Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos
Year: 2025
Type: article
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})$$ exp ( ∑ k = 1 ∞ X k z k / k ) , where $$\{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
Authors: Haotian Gu, Zhenyuan Zhang
Institutions: University of California, Los Angeles, Stanford University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: National Science Foundation
OpenAlex ID: W4415880797