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March 2020 The distribution of Gaussian multiplicative chaos on the unit interval
Guillaume Remy, Tunan Zhu
Ann. Probab. 48(2): 872-915 (March 2020). DOI: 10.1214/19-AOP1377


We consider a subcritical 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 nontrivial 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.


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Guillaume Remy. Tunan Zhu. "The distribution of Gaussian multiplicative chaos on the unit interval." Ann. Probab. 48 (2) 872 - 915, March 2020.


Received: 1 August 2018; Revised: 1 May 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199864
MathSciNet: MR4089497
Digital Object Identifier: 10.1214/19-AOP1377

Primary: 60G57
Secondary: 60G15, 60G60, 60G70, 82B23

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 2 • March 2020
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