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

TL;DR

This paper studies the Fourier coefficients of holomorphic multiplicative chaos, showing their normalized convergence to complex normal variables and extending results to joint distributions and secular coefficients in circular-$eta$-ensembles.

Contribution

It introduces the convergence behavior of Fourier coefficients of HMC in the large frequency limit, connecting it to Gaussian multiplicative chaos and circular-$eta$-ensembles.

Findings

Normalized Fourier coefficients converge to complex normal variables.

Joint convergence of consecutive Fourier coefficients established.

Secular coefficients of circular-$eta$-ensembles converge in law for all $eta > 2$.

Abstract

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 $\beta > 2$.