The Liouville quantum gravity scaling-limit conjecture for quadrangulations of the annulus

Let $Q_{n,p}$ be the set of quadrangulations with the topology of an annulus, $n$ inner faces, perimeter $2p$, and one marked point on the boundary. For $a>0$, let $( u_a, u_a^{\partial})$ be the random bulk and boundary measures on the reference annulus obtained by assigning mass $a^2$ to each face and length $a$ to each boundary edge, with weights determined by the cosmological constants described above. Let $Z$ and $Z^{\partial}$ denote the Liouville quantum gravity bulk and boundary measures on the annulus.

Liouville quantum gravity scaling-limit conjecture. The limit in law as $a\to0$ of $( u_a,\nu_a^{\partial})$ \exists in the space of Radon measures equipped with the topology of weak convergence and is given by $(Z,Z^{\partial})$ with $\gamma=\sqrt{\frac{8}{3}}$ and suitable cosmological constants $\mu$ and $\mu_{\partial}$. More precisely, for suitable functionals $F$,

$$\lim_{a\to0}\mathbb{E}^a\left[F(\nu_a,\nu_a^{\partial})\right]=\mathbb{E}^{\sqrt{\frac{8}{3}}}_{\mu,\mu_{\partial}}\left[F(Z,Z^{\partial})\right].$$

This conjecture proposes that suitably weighted random quadrangulations of the annulus converge, after rescaling, to Liouville quantum gravity with parameter $\gamma=\sqrt{8/3}$. Establishing this convergence, including the appropriate cosmological constants, is left open by the source.