In this talk, I will introduce a new algebraic invariant of a Fano manifold X - the Gibbs polystability threshold of X. It is conjecturally equal to a previously studied analytic polystability threshold, which quantifies the coercivity of the Mabuchi functional modulo the action of the automorphism group of X. The invariant leads to an effective sufficient criterion for the existence of a Kähler–Einstein metric on X, reducing the problem to the computation of finitely many log canonical thresholds; the criterion is also necessary if the conjectural equality holds. The motivation comes from the probabilistic approach to constructing Kähler–Einstein metrics. This is joint work with Rolf Andreasson and Ludvig Svensson.