  The Patterson-Sullivan construction is proved almost surely to recover every
harmonic function in a certain Banach space from its values on the zero set of
a Gaussian analytic function on the disk. The argument relies on the slow
growth of variance for linear statistics of the concerned point process. As a
corollary of reconstruction result in general abstract setting,
Patterson-Sullivan reconstruction of harmonic functions is obtained in real and
complex hyperbolic spaces of arbitrary dimension.
