Square-lattice FK interfaces and nested loops for 1 <= q < 4
For every fixed $1\le q\lt 4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, confirming the Rohde–Schramm prediction in this parameter range. This holds in every bounded Jordan domain under uniform marked boundary approximation. The complete nested plane loop collections converge to whole-plane $\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}$ in a spherical matching topology retaining multiplicities and traversals. At q = 1 we obtain Cardy's…
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