20 problems

Let $G_n$ be the partially wired square grid described in the source, let $\omega$ have the critical random-cluster measure with $q\in(0,4)$, let $\omega^\dagger$ be its dual confi…

Random-cluster p-NC conjecture. The measures $\phi_{p,q}$ satisfy the p-NC property.

Let $G$ be any graph and $0\leq q\leq1$. Let $\mathcal{M}_{ef}(q)$ be the random-cluster polynomial associated with distinct edges $e,f$, and let $A_{\beta,\gamma}$ be the set of…

Positivity conjecture for $\mathcal{M}_{ef}(q)$. For $0\leq q\leq1$ and positive edgeweights $x_e$, one has

Let $G=(V,E)$ be a graph, let $boldsymbol{x}=xe:e otin E$ be positive edgeweights, and let the random cluster model on $G$ have parameter $q$. For distinct edges $e,f otin E$, writ…

Hexagonal-lattice random-current conjecture. All phase transitions of the random current and random-cluster model on the hexagonal lattice coincide:

For $q\in[1,\infty)$, let $p_c(q,h)$ be the critical edge parameter and let $\phi^1_{p,q,h,\mathbb{Z}^d}$ denote the wired random-cluster measure. The wired-percolation monotonicit…

For parameters $q\geq1$ and $h\geq0$, let $\mathfrak{Z}$ denote the relevant pressure function, and consider the phase transition as a function of $p$. The Kertész-line discontinui…

Critical random-cluster interface conjecture. As $\delta\to 0$, the interface in $(\Omega^{\delta};x^{\delta},y^{\delta})$ converges weakly to the chordal…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

Gumbel limiting-distribution conjecture.

Mean-dominates-standard-deviation conjecture.

Off-critical variance conjecture. There exists $C(p,q,d)\geq1$ such that, as $L\to\infty$,

Off-critical mean conjecture. There exists $C(p,q,d)\geq1$ such that, as $L\to\infty$,

Let $\mathcal{L}\in\{\mathbb{Z}^2,\mathbb{H}\}$ be either the square or hexagonal lattice, let $p_c^{\mathcal{L}}$ denote its critical percolation parameter, and let…

Let $d\geq 2$ and $q\geq 1$. For the random-cluster model on $\mathbb{L}^d$, let $\phi_{p,q}^0$ and $\phi_{p,q}^1$ be the infinite-volume random-cluster measures with free and wire…

Let $G$ be a finite graph and consider a random cluster measure on its edge sets with parameter $q<1$. Random-cluster conjecture. Every such random cluster measure is NA+. Equivale…

Let $\tilde X(p,q,r)$ be the divide-and-colour model associated with a random-cluster measure with parameters $q>0$ and $p<p_c(q)$, and consider it on the triangular lattice at…

The one-dimensional continuum random-cluster critical-value conjecture. For $q\geq1$,