Interface scaling limit for the critical planar Ising model perturbed by a magnetic field

We prove that the interface separating $+1$ and $-1$ spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field has a scaling limit. This result holds when the Ising model is defined on a bounded and simply connected subgraph of $\delta {\mathbb{Z}}^2$, with $\delta >0$. We show that if the scaling of the external field is of order ${\delta }^{15\mathrm{/}8}$, then, as $\delta \to 0$, the interface converges in law to a random curve whose law is conformally covariant and absolutely continuous with respect to SLE${}_3$. This limiting law is a massive version of SLE${}_3$ in the sense of Makarov and Smirnov and we give an explicit expression for its Radon-Nikodym derivative with respect to SLE${}_3$. We also prove that if the scaling of the external field is of order ${\delta }^{15\mathrm{/}8}{g}_1(\delta )$ with $g_1(\delta )\to 0$, then the interface converges in law to SLE${}_3$. In contrast, we show that if the scaling of the external field is of order ${\delta }^{15\mathrm{/}8}{g}_2(\delta )$ with $g_2(\delta )\to \mathrm{\infty }$, then the interface degenerates to a boundary arc.