The triangulation-to-Liouville quantum gravity convergence conjecture

Let $\mathcal{T}_{N}$ be the set of triangulations of the sphere with $N$ faces, and let $\mathcal{T}_{N,3}$ be the set of triangulations with $N$ faces and three marked faces. Equip each triangulation with the standard conformal structure in which every triangle has volume $1/N$, and use uniformization, with the three roots sent to prescribed points $z_1,z_2,z_3\in\mathbb{S}$, to obtain the deterministic unit-volume measure $\nu_{T,N}$ on $\mathbb{S}$. Define the random measure $\nu_N$ by

$$\mathds{E}^{N}[F(\nu_N)]=\frac{1}{Z_N}\sum_{T\in\mathcal{T}_{N,3}}F(\nu_{T,N}),$$

for positive bounded functions $F$, where $Z_N=\#\mathcal{T}_{N,3}$, and denote its law by $\mathds{P}^{N}$. Let the unit-volume Liouville measure be the measure given by the construction referenced in the source, with parameters $\gamma=\sqrt{\frac{8}{3}}$, $n=3$, and $(z_i,\alpha_i)=(z_i,\gamma)$. The triangulation-to-Liouville quantum gravity convergence conjecture. Under $\mathds{P}^{N}$, the family of random measures $(\nu_N)_{N\;\geqslant\; 1}$ converges in law as $N\to\infty$, in the space of Radon measures equipped with the topology of weak convergence, to the law of this unit-volume Liouville measure. This conjecture gives a precise mathematical formulation of the link between discrete gravity and Liouville quantum gravity, a connection understood by physicists since the 1980s. It was first stated in the cited work of David, Kupiainen, Rhodes, and Vargas; no resolution is supplied in the source.