55 problems

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…

First-order phase transition conjecture. The phase transition on random $\Delta$-regular graphs is of first order, and

Let $q\geq 1$ and consider the random-cluster model on the square lattice $\mathbb{Z}^2$, with critical threshold … Let $\theta^{\mathrm{w}}(p,q)$ denote the probability that the o…

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 ${\bf a}$…

Häggström's uniqueness conjecture. For $p>p_{\mathsf{s}}$, the random-cluster model has a unique infinite-volume Gibbs measure on the wired tree.

Let $\Delta\geq 3$ and $q>2$, and consider the random cluster model on a random $\Delta$-regular graph. Denote its critical point by $\beta_c(q,\Delta)$. Critical-point formula con…

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 ${\bf a}$…

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

Two-dimensional FK criticality conjecture. In two dimensions, at the critical value $p_c$, the FK measure exhibits nice large-scale properties only up to $q=4$. This is the random-…

Let $d\geq 2$, let $\phi_{p,q}$ be the free-boundary random-cluster measure on the cubic lattice $\mathbb{Z}^d$, and define the percolation probability and critical point by … … Le…

Equality of random-cluster thresholds. The threshold $p_g^{*}$ is conjectured to equal $p_c$.

Baxter's numerical self-dual-point conjecture. The critical point is predicted to satisfy

Rohde–Schramm's conjecture. The random cluster representation of the Potts model for $0<q<4$ is related to the SLE process.

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…

Let $p_c(q)$ denote the critical percolation probability of the random cluster model with cluster-weight parameter $q>0$ on the square lattice $\mathbb{Z}^2$. Self-dual critical pr…

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

Self-dual criticality conjecture. The self-dual point $p_{\operatorname{sd}}(q)$ is critical in the $q<1$ regime. For $q\geq 1$, the corresponding criticality and sharp phase-trans…

Let $G$ be a graph and let $a,b,c,d$ be vertices of $G$. For the arboreal gas measure $\mu^{\text{F}}_{G,\lambda}$, the notation $\mu^{\text{F}}_{G,\lambda}(a,b)$ and similar expre…

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…

Let $G$ be a graph with distinct edges $e,f$, and let $\mathcal{M}_{ef}(q)$ be the polynomial defined from the random-cluster partition-function sums. For edge subsets $\beta$ and…

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

Let $G_n$ be a sequence of graphs converging locally to the $d$-regular tree, let $\beta$ and $B$ denote Potts-model parameters, and let $w$ and $B$ denote the corresponding random…

Slab-threshold equality conjecture. It is conjectured that

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

Random-cluster initialization conjecture. On random regular graphs, the multimodality and associated bottlenecks of the random cluster model should be circumvented by initializing…