124 problems

Let $\gamma\bin(0,2]$ and let $\eta$ be such that quantum natural time $\text{QLE}(\gamma^2,\eta)$ has been constructed. Let $\eta$-DBM denote diffusion-limited aggregation with pa…

Liouville quantum gravity scaling-limit conjecture. The limit in law as $a\to0$ of $( ua,nua^{partial})$ exists in the space of Radon measures equipped with the topology of weak co…

Fix parameters $\rho$ and $q$ in the range for which the skew Brownian permuton construction gives corresponding parameters $\gamma$ and $\theta(q)$ through the relations stated in…

Let $h$ be a variant of the Gaussian free field on a planar domain, let $dx^2+dy^2$ be the Euclidean metric tensor, and let $\gamma\in(0,2)$. The associated Liouville quantum gravi…

Di Francesco–Golinelli–Guitter meander asymptotic conjecture. There exists a positive constant $R$ such that

Let $\mathfrak{h}$ be a unit-area $1$-LQG sphere independent of the three whole-plane space-filling curves $(\eta^{\mathrm{b}},\eta^{\mathrm{r}},\eta^{\mathrm{g}})$ obtained from t…

Let $(\Omega,h)$ be a quantum disc as defined in the cited work, and let $\mathbf{K}(t)$ denote the corresponding heat-content quantity. Quantum-disc heat-content conjecture. As…

Let $(\Omega,h)$ be a quantum disc as above, and let $\mathbf{H}(t)$ denote its heat trace, with $c_\gamma$ the leading Weyl-law constant and $\mu(\Omega)$ its quantum area. Quantu…

For $\gamma\in(0,2)$, consider the $\gamma$-Liouville quantum gravity ($\gamma$-LQG) sphere equipped with its associated metric, whose construction is known in the generality descr…

Fix $\textbackslash gamma\textbackslash in(0,2)$, and let $\textbackslash mathbf f_n$ be the LQG eigenfunctions, normalised to have unit…

Let $\textbackslash Sigma$ be an LQG domain, let $\textbackslash N(\textbackslash lambda)$ denote its eigenvalue counting function, and consider the next-order term beyond the lead…

Let $\textbackslash Sigma$ be a domain, let $\textbackslash varphi$ be the underlying Gaussian free field, and let…

Let $\textbackslash varphi$ be the Gaussian free field defining Liouville quantum gravity, let $\textbackslash mu_\textbackslash varphi$ be the associated Liouville measure, and sa…

Scaling-limit conjecture. LQG surfaces should arise as scaling limits of random planar maps.

Let $D_\Phi$ be the $\gamma$-Liouville quantum gravity metric on a $\gamma$-LQG surface, and consider random planar maps equipped with their graph distance. LQG metric scaling-limi…

Let $D_\Phi$ be the $\gamma$-Liouville quantum gravity metric on a $\gamma$-LQG surface, and let $\zeta$ denote its Weyl scaling exponent. Weyl-scaling exponent conjecture. The exp…

Set $\alpha \defeq 1/2+4/\kappa$. Let $(\mathcal{D},\Phi,0,\infty)$ be a $\gamma$-Liouville quantum gravity disk, and let $\Gamma$ be an independent non-nested…

For $\gamma\in(0,2)$ and $\widetilde\theta\in(0,\pi)$, let $\mathfrak L_\Phi^{\operatorname{LDP}}$ and $\mathfrak L_\Phi^{\operatorname{SDP}}$ denote candidate random functions obt…

For each $n\in{\bf N}$, let $M_n$ be a random planar map sampled as in the planar-map model above, and embed it into $\mathbbm S^2$ or a subset of $\mathbbm C$ by a discrete confor…

Let $M_n$ be a random planar map with $n$ edges whose law is given, for each fixed map $\mathfrak m$ with $n$ edges, by … where $\Delta_{\mathfrak m}$ is the graph Laplacian and…

Let the Brownian annulus be conformally embedded, and condition its random geometry on the modulus of the annulus. Liouville CFT conjecture. The resulting random geometry is given…

Brownian separable permuton construction conjecture. Brownian separable permutons can be constructed using $\text{SLE}_4/\text{CLE}_4$ and critical Liouville quantum gravity with…

Let $Q\in(0,2)$ and let $p_c(Q)\in(0,1)$ be the critical probability from the non-trivial percolation transition conjecture. Consider critical site percolation on $\mathcal M_k$ wi…

Fix $Q\in(0,2)$ and consider site percolation on $\mathcal M_k$ with open boundary condition: boundary vertices are open, while each interior vertex is independently open with prob…

Let $\mathcal M_k$ be the supercritical random planar map, and let its free uniform spanning forest (FUSF) be the forest obtained from the free boundary condition. FUSF connectedne…