The probabilistic–variational correspondence for negative-temperature point processes

Let $X$ be a Fano manifold with volume form $dV$, let $\beta_0<0$, and let $\beta$ range over the real numbers satisfying $\beta>\beta_0$. For each $N$, let $\mathcal{Z}_N(\beta)$ be the partition function, let $\mathcal{M}_\beta$ be the twisted Mabuchi functional on the space $\mathcal{H}$ of Kähler metrics representing the first Chern class of $X$, and let $\delta_N$ be the empirical measure of the ensemble $(X^N,\mu_\beta^{(N)})$. Probabilistic–variational correspondence conjecture. The following are equivalent: (i) for every $\beta>\beta_0$, $\mathcal{Z}_N(\beta)$ is finite for all sufficiently large $N$; (ii) for every $\beta>\beta_0$, $\mathcal{M}_\beta$ admits a minimizer in $\mathcal{H}$. Moreover, if the first condition holds for $\beta_0$, then for every $\beta>\beta_0$, $\delta_N$ converges in probability, possibly after passing to a subsequence, as $N\to\frac{}{}\infty$ to a volume form $dV_\beta$ whose corresponding Kähler metric $\omega_\beta$ minimizes $\mathcal{M}_\beta$ on $\mathcal{H}$. This conjecture links finiteness of the partition function and existence of variational minimizers with convergence of the associated random point processes; its status is not resolved in the supplied source.