  This is an invitation to the probabilistic approach for constructing
K\"ahler-Einstein metrics on complex projective algebraic manifolds X. The
metrics in question emerge in the large N-limit from a canonical way of
sampling N points on X, i.e. from random point processes on X, defined in terms
of algebro-geometric data. The proof of the convergence towards
K\"ahler-Einstein metrics with negative Ricci curvature is explained. In the
case of positive Ricci curvature a variational approach is introduced to prove
the conjectural convergence, which can be viewed as a probabilistic
constructive analog of the Yau-Tian-Donaldson conjecture. The variational
approach reveals, in particular, that the convergence holds under the
hypothesis that there is no phase transition, which - from the
algebro-geometric point of view - amounts to an analytic property of a certain
Archimedean zeta function.
