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Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces

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OpenAI Mathopenai-math/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-InterfacesOct 5, 2026math.PRAI-generatedscore 0

For every fixed $1\le q\le4$, critical square-lattice random-cluster Dobrushin interfaces converge to chordal $\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, and the complete nested plane loop collections converge to whole-plane $\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}$. The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

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