Secular Coefficients and the Holomorphic Multiplicative Chaos

TL;DR

This paper investigates the asymptotic behavior of secular coefficients of random unitary matrices from the Circular β-Ensemble, revealing new limiting distributions linked to Gaussian multiplicative chaos and introducing the Holomorphic Multiplicative Chaos as a novel stochastic object.

Contribution

It extends the magic square formula for secular coefficients to all β>0, analyzes their moments, and introduces the Holomorphic Multiplicative Chaos with regularity results.

Findings

New limiting distributions for β>4

Middle coefficient tends to zero for β=2

Sharp estimates for secular coefficients for all β≥2

Abstract

We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from the Circular $\beta$-Ensemble, which are defined as the coefficients of $\{z^n\}$ in the characteristic polynomial $\det(1-zU_{N}^{*})$. When $\beta > 4$ 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 by showing that for $\beta=2$, the middle coefficient tends to zero as $N \to \infty$. 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 Diaconis and Gamburd for the moments of secular coefficients to all $\beta>0$ and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all $\beta > 0,$ and these estimates are sharp when $\beta \geq 2$. 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.