Universality and phase transitions of holomorphic multiplicative chaos

Critical holomorphic multiplicative chaos (HMC) arises naturally from the studies of characteristic functions of CUE and partial sums of random multiplicative functions. We investigate the low moments of secular (Fourier) coefficients of the critical HMC. We establish:

1. universality for non-Gaussian HMC, generalizing results of Gaussian HMC (Najnudel–Paquette–Simm, 2023 and Soundararajan–Zaman, 2022);
2. a double-layer phase transition on the tail of the input of the non-Gaussian HMC, which exhibits a completely different mechanism from the Gaussian universality regime.

I will briefly survey recent works and outline the ideas underlying the phase transition.

This is based on joint work with Haotian Gu (UCLA).