Abstract: In joint work with Alexander Bufetov, we show that the classical Patterson-Sullivan construction can be generalized to the random setting in the theory of point processes. This construction allows us to recover the value of harmonic functions with sufficient regularity at any point of the disc from its restriction to a random configuration of the determinant point process with the Bergman kernel. This extrapolation result is then extended to real and complex hyperbolic spaces of higher dimension. Recovering continuous functions by the Patterson-Sullivan construction is also shown to be possible in more general Gromov hyperbolic spaces.
