The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency

The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular-$\beta$-ensemble, random matrices. We consider the Fourier coefficients of the holomorphic multiplicative chaos in the $L^1$-phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular-$\beta$-ensemble for all $\beta > 2$.