Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions
Résumé
The Patterson-Sullivan construction is proved almost surely to recover every Hardy function from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the conformal invariance and the slow growth of variance of the linear statistics for the underlying point process. Patterson-Sullivan reconstruction of Hardy functions is obtained in real and complex hyperbolic spaces of arbitrary dimension, while reconstruction of continuous functions is shown to hold in general $\mathrm{CAT}(-1)$ spaces.