The negative-temperature equivalence conjecture for Gibbs measures

Let $(X,L)$ be a polarized manifold with a Hermitian metric $\left\Vert\cdot\right\Vert$ on $L$ and a volume form $dV$ on $X$. Let $\mathcal{M}_{1}(X)$ be the space of probability measures on $X$, let $Z_{N_k,\beta}$ be the partition functions, let $H^{(N_k)}$ be the Hamiltonians, and let $\delta_N$ be the empirical-measure map. For a negative number $\beta_0$, write $E_{\omega_0}$ for the pluricomplex energy relative to the curvature form $\omega_0$ of $\left\Vert\cdot\right\Vert$, and $D_{dV}$ for the entropy term.

Negative-temperature equivalence conjecture. For every $\beta>\beta_0$, the following are equivalent: the partition functions $Z_{N_k,\beta}$ are finite for all sufficiently large $k$; the functional $\beta F_\beta$ admits a minimizer on $\mathcal{M}_1(X)$; and the measures

$$(\delta_N)_*\left(e^{-\beta H^{(N_k)}}dV^{\otimes N_k}\right)$$

satisfy a large deviation principle with speed $N$ and rate functional

$$\beta F_\beta(\mu)=\beta E_{\omega_0}(\mu)+D_{dV}(\mu).$$

This is proposed as a probabilistic analogue of the Yau–Tian–Donaldson picture: the large deviation principle, existence of minimizers, and finiteness of the partition functions should coincide. The source notes partial finiteness results near zero and explains consequences if the conjectural large deviation principle holds, but does not state a resolution.