 Self-dual parameter of the random-cluster model

Planar random-cluster duality relates edge odds by $[p/(1-p)][p^{\ast }/(1-p^{\ast })]=q$. The fixed point $p=p^{\ast }$ has odds $\sqrt{q}$, giving the displayed parameter. A self-dual parameter does not by itself establish a critical threshold on an arbitrary planar graph. At this parameter the symmetric cluster weight at the self-dual parameter makes the primal-dual cluster counts explicit.