Universality and Phase Transitions in Low Moments of Secular Coefficients of Critical Holomorphic Multiplicative Chaos

2024/01/11 by Gu, Haotian, Zhang, Zhenyuan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2401.05681

Abstract

We investigate the low moments 𝔼[|A_N|^2q], 02q. 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.

Related

• Secular Coefficients and the Holomorphic Multiplicative Chaos
• The Fourier coefficients of the critical holomorphic multiplicative chaos
• Tail universality of critical Gaussian multiplicative chaos
• On the Fourier Coefficients of critical Gaussian multiplicative chaos
• The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency