Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

Gu, Haotian; Zhang, Zhenyuan

doi:10.1007/s00440-025-01443-z

Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0

E [| A_{N} |^{2 q}], 0 < q ⩽ 1

of secular coefficients$$A_N$$

A_{N}

of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$

z^{N}

in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$

exp (\sum_{k = 1}^{\infty} X_{k} z^{k} / \sqrt{k})

, where$$\{X_k\}_{k\geqslant 1}$$

{X_{k}}_{k ⩾ 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$$X_k$$

X_{k}

is standard complex Gaussian,$$A_N$$

A_{N}

features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$

E [| A_{N} |^{2}] = 1

and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$

E [| A_{N} |^{2 q}] ≍ {(log N)}^{- q / 2}

for fixed$$q\in (0,1)$$

q \in (0, 1)

as$$N\rightarrow \infty $$

N \to \infty

. We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$

E [e^{γ | X_{k} |}] < \infty

for some$$\gamma >2q$$

γ > 2 q

. As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$

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$$\mathbb {E}[|A_N|^{2q}]$$

E [| A_{N} |^{2 q}]

for$$|X_k|$$

| 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$$A_N$$

A_{N}

.

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