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Harmonic analysis of Gaussian multiplicative chaos on the circle

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arxiv 2311.04027 v2 pith:OB7MGP3P submitted 2023-11-07 math.PR math-phmath.FAmath.MP

Christophe Garban , Vincent Vargas This is my paper · ORCID
classification math.PRmath-phmath.FAmath.MP
keywords coefficientsfourieranalysischaoscirclegaussianharmonicmultiplicative
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abstract

In this paper, we initiate the harmonic analysis of Gaussian multiplicative chaos (GMC) on the circle, i.e. the study of its Fourier coefficients. In particular, we show that almost surely GMC is a so-called Rajchman measure which means that its Fourier coefficients converge to $0$ when the frequency goes to infinity. We supplement this result with a convergence in law result for the rescaled Fourier coefficients.

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  1. Exact values of Fourier dimensions of Gaussian multiplicative chaos on high dimensional torus

    math.PR 2025-07 conditional novelty 8.0 of 10

    For a specially constructed log-correlated field on T^d, the GMC measure almost surely has Fourier dimension d-γ^2 when γ<√(2d)/2 and (√(2d)-γ)^2 when √(2d)/2<=γ<√(2d), for all d>=1.

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