15 problems

Let CLE$_4$ denote the conformal loop ensemble with parameter $4$. The double-dimer model is a random collection of loops, and the loop $O(n)$ model is parametrized here by $n=2$ a…

Potts-to-simple-CLE conjecture. The scaling limit of the critical $q$-state Potts model can be described by simple $\mathrm{CLE}_\kappa$. The conjecture identifies simple CLE as th…

The Ashkin–Teller model on a box $\Lambda_n$ has configurations $(\sigma_1,\sigma_2):\Lambda_n\to\{-1,+1\}^2$ with density proportional to … On the self-dual line, assume $J\geq0$,…

Consider the continuum limit of the Ising spin model, with spins taking values in $\{-1,+1\}$. A CLE$_4$-based FK representation is a representation of this limiting spin field usi…

For $\kappa\in(8/3,8]$, let CLE$_\kappa$ be the conformal loop ensemble with parameter $\kappa$, and let $c$ denote the central charge of a conformal field theory. CLE–CFT correspo…

Mixing-rate conjecture. The mixing rate satisfies $\iota>\frac12$ for $1<q<4$, while equality holds only at $q=4$. Consequently, the multiplicative correction to $\mathrm{W}(N)$ is…

Fix $q\in[1,4]$. Let $\phi_{\delta\mathbb{Z}^2,p_c,q}$ be the critical FK-percolation measure on the rescaled lattice $\delta\mathbb{Z}^2$, and let $\operatorname{CLE}_\kappa$ deno…

Conformal-invariance conjecture. The collections $\Gamma_n$ converge in distribution to $\Gamma$ with respect to the metric $d_{\mathfrak{L}}$. This conjecture describes the expect…

For each $\kappa' \in (4,8)$, consider critical FK percolation with cluster weight … Its loop ensemble consists of the outer and inner boundaries of FK clusters, alternating betwee…

Let $\mathrm{CLE}_{\kappa}$ denote the conformal loop ensemble with $\kappa\in(8/3,8)$, and consider critical loop $O(n)$ loops with $n\in(0,2]$ and $N\geq2$ alternating boundary i…

Let $q\in (0,4]$ and let … Suppose that $D$ is a Jordan domain, $\epsilon_n\to 0$ as $n\to\infty$, and that $\mathcal{D}_n$ is a discrete domain in $\epsilon_n\mathbb{Z}^2$ for eac…

Let $\Gamma$ be a uniform $\operatorname{CLE}_4$ in the unit disk, and consider the conformally invariant metric on $\Gamma\cup\{\partial\mathbb{D}\}$ whose distance from a loop to…

Let $(\mathfrak{cle},\mathfrak{lqg},\mathfrak{be})$ be the coupling considered in the paper, where $\mathfrak{be}$ denotes the branching structure and…

Condition on the event $|{\bf t}(r)|=n$, and let ${\sf Canonical}({\bf t}^{<n}(r))$ be the canonical embedding of the root component obtained by edge evaporation. The massive…

Loop-cluster metric conjecture. There is a metric on the clusters of $\mathcal{L}_{g,1/2}$ inside $\Sigma$, measurable with respect to $\mathcal{L}_{g,1/2}$, obtained as the limit…