Self-dual parameter of the random-cluster model ($p_{\mathrm{sd}}(q)=\sqrt{q}/(1+\sqrt{q})$)

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.

The planar cluster-count identity rewrites $o(\omega )+2k(\omega )$ as $|V|-1+k(\omega )+k({\bar{\omega }}^{\ast })$. At the self-dual parameter of the random-cluster model, $p/(1-p)=\sqrt{q}$, so its weight is $(\sqrt{q}{)}^{o(\omega )+2k(\omega )}$ up to a configuration-independent factor. Absorbing $(\sqrt{q}{)}^{|V|-1}$ into normalization proves the symmetric weight, with the unbounded dual face counted.