Critical measures and coefficient limits for non-Gaussian holomorphic chaos

Source abstract

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 $\mathbb{E}[R^2\log(2+R)] u)\sim-λu^α$, $0<α<1$, 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 $0<p\le2$ and the boundary Sobolev threshold $s=-1/2$.