  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).

Complex Variables;Functional Analysis;Probability