Speaker
Najnudel, University of Bristol
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 matrices, and more generally, random matrices following the Circular-Beta-Ensemble. In a previous article with Paquette and Simm, we prove that in the \(L2\) 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 \(L1\) phase in our joint paper with Paquette, Simm and Vu. In a joint work with Christopher Atherfold, 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.