Year: 2026
Type: preprint
Abstract: We study critical holomorphic multiplicative chaos with circular iid inputs of unit variance. Writing \(R=|X_1|\), the radially normalized boundary measures converge in probability to a finite measure with almost surely positive mass exactly when \(\mathbb{E}[R^2\log(2+R)]<\infty\); otherwise they converge to zero. Radial and Fourier--radial approximations have distinct sharp \(L^q\) convergence criteria at each \(0u)\sim-λu^α\), \(0<α<1\) and \(λ>0\), the normalization and Gaussian-mixture form persist, while every positive limit moment diverges; we determine the logarithmic tails. Finite limiting moments need not imply convergence of the original coefficient moments. In the positive-limit class, coefficient periodograms recover the critical measure and energy normalization gives Gaussian limits. Hardy nonmembership holds under finite variance alone; the positive-limit class has sharp weighted Bergman criteria for \(0
Source: arXiv (Cornell University)
Author: Guo, Feng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223943161