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

TL;DR

This paper demonstrates that Patterson-Sullivan measures can almost surely reconstruct harmonic functions from zero sets of Gaussian analytic functions, leveraging variance growth properties, with applications in hyperbolic spaces.

Contribution

It introduces a novel reconstruction method for harmonic functions using Patterson-Sullivan measures based on point processes, extending to hyperbolic spaces.

Findings

Almost sure reconstruction of harmonic functions from zero sets

Reconstruction applies to real and complex hyperbolic spaces

Utilizes slow variance growth of linear statistics

Abstract

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.