The Fourier coefficients of the critical holomorphic multiplicative chaos

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 matrices, and more generally, random matrices following the Circular-$\beta$-Ensemble. In a previous article, Najnudel, Paquette and Simm prove that in the $L^2$ phase $\beta >4$, the appropriately normalized Fourier coefficient of the HMC converges in distribution to the square root of the total mass of the Gaussian multiplicative chaos on the unit circle, multiplied by an independent complex normal random variable. This convergence has been extended to the $L^1$ phase by Najnudel, Paquette, Simm and Vu. In the present article, we prove that this convergence further extends to the critical case $\beta =2$, which corresponds to the limiting coefficients of the characteristic polynomial of the Circular Unitary Ensemble. We also prove the joint convergence of consecutive Fourier coefficients, and we derive convergence in distribution of the secular coefficients of the Circular Unitary Ensemble with index growing sufficiently slowly with the dimension.