Random-cluster critical-point conjecture for the discontinuity set

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 wired boundary conditions, respectively, and define

$$\theta^1(p,q)=\phi_{p,q}^1(0\leftrightarrow\infty),$$ $$p_{\text{c}}(q)=\sup\{p:\theta^1(p,q)=0\},$$

and let ${\mathcal D}_{d,q}\subseteq[0,1]$ be the countable set such that $\phi_{p,q}^0=\phi_{p,q}^1$ exactly when $p\notin{\mathcal D}_{d,q}$. Random-cluster critical-point conjecture. There exists $Q(d)$ such that:

• if $q<Q(d)$, then $\theta^1(p_{\text{c}},q)=0$ and ${\mathcal D}_{d,q}=\varnothing$;
• if $q>Q(d)$, then $\theta^1(p_{\text{c}},q)>0$ and ${\mathcal D}_{d,q}=\{p_{\text{c}}\}$.

This conjecture predicts a threshold in the cluster-weight parameter separating a continuous regime, with no discontinuity points, from a regime in which the sole discontinuity occurs at the critical point. The statement does not specify what happens at $q=Q(d)$.