Gaussian zero-count fluctuations in annuli

Source abstract

We determine joint zero-count fluctuations of the self-dual weighted Szegő Gaussian analytic function in fixed and thinning annuli. For a fixed annulus, the counts near its two boundary components converge jointly to independent standard normal variables after centering and marginal normalization, at arbitrary relative rates of boundary approach. This answers the simultaneous central limit question of Katori and Shirai for this model. The unscaled covariance converges to a negative constant, which we evaluate in complete elliptic integrals. A sharp variance asymptotic for the difference between an annular winding number and its coupled disk count yields quantitative normal approximation bounds. For thinning annuli with inner radius $q=e^{-h}$, the counts in every fixed finite family of interior transverse bands converge jointly to a Gaussian vector after centering and multiplication by $\sqrt h$. We give its spectral covariance matrix, prove strict negative covariance for disjoint bands, and identify its nullspace from the interval indicators. The fixed-annulus proof couples the Laurent series to two independent Gaussian power series. For thinning annuli, we combine exact periodization of a strip kernel, a uniform Laguerre-chaos tail bound, and connected Gaussian diagrams.