Secular Coefficients and the Holomorphic Multiplicative Chaos

We study the secular coefficients of N × N random unitary matrices U_N drawn from the Circular β-Ensemble which are defined as the coefficients of {zⁿ} in the characteristic polynomial det(1 − zU∗/N). When β > 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 β = 2, the middle coefficient of degree n = ⌊N / 2⌋ tends to zero as N → ∞. 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 β > 0 and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all β > 0, and we prove these estimates are sharp when β ≥ 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.