We investigate the low moments \(\mathbb {E}[|A_N|^{2q}],\, 0<q\leqslant 1\) 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 \(\{X_k\}_{k\geqslant 1}\) 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 \(N\rightarrow \infty \) . 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 \(A_N\) .
