{"id":"openai-math/Self-dual-random-cluster-interfaces-below-one","pdf":"https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026","tags":["probability","random-cluster-model","sle"],"score":0,"title":"Self-dual random-cluster interfaces below one","domain":"Probability and statistical mechanics","family":"223","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"For every fixed $`0\\lt q\\lt 1`$, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLE κ , where $`\\kappa=4\\pi/\\arccos(-\\sqrt q/2)\\in(6,8)`$. The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection…","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"}