Secular coefficients and the holomorphic multiplicative chaos

Abstract

We study the secular coefficients of $\mathit{N}\times \mathit{N}$ random unitary matrices ${\mathit{U}}_{\mathit{N}}$ drawn from the Circular β-Ensemble which are defined as the coefficients of $\{{\mathit{z}}^{\mathit{n}}\}$ in the characteristic polynomial $det(1-\mathit{z}{\mathit{U}}_{\mathit{N}}^{\ast })$. When $\mathit{\beta }>4$, we obtain a new class of limiting distributions that arise when both n and N tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd (Electron. J. Combin. 11 (2004/06) 2) by showing that, for $\mathit{\beta }=2$, the middle coefficient of degree $\mathit{n}=\lfloor \frac{\mathit{N}}{2}\rfloor$ tends to zero as $\mathit{N}\to \infty$. We show how the theory of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems and in the explicit description of the obtained limiting distributions. We extend the remarkable magic square formula of (Electron. J. Combin. 11 (2004/06) 2) for the moments of secular coefficients to all $\mathit{\beta }>0$ and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all $\mathit{\beta }>0$, and we prove these estimates are sharp when $\mathit{\beta }\ge 2$ and N is sufficiently large with respect to n. These insights motivated us to introduce a new stochastic object associated with the secular coefficients, which we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel connections with other areas, including random permutations, Tauberian theorems and combinatorics.