We extend the probabilistic approach for constructing Kahler-Einstein metrics on log Fano manifolds X - involving random point processes - to the case of non-discrete automorphism groups, by breaking the symmetry using a moment map constraint. In particular, an algebraic notion of Gibbs polystability is introduced, ensuring that the corresponding point processes on X are well-defined. We conjecture that the Gibbs polystability of X is equivalent to the existence of a Kahler-Einstein metric and that the unique such metric with vanishing moment emerges when sampling a large number of N points on X. The definition of Gibbs polystability involves a limit of log canonical thresholds on the GIT semistable locus of the N-fold products of X, that we conjecture coincides - as N tends to infinity - with an analytic reduced stability threshold, encoding the coercivity of the K-energy functional modulo automorphisms. These conjectures follow from an overarching conjectural Large Deviation Principle for the large N-limit. We prove several of our conjectures on log Fano curves and derive a strengthened form of the sharp logarithmic Hardy-Littlewood-Sobolev (HLS) inequality on the two-sphere, under a moment constraint. It yields quantitative stability results for the sharp logarithmic HLS inequality with optimal stability constants. Furthermore, we show that any log Fano manifold that is strongly uniformly Gibbs polystable admits a Kahler-Einstein metric. In companion papers we will present applications to Onsager's point vortex model on the two-sphere and the AdS/CFT correspondence.