The Rohde-Schramm conjecture: critical square-lattice FK interfaces converge to SLE for 1 <= q < 4 (and 0 < q < 1 in smooth domains)

For every fixed $1\le q<4$ (September 23), and $1\le q\le4$ (October 5 companion), critical square-lattice FK Dobrushin interfaces converge to chordal $\mathrm{SLE}_{\kappa(q)}$ in every bounded Jordan domain with uniformly approximated marks, and the complete nested loop collection of the free infinite-volume measure converges to whole-plane $\mathrm{CLE}_{\kappa(q)}$, keeping multiplicities and traversals. For $0<q<1$ the interface converges to $\mathrm{SLE}_\kappa$, $\kappa\in(6,8)$, in smooth Jordan domains only. No uniformity in $q$ is asserted. Not covered: $0<q<1$ in nonsmooth domains such as the square, CLE for $q<1$, and other lattices.

For every fixed $1\le q<4$ (September 23), and $1\le q\le4$ (October 5 companion), critical square-lattice FK Dobrushin interfaces converge to chordal $\mathrm{SLE}_{\kappa(q)}$ in every bounded Jordan domain with uniformly approximated marks, and the complete nested loop collection of the free infinite-volume measure converges to whole-plane $\mathrm{CLE}_{\kappa(q)}$, keeping multiplicities and traversals. For $0<q<1$ the interface converges to $\mathrm{SLE}_\kappa$, $\kappa\in(6,8)$, in smooth Jordan domains only. No uniformity in $q$ is asserted. Not covered: $0<q<1$ in nonsmooth domains such as the square, CLE for $q<1$, and other lattices.