38 problems

Let $\kappa\in(4,8)$, and consider a statistical mechanics model in two dimensions whose cluster converges to the ${\rm CLE}_{\kappa}$ gasket. Equip the discrete cluster with its e…

CLE scaling-limit conjecture. CLE with parameter $\kappa$ is conjectured to be the scaling limit of the percolation interfaces in the critical $Q$-Potts random cluster model.

Recall that the Fortuin–Kasteleyn (FK) random-cluster model has cluster weight $q=1$. More generally, let $q\in(0,4]$, and set … For $\kappa\in[4,8)$, let…

Consider the FK cluster model on a planar lattice, with cluster weight $q$ and edge parameter $\beta$, and let $\beta_c$ be its critical value. The edge subsets determine collectio…

Let $\gamma$ be a path sampled from the relevant SLE process, and define its trunk by the set of points where the driving function and force-point process agree. Let $\beta$ be the…

Let $\operatorname{CLE}_\kappa$ be a conformal loop ensemble and consider general multichordal configurations with prescribed boundary connectivity data. Coulomb gas integrals are…

Fix $mathbbm{H}$ as the upper half-plane and let $\operatorname{CLE}_\kappa$ denote conformal loop ensemble with parameter $kappa$. For critical FK-$q$ percolation, the parameter…

Let $\kappa\in(4,8)$, and consider statistical mechanics models in two dimensions that converge to ${\rm CLE}_{\kappa}$, together with their clusters and simple random walks on tho…

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…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant probability measure…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant measure $\mu_X$, norma…

For $\kappa \in (8/3,4)$, let $\theta(\kappa)$ denote the distance exponent associated with the canonical geodesic metric on the $\operatorname{CLE}_\kappa$ carpet. Distance-expone…

Let $\kappa \in (8/3,4)$, and let a discrete loop model converge to $\operatorname{CLE}_\kappa$ in the scaling limit. Let the chemical distance metric be the graph metric associate…

Let $q\in(0,4]$, and consider critical Fortuin–Kasteleyn percolation on the discretized box … with wired boundary conditions. The parameter associated with $q$ is … The interfaces…

The meandric-system scaling-limit conjecture. Uniform meandric systems converge in the scaling limit to the LQG sphere with central charge $\mathbf c_{\mathrm L}=28$, together with…

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 ……

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…

A supercritical LQG disk is a random planar surface with disk topology and central charge parameter $\mathbf c_{\mathrm L}\in(1,25)$, coupled with a conformal loop ensemble CLE$_4$…

Gasket convergence conjecture. For $n=2$, the gasket should converge in some sense to the $\mathrm{CLE}_4$ ensemble.

Ising cluster-measure conjecture. The cluster measure of the Ising model should be a natural measure on the ${\rm CLE}_3$ carpet.

Nesting-reweighted scaling-limit conjecture. The disk obtained via the reweighting describes the scaling limit of random planar maps with a loop $O(n)$ model whose law has been rew…

Let CLE be a conformal loop ensemble on the sphere, and consider loops surrounding three marked points. For each such loop, let its conformal radius about the corresponding point b…

Let $\Gamma$ be a conformal loop ensemble with simple loops, ${\rm CLE}_\kappa$ for $\kappa \in (8/3,4)$, in a simply connected domain $D$, and let $\Upsilon$ be its carpet. For…

Let $P_n(x_1,\ldots,x_n)$ denote the probability that $n$ points belong to the same cluster in a random cluster model, and let $\operatorname{CLE}_\kappa$ be the associated conform…

Let $[\mathrm{CLE}_\kappa(\Omega)$ be a nested conformal loop ensemble in a simply connected domain $[\Omega$ with $[8/3<\kappa\le4$. Let the analogue of the crossing-count quantit…