In the probabilistic construction of Kähler-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 Kähler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler-Einstein geometry.