Fourier dimension of imaginary Gaussian multiplicative chaos

TL;DR

This work analyzes the Fourier analytic properties of imaginary Gaussian multiplicative chaos on the unit circle in the subcritical regime $\beta\in(0,1)$. By representing moments of the Fourier coefficients via Coulomb-gas integrals and expanding them in Jack polynomials, the authors derive precise asymptotics and identify the limiting Gaussian fluctuations: $n^{(1-\beta^2)/2}\,c_n$ converges to a complex Gaussian with variance $\kappa(\beta)$, and finite-length vectors converge to independent complex Gaussians. They further show tightness and convergence of the rescaled chaos itself to complex white noise in suitable Sobolev spaces, establishing a harmonic-analysis description of imaginary GMC on the circle. The results provide explicit Fourier-dimension $1-\beta^2$ for $M_{i\beta}$ and a concrete CLT and white-noise limit, highlighting the monofractal, non-positive nature of imaginary chaos and its distinct Fourier-analytic behavior from real GMC.

Abstract

We study the Fourier coefficients of imaginary Gaussian multiplicative chaos (GMC) on the unit circle. Under the subcritical phase $β\in(0,1)$, we show that the Fourier dimension is $1-β^2$ and prove a central limit theorem for the rescaled coefficients.

Figures (4)

• Figure 1: An example of partition $\nu=\lambda + \sigma$ obtained by adding $\sigma$ of shape $N_1=3$ and $N_2=2$.
• Figure 2: The Young diagram of a partition $\lambda$ with $a_\lambda(s)=5$ and $l_\lambda(s)=2$.
• Figure 3: The skew diagram obtained from two partitions $\lambda$ and $\mu$ represented by the blue cells.
• Figure :

Theorems & Definitions (20)

• Theorem 1.1: Fourier dimension
• Theorem 1.2: CLT for the rescaled coefficients
• Corollary 1.3: Convergence toward a complex white noise
• Conjecture 1.4
• Proposition 2.1
• proof
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Lemma 3.3