  In the present paper and the companion paper [9] a probabilistic
(statistical-mechanical) approach to the construction of canonical metrics on a
complex algebraic varieties X is introduced, by sampling "temperature deformed"
determinantal point processes. The main new ingredient is a large deviation
principle for Gibbs measures with singular Hamiltonians, which is proved in the
present paper. As an application we show that the unique Kahler-Einstein metric
with negative Ricci curvature on a canonically polarized algebraic manifold X
emerges in the many particle limit of the canonical point processes on X. In
the companion paper [9] the extension to algebraic varieties X with positive
Kodaira dimension is given and a conjectural picture relating negative
temperature states to the existence problem for Kahler-Einstein metrics with
positive Ricci curvature is developed.
