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

Abstract

We investigate the low moments $\mathbb{E}[|A_N|^{2q}], 0<q\leq 1$ of {secular coefficients} $A_N$ of the {critical non-Gaussian holomorphic multiplicative chaos}, i.e. coefficients of $z^N$ in the power series expansion of $\exp(\sum_{k=1}^\infty X_kz^k/\sqrt{k})$, where $\{X_k\}_{k\geq 1}$ 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, $A_N$ features better-than-square-root cancellation: $\mathbb{E}[|A_N|^2]=1$ and $\mathbb{E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$ for fixed $q\in(0,1)$ as $N\to\infty$. We show that this asymptotics holds universally if $\mathbb{E}[e^{\gamma|X_k|}]<\infty$ 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 $\mathbb{E}[|A_N|^{2q}]$ 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 $A_N$.