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The Patterson-Sullivan construction is proved almost surely to recover aBergman function from its values on a random discrete subset sampled with thedeterminantal point process induced by the Bergman kernel on the unit ball$\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces,the interpolation is uniform when the functions range over the unit ball of theweighted Bergman space. As main results, we obtain a necessary and sufficientcondition for interpolation of a fixed pluriharmonic function in the complexhyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11);optimal simultaneous uniform interpolation for weighted Bergman spaces (cf.Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniforminterpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem4.15); and establish the impossibility of the uniform simultaneousinterpolation for the Bergman space $A^2(\mathbb{D}_d)$ on $\mathbb{D}_d$ (cf.Theorem 1.12 and Theorem 6.7).

The Patterson-Sullivan Interpolation of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces