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May 2022 Convergence in law for complex Gaussian multiplicative chaos in phase III
Hubert Lacoin
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Ann. Probab. 50(3): 950-983 (May 2022). DOI: 10.1214/21-AOP1551

Abstract

Gaussian multiplicative chaos (GMC) is informally defined as a random measure eγXdx where X is Gaussian field on Rd (or an open subset of it) whose correlation function is of the form K(x,y)=log1|yx|+L(x,y), where L is a continuous function of x and y and γ=α+iβ is a complex parameter. In the present paper we consider the case γPIII, where

PIII:={α+iβ:α,γR,|α|<d/2,α2+β2d}.

We prove that if X is replaced by an approximation Xε obtained by convolution with a smooth kernel, then the random distribution eγXεdx, when properly rescaled, has an explicit nontrivial limit in law when ε goes to zero. This limit does not depend on the specific convolution kernel which is used to define Xε and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter 2α.

Funding Statement

This work was realized during the author’s extended stay in Aix-Marseille University funded by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 837793.

Acknowledgements

The author is grateful to Paul Gassiat for indicating the reference [8] for the proof of Theorem 2.5 and letting him know about the notion of stable convergence which is the adequate framework to present the main result of this paper. He thanks J. F. Le Gall, Rémi Rhodes and Vincent Vargas for enlightening comments.

Citation

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Hubert Lacoin. "Convergence in law for complex Gaussian multiplicative chaos in phase III." Ann. Probab. 50 (3) 950 - 983, May 2022. https://doi.org/10.1214/21-AOP1551

Information

Received: 1 December 2020; Revised: 1 October 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1551

Subjects:
Primary: 60F99 , 60G15 , 82B99

Keywords: Gaussian multiplicative chaos , log-correlated fields , random distributions

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • May 2022
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