The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal 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 the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic 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 uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(mathbb{D}_d)$ on $mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).
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