Secular coefficients and the holomorphic multiplicative chaos

We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from Circular $\beta$-Ensemble, which are defined as $\{z^n\}$ in characteristic polynomial $\det(1-zU_{N}^{*})$. When $\beta > 4$ we obtain a new class limiting distributions that arise when both $n$ and $N$ tend to infinity simultaneously. solve an open problem Diaconis Gamburd by showing for $\beta=2$, middle coefficient tends zero \to \infty$. show how theory Gaussian multiplicative chaos (GMC) plays prominent role these problems explicit description obtained distributions. extend remarkable magic square formula moments all $\beta>0$ analyse asymptotic behaviour moments. estimates on order magnitude 0,$ sharp \geq 2$. These insights motivated us introduce stochastic object associated with coefficients, call Holomorphic Multiplicative Chaos (HMC). Viewing HMC distribution, prove result about its regularity appropriate Sobolev space. Our proofs expose exploit several novel connections other areas, including permutations, Tauberian theorems combinatorics.