{"id":"openai-math/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4","pdf":"https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026","tags":["probability","percolation","sle-curves","conformal-invariance"],"score":0,"title":"Square-lattice FK interfaces and nested loops for 1 <= q < 4","domain":"Probability and statistical mechanics","family":"223","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"For every fixed $`1\\le q\\lt 4`$, 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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":false,"family_title":"Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling"}