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

TL;DR

This paper studies the universal behavior and phase transitions of low moments of secular coefficients in critical holomorphic multiplicative chaos, revealing new universality results and phase transition phenomena beyond Gaussian cases.

Contribution

It establishes universality of low moments for non-Gaussian chaos under certain conditions and characterizes phase transitions for distributions with stretched exponential tails.

Findings

Universality of low moments for a broad class of non-Gaussian variables.

Identification of a double-layer phase transition related to tail behavior.

Almost sure regularity of the chaos in Sobolev spaces.

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 $A_N$.