18 problems

Conformal invariance conjecture. The scaling limit of the model is conformally invariant.

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…

For the $q$-state Potts model, let $\theta^w(\beta)$ and $\theta^f(\beta)$ denote the wired and free FK-percolation probabilities at inverse temperature $\beta$, and let…

Consider FK percolation with parameter $q>4$ at its critical probability. A phase transition is called first-order when an infinite open cluster can exist at the critical probabili…

For $q>0$ and $p\in[0,1]$, let $\operatorname{FK}(p,q)$ denote the set of Fortuin–Kasteleyn random-cluster measures, with edge parameter $p' = p/(p+q(1-p))$. In the limit $q\to0$ w…

Polynomial decay conjecture. There exists $c>0$ such that

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…

For a planar FK-percolation model with cluster-weight parameter $q\geq 1$, let $p_c$ denote its critical parameter and let … denote the self-dual point. Self-dual critical-point co…

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 $M_n$ be an FK-weighted planar map sampled from the measure defined in the source, and let $\mu_n$ be the measure that assigns mass $1/N$ to each embedded vertex, where $N$ is…

For each $d\ge 2$ and all $q\ge 1$ not too large, consider the space of $d$-dimensional weighted graphs modulo equivalence and the Markov process described in the source. Let…

Fix $q\ge 1$ and a spatial dimension $d$ below the critical dimension associated with $q$. Critical FK-percolation models may be defined on different $d$-dimensional grids and have…

Fix $q\in[0,4]$. A critical FK interface is the exploration interface in a Dobrushin domain for FK percolation with cluster-weight $q$. Schramm's conjecture. The law of critical FK…

Let $q\leq4$, and consider critical FK interfaces with cluster-weight $q$. FK interface convergence conjecture. Their law converges, in the scaling limit, to Schramm–Loewner Evolut…

Let $q\leq 4$, and let $(\Omega,a,b)$ be a simply connected domain with two boundary points. For each $z\in\Omega$, let $F_\delta(z)$ denote the FK parafermionic observable at…

Let the FK percolation model have cluster-weight $q\in(0,\infty)$ and edge parameter $p$. FK criticality conjecture. The model is critical when … This extends the result known in t…