Mabuchi energy expansion conjecture for polarized Riemann surfaces

Let $(X,L)$ be a polarized compact Riemann surface. For a metric $\phi$ on $L$ and a function $u$ on $X$, let $\mu_{\phi}$ denote the associated equilibrium measure, $P(\phi+u)$ and $P\phi$ the corresponding equilibrium envelopes, $\mathcal{F}$ the leading-order free-energy functional, and $\mathcal{M}$ Mabuchi's K-energy functional. The weak form assumes that $\phi$ and $u$ are smooth, that $S_{\phi}$ and $S_{\phi+u}$ are domains with smooth boundaries, and that $\mu_{\phi+u}$ and $\mu_{\phi}$ are strictly positive on their supports.

Mabuchi energy expansion conjecture. Under these assumptions,

$$\lim_{k\rightarrow\infty}\left(\frac{1}{k}\log\mathbb{E}(e^{-kN_{k}U_{N_{k}}})+N_{k}\left(\mathcal{F}(\phi+u)-\mathcal{F}(\phi)\right)\right)=-\mathcal{M}(P(\phi+u))+\mathcal{M}(P\phi).$$

The stronger form asserts the same convergence when $\mu_{\phi+u}$ and $\mu_{\phi}$ have $L^{\infty}$ densities, while the strongest form assumes only that they have finite entropy. The conjecture refines the leading-order large-deviation expansion by identifying the next-order term with Mabuchi's K-energy and is intended to extend to higher-dimensional polarized compact complex manifolds and certain $\beta$-ensembles.