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 asymptotic holds universally if \(\mathbb{E}[e^{\gamma|X_k|}]<\infty\) for some \(\gamma>2q\). As a consequence, we establish the universality for the sharp tightness of the normalized secular coefficients \(A_N(\log(1+N))^{1/4}\) of critical holomorphic chaos, generalizing a result of Najnudel, Paquette, and Simm. Moreover, we completely characterize the asymptotic 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 universality regime. As a result, we unveil a double-layer phase transition, occurring at exponential-type tails and exponential tails of parameter \(2q\). Our proofs combine the robustness of Harper's multiplicative chaos approach and a careful analysis of the (possibly random) leading terms in the monomial decomposition of \(A_N\).