29 problems

Thermodynamic limit of the random-cluster partition function. As $n\rightarrow\infty$,

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…

Simon-type conjecture. The displayed inequality holds for every $q\in[1,2]$.

Let $q\ge 1$. Write ${\cal R}^1_{p,q}$ for the set of probability measures under consideration, and let $p_{\text{c}}^0(q)$ and $p_{\text{c}}^1(q)$ denote the corresponding critica…

Let FK percolation be a random-cluster model with a super-critical phase, and let slab percolation mean percolation occurring in a slab of finite thickness. Pisztora's conjecture.…

Simultaneous uniqueness conjecture. The asserted simultaneous cluster-containment property holds for all three pairs of random variables and all quadruples in $\mathcal D$. This co…

The functions $h_1(\xi)$, $h_2(\xi)$, and $h_3(\xi)$ are combined as in the preceding construction, with coefficients $c_2$ and $d_1$ defined by the displayed formulas, and the str…

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 $q>0$ 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…

Square-lattice random-cluster critical-point conjecture. The critical point of the random-cluster model is the self-dual point

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…

The bunkbed graph $G=(V,E)$ of a graph $\widetilde{G}=(\widetilde{V},\widetilde{E})$ is formed by superposing two identical copies of $\widetilde{G}$ and adding edges between corre…

An FK random-cluster model with cluster-weight parameter $q>0$ induces a color process by assigning independent colors to its percolation clusters. Positive-association conjecture.…

Scaling-limit conjecture. Each non-simple conformal loop-ensemble is conjectured to be the scaling limit of the critical FK$_q$ random-cluster model for $q=q(\kappa')\in(0,4)$.

Let $q$ be the random-cluster model parameter. The phase transition is the change in behaviour of the model at its critical point $p=p_c(q)$ as the edge-occupation parameter $p$ va…

Large-activity uniqueness conjecture. There exists $z_1>0$ such that, for all $z>z_1$, there exists a unique stationary $\operatorname{CRCM}(z,Q,q)$, and this measure is…

Universality conjecture. The methods of the cited work may be extended to obtain universality for the random-cluster model on isoradial graphs.

Fix $q\in[1,4]$ and let $(\omega_t^{\mathrm{FK}_q})_{t\geq0}$ be the critical infinite-volume heat-bath dynamics on $\mathbb Z^2$. Let $\mathcal E_q$ be the random set of times at…

Let $\mathbb T_n:=\mathbb Z^2/n\mathbb Z^2$, and let $A_n$ be the event that there is an open circuit with non-trivial homotopy in $\mathbb T_n$. For an edge $e$, write…

Let $\mathbb T_n:=\mathbb Z^2/n\mathbb Z^2$, and let $A_n$ be the event that there is an open circuit with non-trivial homotopy in $\mathbb T_n$. For an edge $e$, write…

Consider planar FK random-cluster models with cluster-weight $q>4$. First-order phase-transition conjecture. The phase transition is of first order; the near-critical regime has on…

Consider the planar FK random-cluster model with cluster-weight $q ame [1,4]$. A phase transition is continuous when there is a unique infinite-volume measure at criticality an…

For a critical random-cluster model with cluster-weight $q$, let macroscopic pivotals mean pivotal edges whose influence persists at macroscopic scales, and let exceptional times m…