44 problems

Fix $\gamma\in(0,2)$, a whole-plane Gaussian free field $h$, and distinct points $z_1,z_2,z_3,z_4\in\mathbb{C}$. Let $\Gamma_{z_i,z_j}$ denote a geodesic from $z_i$ to $z_j$ in the…

Fix a whole-plane Gaussian free field $h$ and let $D_h$ be the associated $\gamma$-Liouville quantum gravity metric for $\gamma\in(0,2)$. Let $\mathfrak{S}_3\subseteq\mathbb{C}$ de…

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…

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

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 $Q\in(0,2)$ 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…

Let $Q=\sqrt{({\mathbf{c}}-1)/6}$ and let $X^k$ be the continuous-time random walk on $\mathcal M_k$ reflected off infinity, stopped when it first hits the boundary at time…

Let $\mathcal M_k$ be the supercritical random planar map and let $H_k$ be its Tutte embedding into $\overline{\mathbbm D}$. For an end $\omega$ of $\mathcal M_k$, define its image…

Let $\{M_n\}_{n\geq1}$ be a sequence of infinite random planar maps believed to lie in the universality class of $\gamma$-LQG, such as uniform infinite triangulations when…

Let $q\text{-FK}$ planar maps denote the critical Fortuin–Kasteleyn planar maps with parameter $q$, and let $\gamma$ be a parameter in $[\sqrt 2,2)$ satisfying … Scaling-limit conjec…

The spanning-tree and GFF scaling-limit conjecture. Under an appropriate scaling as $n\to\infty$, the 4-tuple $(M,P,\phi_1,\phi_2)$ converges in law to a unit boundary length…

Fix $s>1$, and let $(M_k',L_k')$ be sampled from the modified iteration procedure. Let $\mathbf c_{\mathrm L}\in(1,25)$ be the central charge parameter determined by $s$ through ……

Fix $s>1$. In the modified iteration procedure, each hole of perimeter $m$ is filled using a ring whose inner boundary has length either $\lfloor sm\rfloor$ or…

For $k\in\mathbb N$, let $(M_k,L_k)$ be sampled from the probability measure on triangulations of the disk decorated by fully packed $O(2)$ loop configurations, at the critical par…

Let $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on $D$, and let $(\boldsymbol{\lambda}_j)_{j\ge1}$ be the LQG eigenvalues. Write $F_{\mathrm{GOE}}(x)$ for the GOE l…

Fix $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on a bounded domain $D$, and let $(\mathbf{f}_n)$ be eigenfunctions normalised to have unit $L^2(\mu_\gamma)$ norm.…

Let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalue sequence of Liouville Brownian motion on a random surface $(D,h)$, where $D$ is the domain and $h$ is the Gaussian free fi…

Let $D$ be a fixed bounded domain, let $h$ be the Gaussian free field on $D$, and let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalues of Liouville Brownian motion associated…