For every sub-critical 1D Gaussian multiplicative chaos measure, the Fourier dimension equals the correlation dimension: 1-γ² for small γ, (√2-γ)² for large γ.
Random analytic functions via Gaussian multiplicative chaos
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abstract
We define a random analytic function $\varphi$ on the unit disc by letting a Gaussian multiplicative measure to be one of its Clark measures. We show that $\varphi$ is almost surely a Blaschke product and we provide rather sharp estimates for the density of its zeroes.
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Harmonic analysis of multiplicative chaos Part I: the proof of Garban-Vargas conjecture for 1D GMC
For every sub-critical 1D Gaussian multiplicative chaos measure, the Fourier dimension equals the correlation dimension: 1-γ² for small γ, (√2-γ)² for large γ.