A Functional Central Limit Theorem for Window Counts of Hardy--Szeg\H{o} Zeros
The Hardy--Szeg\H{o} zero process, investigated in the disk by Peres and Vir\'ag through the independent identically distributed Gaussian analytic function, is a canonical conformally invariant determinantal point process. This paper studies its upper half-plane realization. Although conformally equivalent to the disk model, this realization has its own natural geometry: real-translation invariance turns the process into a stationary object along the boundary and makes long horizontal windows the natural observables. For every admissible height window, we prove a Donsker-type functional central limit theorem for the centered zero counts in expanding horizontal windows, with an explicit intensity and variance depending on the height window. The proof is based on factorial cumulants and Brillinger mixing. The main technical input is a family of all-order integrability estimates for reduced cumulant densities, obtained by exploiting the determinantal cycle structure before integrating over the height variables. As further consequences, we derive an explicit covariance density, an asymptotic variance formula, and a macroscopic Gaussian white-noise limit for linear statistics.