Polyakov triangulation conjecture for Liouville measure on higher-genus surfaces

Let $M$ be a surface of genus ${\bf g}$, let $\mathcal{T}_{N,\mathbf{g}}$ denote the set of triangulations of size $N$, and let $\nu_a$ be the random measure defined by

$$\mathbb{E}^{a}[F(\nu_a)]=\frac{1}{Z_a}\sum_{N\geqslant 1}e^{-\bar\mu N}\sum_{T\in\mathcal{T}_{N,\mathbf{g}}}F(e^{\varphi_T}\,d\operatorname{v}_{g_{\tau_T}}),$$

for positive bounded functions $F$, where $Z_a$ normalizes the expectation and $\mathbb{P}^a$ is the associated probability law. Set

$$\bar\mu=\mu_c+a^2\mu,$$

with fixed $\mu>0$. Polyakov's triangulation conjecture. Under $\mathbb{P}^a$, the random measure $\nu_a$ converges in law as $a\to0$, in the space of Radon measures with the topology of weak convergence, to the Liouville measure $\mathcal{L}_\gamma$ under $\mathbb{E}_{(g_\tau)_\tau,\mu}[\cdot]$, with parameter $\gamma=\sqrt{\frac{8}{3}}$. This conjecture identifies the continuum scaling limit of random triangulated surfaces with Liouville quantum gravity; analogous conjectures for the sphere, disk, and torus are described as completely open in the source, although partial progress is known under an unproved condition.