Discrete Gaussian free field level-line convergence to SLE4

Let $\phi_n$ be the discrete Gaussian free field, let $u_n$ be the auxiliary function appearing in the construction, and let $\eta^{(n)}$ be the original level line. For sufficiently small $T$, define $\eta^{(n),T}$ to be the level line of the imaginary part of $\exp(iT(\phi_n+u_n))$, with $T\lambda<\pi$. Discrete-Gaussian-free-field level-line conjecture. There exists a sufficiently small $T_c$ such that, for every $T<T_c$, $\eta^{(n),T}$ converges in law to an $\mathrm{SLE}_4$; moreover, $\eta^{(n)}$ and $\eta^{(n),T}$ converge to the same limit.

The conjecture proposes a local recovery of Gaussian-free-field level lines from the exponential field. The preceding corollary gives a nonlocal recovery result, while the local convergence assertion is not proved in the supplied text.