  In the probabilistic construction of K\"ahler-Einstein metrics on a complex
projective algebraic manifold X - involving random point processes on X - a key
role is played by the partition function. In this work a new quantitative bound
on the partition function is obtained. It yields, in particular, a new direct
analytic proof that X admits a K\"ahler-Einstein metrics if it is uniformly
Gibbs stable. The proof makes contact with the quantization approach to
K\"ahler-Einstein geometry.
