Critical measures and coefficient limits for non-Gaussian holomorphic chaos
Abstract
We study critical holomorphic multiplicative chaos with circular iid inputs of unit variance. Writing R=|X1|, we prove that the radially normalized boundary measures converge in probability to a finite measure with almost surely positive mass if and only if 𝔼[R2log(2+R)]<∞; otherwise they converge to zero. We obtain distinct sharp criteria for convergence at each fractional order 0<q<1 for radial approximations and for the full Fourier-radial family. Convergence at every such order is equivalent to an exponential-moment radius of at least two. In the positive-limit class, coefficient vectors normalized by (log n)1/4 and their analytic windows converge stably to Gaussian mixtures determined by the same critical measure. This measure is nonatomic, has full support, and is carried by a set of Hausdorff dimension zero. Coefficient periodograms recover the measure, while scalar and matrix energy normalizations give universal Gaussian limits. Exact weighted tails under 𝔼 I0(2R)<∞, where I0 is the modified Bessel function, and negative mass moments distinguish uncountably many coefficient-limit laws, even modulo deterministic scalar rescaling, within a uniformly bounded input family. This disproves the common-law clause of Gu-Zhang's Question 2 at its fixed normalization. We give conditions for convergence of the original coefficient moments and show that finiteness of the limiting moments alone does not suffice. For inputs with logℙ(R>u)∼−λuα, 0<α<1, the coefficient normalization and Gaussian-mixture form persist, but the limit has no finite positive moments; we determine its logarithmic tails. Finally, we prove Hardy nonmembership under finite variance alone and, in the positive-limit class, sharp weighted Bergman criteria for 0<p≤2 and the boundary Sobolev threshold s=−1/2.
Related
- Almost sure bounds and tail phases for critical non-Gaussian holomorphic chaos
- Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations
- Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces
- Invariant probability measures on holomorphic functions on the unit disk
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
BibTeX
@misc{guo2026critical,
title = {{Critical measures and coefficient limits for non-Gaussian holomorphic chaos}},
author = {Feng Guo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34371},
eprint = {2610.34371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34371},
}
RIS
TY - UNPB
TI - Critical measures and coefficient limits for non-Gaussian holomorphic chaos
AU - Feng Guo
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34371
UR - https://arxiv.org/abs/2610.34371
AB - We study critical holomorphic multiplicative chaos with circular iid inputs of unit variance. Writing R=|X_1|, we prove that the radially normalized boundary measures converge in probability to a finite measure with almost surely positive mass if and only if 𝔼[R^2log(2+R)]<∞; otherwise they converge to zero. We obtain distinct sharp criteria for convergence at each fractional order 0<q<1 for radial approximations and for the full Fourier-radial family. Convergence at every such order is equivalent to an exponential-moment radius of at least two. In the positive-limit class, coefficient vectors normalized by (log n)^1/4 and their analytic windows converge stably to Gaussian mixtures determined by the same critical measure. This measure is nonatomic, has full support, and is carried by a set of Hausdorff dimension zero. Coefficient periodograms recover the measure, while scalar and matrix energy normalizations give universal Gaussian limits. Exact weighted tails under 𝔼 I_0(2R)<∞, where I_0 is the modified Bessel function, and negative mass moments distinguish uncountably many coefficient-limit laws, even modulo deterministic scalar rescaling, within a uniformly bounded input family. This disproves the common-law clause of Gu-Zhang's Question 2 at its fixed normalization. We give conditions for convergence of the original coefficient moments and show that finiteness of the limiting moments alone does not suffice. For inputs with logℙ(R>u)∼−λu^α, 0<α<1, the coefficient normalization and Gaussian-mixture form persist, but the limit has no finite positive moments; we determine its logarithmic tails. Finally, we prove Hardy nonmembership under finite variance alone and, in the positive-limit class, sharp weighted Bergman criteria for 0<p≤2 and the boundary Sobolev threshold s=−1/2.
ER -