distribution (mathematics)
quantum gravity
60G15
fractional
60G57
82B23
Statistics
Probability and Uncertainty
Mathematics - Probability
reflection
Unit interval
Zdroj: Annals Probab.
Annals Probab., 2020, 48 (2), pp.872-915. ⟨10.1214/19-AOP1377⟩
Ann. Probab. 48, no. 2 (2020), 872-915
ISSN: 0091-1798
DOI: 10.1214/19-aop1377
Popis: We consider a sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in 0 and 1, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides non-trivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.
Comment: 54 pages, 1 figure
Databáze: OpenAIRE
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