Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

• 成果类型: Article
• 署名作者: Gu, Haotian; Zhang, Zhenyuan
• 署名单位: University of California System; University of California Los Angeles; Stanford University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01443-z
• 发表日期: 2026-02
• 页码: 613-716
• 关键词: Holomorphic multiplicative chaos; Secular coefficients; Low moments; UNIVERSALITY; Phase Transition

摘要

We investigate the low moments E[|A(N) |(2q)], 0 < q <= 1 of secular coefficients A(N) of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of zN in the power series expansion of exp(Sigma(infinity)(k = 1) X-k z(k)/ root k), where {X-k} (k >= 1) are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper's remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each X-k is standard complex Gaussian, AN features better-than-square-root cancellation: E[|A(N)|(2)] = 1 and E[|A(N)|(2q)] asymptotic to (log N)(-q/2) for fixed q is an element of (0, 1) as N -> infinity. We show that this asymptotics holds universally if E[e(gamma) (|X)(k)|] < infinity for some gamma > 2q. As a consequence, we establish the universality for the tightness of the normalized secular coefficients A(N) (log(1 + N))(1/4), generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of E[|A(N) |(2q)] for|X-k| following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper's robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of A(N).