SECULAR COEFFICIENTS AND THE HOLOMORPHIC MULTIPLICATIVE CHAOS

• 成果类型: Article
• 署名作者: Najnudel, Joseph; Paquette, Elliot; Simm, Nick
• 署名单位: University of Bristol; McGill University; University of Sussex
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1616
• 发表日期: 2023
• 页码: 1193-1248
• 关键词: riemann zeta-function; CONVERGENCE; maximum; moments; eigenvalues

摘要

We study the secular coefficients of N x N random unitary matrices UN drawn from the Circular-Ensemble which are defined as the coefficients of {zn} in the characteristic polynomial det(1 - zU*N). When > 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 = 2, the mid-dle coefficient of degree n = N2 j tends to zero as N -oo. 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 dis-tributions. We extend the remarkable magic square formula of (Electron. J. Combin. 11 (2004/06) 2) for the moments of secular coefficients to all > 0 and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all > 0, and we prove these estimates are sharp when > 2 and N is sufficiently large with respect to n. These insights motivated us to introduce a new stochastic object associ-ated 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 per-mutations, Tauberian theorems and combinatorics.