Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 223-01. Square-lattice FK interfaces and nested loops for 1 <= q < 4
Manuscript map · lines 6565–6574
OpenAI@2026-10-07#rem:oai-p-223-01
Authors: OpenAI. Family 223.
Title: Square-lattice FK interfaces and nested loops for 1 <= q < 4.
Abstract: 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 formula for square-lattice bond percolation with free boundary edges.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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