Probabilistic Yau–Tian–Donaldson conjecture for Fano manifolds

Let $X$ be a Fano manifold. Let $N_k:=\dim H^0(X,-kK_X)$, and let the corresponding canonical point processes be defined by the Gibbs measures associated with determinants of bases of $H^0(X,-kK_X)$. Their empirical measures are denoted by $\delta_N$. Probabilistic Yau–Tian–Donaldson conjecture. $X$ admits a unique Kähler–Einstein metric $\omega_{KE}$ if and only if $X$ is Gibbs stable. If $X$ is Gibbs stable, the empirical measures $\delta_N$ of the corresponding canonical point processes converge in probability to the normalized volume form of $\omega_{KE}$. This is proposed as a probabilistic analogue of the Yau–Tian–Donaldson conjecture for Fano manifolds; the source does not state a resolution of either assertion.