Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions

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 of the point process and the slow growth of variance for linear statistics. Patterson-Sullivan reconstruction of Hardy functions is obtained in real and complex hyperbolic spaces of arbitrary dimension, while reconstruction of continuous functions is established in general CAT(-1) spaces.
Based on joint work with Yanqi Qiu, https://arxiv.org/abs/1806.02306.