{"id":"openai-math/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces","pdf":"https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026","tags":["statistical-mechanics","schramm-loewner-evolution","percolation"],"score":0,"title":"Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces","domain":"Probability and statistical mechanics","family":"223","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling"}