Marginal irrelevance of weak bond disorder at the planar Ising critical point: quenched SLE3 interfaces for weak random bonds

Theorem 1.1: for any bounded mean-zero nondegenerate law $\rho$ on $[-1,1]$ there is $\varepsilon_0(\rho)>0$ such that for fixed $0<\varepsilon<\varepsilon_0$, at the spontaneous-magnetization threshold $\beta_c^\rho(\varepsilon)=\tfrac12\log(1+\sqrt2)+O(\varepsilon^2)$, the quenched Dobrushin spin interface in any bounded Jordan domain converges to chordal $\mathrm{SLE}_3$ in probability over environments, in the bounded-Lipschitz metric on oriented curves. A companion proves, conditionally on stated deterministic reference estimates, that the relative second moment of the quenched spin correlation grows as $(\log r)^{1/4+o(1)}$ for fair $\pm\varepsilon$ bonds. Not shown: strong disorder, the double-logarithmic specific heat, or unconditional logarithmic corrections.

Theorem 1.1: for any bounded mean-zero nondegenerate law $\rho$ on $[-1,1]$ there is $\varepsilon_0(\rho)>0$ such that for fixed $0<\varepsilon<\varepsilon_0$, at the spontaneous-magnetization threshold $\beta_c^\rho(\varepsilon)=\tfrac12\log(1+\sqrt2)+O(\varepsilon^2)$, the quenched Dobrushin spin interface in any bounded Jordan domain converges to chordal $\mathrm{SLE}_3$ in probability over environments, in the bounded-Lipschitz metric on oriented curves. A companion proves, conditionally on stated deterministic reference estimates, that the relative second moment of the quenched spin correlation grows as $(\log r)^{1/4+o(1)}$ for fair $\pm\varepsilon$ bonds. Not shown: strong disorder, the double-logarithmic specific heat, or unconditional logarithmic corrections.